Introduction
Let H \mathbb{H} H R 3 \mathbb{R}^{3} R 3 p = ( x , y , z ) p=(x,y,z) p = ( x , y , z ) horizontal distribution
D = span { X , Y } , X = ∂ x − y 2 ∂ z , Y = ∂ y + x 2 ∂ z , (1) \mathcal{D}=\operatorname{span}\{X,Y\},\qquad X=\partial_{x}-\frac{y}{2}\partial_{z},\qquad Y=\partial_{y}+\frac{x}{2}\partial_{z}, \tag*{(1)} D = span { X , Y } , X = ∂ x − 2 y ∂ z , Y = ∂ y + 2 x ∂ z , (1) and the sub-Riemannian structure defined as the one making { X , Y } \{X,Y\} { X , Y } D \mathcal{D} D ξ = ξ X X + ξ Y Y ∈ D \xi=\xi_{X}X+\xi_{Y}Y\in\mathcal{D} ξ = ξ X X + ξ Y Y ∈ D ∣ ξ ∣ = ξ X 2 + ξ Y 2 |\xi|=\sqrt{\xi_{X}^{2}+\xi_{Y}^{2}} ∣ ξ ∣ = ξ X 2 + ξ Y 2
In this paper, we study mean curvature flow evolutions of surfaces in H \mathbb{H} H Ω ⊂ R 3 \Omega\subset\mathbb{R}^{3} Ω ⊂ R 3 C 2 C^{2} C 2 characteristic points of ∂ Ω \partial\Omega ∂ Ω
C ( ∂ Ω ) = { p ∈ ∂ Ω : D p ⊂ T p ∂ Ω } . C(\partial\Omega)=\{p\in\partial\Omega:\mathcal{D}_{p}\subset T_{p}\partial\Omega\}. C ( ∂ Ω ) = { p ∈ ∂ Ω : D p ⊂ T p ∂ Ω } . At non-characteristic points, the Euclidean outer unit normal admits a non-trivial horizontal projection. After normalization, this projection defines the horizontal outer unit normal ν \nu ν D \mathcal{D} D horizontal mean curvature of ∂ Ω \partial\Omega ∂ Ω
H = div ν on ∂ Ω ∖ C ( ∂ Ω ) . H=\operatorname{div}\nu\qquad\text{on }\partial\Omega\setminus C(\partial\Omega). H = div ν on ∂ Ω ∖ C ( ∂ Ω ) . In the Euclidean setting, the mean curvature flow evolves a smooth hypersurface in its normal direction with normal velocity equal to its mean curvature. A formal horizontal analogue of the classical mean curvature flow is obtained by prescribing that a family of hypersurfaces { ∂ Ω t } t ∈ [ 0 , T ) \{\partial\Omega_{t}\}_{t\in[0,T)} { ∂ Ω t } t ∈ [ 0 , T ) T > 0 T>0 T > 0 Γ \Gamma Γ
F : Γ × [ 0 , T ) ⟶ R 3 , ∂ Ω t = F ( Γ , t ) , F:\Gamma\times[0,T)\longrightarrow\mathbb{R}^{3},\qquad\partial\Omega_{t}=F(\Gamma,t), F : Γ × [ 0 , T ) ⟶ R 3 , ∂ Ω t = F ( Γ , t ) , evolves according to
∂ t F ( q , t ) = − H ( F ( q , t ) ) ν ( F ( q , t ) ) , (2) \partial_{t}F(q,t)=-H(F(q,t))\nu(F(q,t)), \tag*{(2)} ∂ t F ( q , t ) = − H ( F ( q , t )) ν ( F ( q , t )) , (2) with initial datum ∂ Ω 0 = ∂ Ω \partial\Omega_{0}=\partial\Omega ∂ Ω 0 = ∂ Ω [9 , 20 , 17 , 16 ] , in analogy with the classical level-set formulation developed in [18 , 13 ] or via variational techniques in the recent preprint [11 ] , in analogy with the classical minimizing movements scheme developed in [3 , 27 ] .
A major difficulty in weak formulations is that fine geometric information along the flow is hard to retain. In this paper, we impose some structure on the initial sets, aiming to obtain more geometric information along the flow. More precisely, we assume that the horizontal mean curvature of the initial set Ω \Omega Ω ∂ Ω ∖ C ( ∂ Ω ) \partial\Omega\setminus C(\partial\Omega) ∂ Ω ∖ C ( ∂ Ω ) (strictly) horizontally mean convex .
The appropriate formulation in this context is given in terms of the arrival time function , introduced in [18 ] . Suppose formally that there exists a function u ∈ C 2 ( R 3 ) u \in C^2(\mathbb{R}^3) u ∈ C 2 ( R 3 ) Ω t = { p ∈ R 3 : u ( p ) > t } \Omega_t = \{p \in\mathbb{R}^3 : u(p) > t\} Ω t = { p ∈ R 3 : u ( p ) > t } t > 0 t > 0 t > 0 u ( p ) u(p) u ( p ) p p p ∂ Ω t \partial\Omega_t ∂ Ω t
ν = − ∇ u ∣ ∇ u ∣ . \nu= -\frac{\nabla u}{|\nabla u|}. ν = − ∣∇ u ∣ ∇ u . where ∇ u \nabla u ∇ u (15) . Differentiating the identity u ( F ( q , t ) ) = t u(F(q,t)) = t u ( F ( q , t )) = t (2) , we formally obtain
1 = ∂ t u ( F ( q , t ) ) = ∇ u ⋅ ( − H ( F ( q , t ) ) ν ( F ( q , t ) ) ) = H ( F ( q , t ) ) ∣ ∇ u ( F ( q , t ) ) ∣ , (3) 1 = \partial_t u(F(q,t)) = \nabla u \cdot\left(-H(F(q,t))\nu(F(q,t))\right) = H(F(q,t))|\nabla u(F(q,t))|, \tag*{(3)} 1 = ∂ t u ( F ( q , t )) = ∇ u ⋅ ( − H ( F ( q , t )) ν ( F ( q , t )) ) = H ( F ( q , t )) ∣∇ u ( F ( q , t )) ∣ , (3) so that, using the explicit expression (17) for H H H u u u
− Δ u + ∇ ∇ u ( ∇ u , ∇ u ) ∣ ∇ u ∣ 2 = 1. (4) -\Delta u + \frac{\nabla\nabla u(\nabla u,\nabla u)}{|\nabla u|^2} = 1. \tag*{(4)} − Δ u + ∣∇ u ∣ 2 ∇∇ u ( ∇ u , ∇ u ) = 1. (4) Here Δ \Delta Δ ∇ ∇ \nabla\nabla ∇∇ § 2.3 . We are therefore led to the boundary value problem
{ − Δ u + ∇ ∇ u ( ∇ u , ∇ u ) ∣ ∇ u ∣ 2 = 1 in Ω , u = 0 on ∂ Ω , \begin{cases}
-\Delta u + \dfrac{\nabla\nabla u(\nabla u,\nabla u)}{|\nabla u|^2} = 1 & \text{in } \Omega,\\
u = 0 & \text{on } \partial\Omega,
\end{cases} ⎩ ⎨ ⎧ − Δ u + ∣∇ u ∣ 2 ∇∇ u ( ∇ u , ∇ u ) = 1 u = 0 in Ω , on ∂ Ω , (MCF)
which is referred to as the level-set formulation of the horizontal mean curvature flow with initial datum Ω \Omega Ω
Identity (3) also explains the role of horizontal mean convexity: wherever the arrival time is smooth and non-characteristic, H = ∣ ∇ u ∣ − 1 > 0 H = |\nabla u|^{-1} > 0 H = ∣∇ u ∣ − 1 > 0
Our first result establishes the existence of a mean-convex horizontal mean curvature flow under two alternative geometric assumptions: either the boundary is non-characteristic and has strictly positive horizontal mean curvature, or the domain is axisymmetric with sphere-like boundary and its horizontal mean curvature admits a strictly positive continuous extension across characteristic points; see § 3 .
The existence statement below involves the right-invariant distance d ~ \tilde d d ~ § 2.4 , which is the sub-Riemannian distance associated with the right-invariant distribution
D ~ = span { X ~ , Y ~ } , X ~ = ∂ x + y 2 ∂ z , Y ~ = ∂ y − x 2 ∂ z . (5) \widetilde{\mathcal{D}} = \operatorname{span}\{\widetilde X,\widetilde Y\}, \qquad\widetilde X = \partial_x + \frac{y}{2}\partial_z, \qquad\widetilde Y = \partial_y - \frac{x}{2}\partial_z. \tag*{(5)} D = span { X , Y } , X = ∂ x + 2 y ∂ z , Y = ∂ y − 2 x ∂ z . (5) Theorem 1.1. Let Ω ⊂ R 3 \Omega\subset\mathbb{R}^3 Ω ⊂ R 3 C 2 C^2 C 2 Ω \Omega Ω
(A) Ω \Omega Ω ∂ Ω \partial\Omega ∂ Ω ∂ Ω \partial\Omega ∂ Ω ∂ Ω \partial\Omega ∂ Ω
(B) ∂ Ω \partial\Omega ∂ Ω Ω \Omega Ω u ∈ C 0 ( Ω ‾ ) u \in C^{0}(\overline{\Omega}) u ∈ C 0 ( Ω ) d ~ \widetilde{d} d
We note that in case (A), right-Lipschitz continuity is equivalent to left-Lipschitz continuity, namely Lipschitz continuity with respect to the sub-Riemannian distance d d d
The two assumptions in Theorem 1.1 are independent. The Pansu sphere, described below, satisfies assumption (A) and has two characteristic points. Conversely, assumption (B) allows for different topologies and does not require axisymmetry. To illustrate this, consider axisymmetric tori, which always have empty characteristic sets but need not to be horizontally mean convex. For instance, the tori T = { ( x , y , z ) : ( r ( x , y ) − r 0 ) 2 + z 2 = a 2 } T=\{(x,y,z):(r(x,y)-r_{0})^{2}+z^{2}=a^{2}\} T = {( x , y , z ) : ( r ( x , y ) − r 0 ) 2 + z 2 = a 2 } 0 < a < r 0 0<a<r_{0} 0 < a < r 0 a / r 0 a/r_{0} a / r 0 C 2 C^{2} C 2
Besides establishing existence for this variety of initial sets, Theorem 1.1 yields an evolution that is non-fattening , in the sense that int { u = t } = ∅ \operatorname{int}\{u=t\}=\varnothing int { u = t } = ∅ t > 0 t>0 t > 0 (4) is manifestly incompatible with u u u (22) easily shows it.
Our proof of existence for (MCF) relies on an approximation procedure. A key technical step, which may be of independent interest, consists in approximating Ω \Omega Ω C 2 C^{2} C 2 § 2.7.1 .
For ε > 0 \varepsilon>0 ε > 0 g ε g_{\varepsilon} g ε
X ε = X , Y ε = Y , Z ε = ε ∂ z (6) X_{\varepsilon}=X,\qquad Y_{\varepsilon}=Y,\qquad Z_{\varepsilon}=\varepsilon\partial_{z} \tag*{(6)} X ε = X , Y ε = Y , Z ε = ε ∂ z (6) an orthonormal frame, and set H ε = H ∂ Ω ( R 3 , g ε ) H_{\varepsilon}=H_{\partial\Omega}(\mathbb{R}^{3},g_{\varepsilon}) H ε = H ∂ Ω ( R 3 , g ε ) ε → 0 + \varepsilon\to0^{+} ε → 0 + g ε g_{\varepsilon} g ε H ε H_{\varepsilon} H ε ∂ Ω \partial\Omega ∂ Ω H H H C ( ∂ Ω ) \mathcal{C}(\partial\Omega) C ( ∂ Ω ) H ε H_{\varepsilon} H ε H H H ε \varepsilon ε
In case (A), instead, horizontal mean convexity does not in general imply strict mean convexity in H ε \mathbb{H}_{\varepsilon} H ε H \mathbb{H} H Pansu sphere [31 ] . For R > 0 R>0 R > 0 R R R
P R = { p = ( x , y , z ) ∈ R 3 : ∣ z ∣ < f R ( r ( p ) ) , r ( p ) < R } (7) P_{R}=\{p=(x,y,z)\in\mathbb{R}^{3}:|z|<f_{R}(r(p)),\ r(p)<R\} \tag*{(7)} P R = { p = ( x , y , z ) ∈ R 3 : ∣ z ∣ < f R ( r ( p )) , r ( p ) < R } (7) where
f R ( r ) = 1 4 ( R 2 arccos ( r R ) + r R 2 − r 2 ) . f_{R}(r)=\frac{1}{4}\left(R^{2}\arccos\left(\frac{r}{R}\right)+r\sqrt{R^{2}-r^{2}}\right). f R ( r ) = 4 1 ( R 2 arccos ( R r ) + r R 2 − r 2 ) . Its boundary has two characteristic points, corresponding to r = 0 r=0 r = 0
H = 2 R on ∂ P R ∖ C ( ∂ P R ) . H=\frac{2}{R}\quad\text{on }\partial P_{R}\setminus\mathcal{C}(\partial P_{R}). H = R 2 on ∂ P R ∖ C ( ∂ P R ) . Nevertheless, its g ε g_{\varepsilon} g ε
H ε = 2 r ( 6 R 2 ε 2 + r 2 ( R 2 − 4 ε 2 ) ) ( 4 R 2 ε 2 + r 2 ( R 2 − 4 ε 2 ) ) 3 / 2 . H_{\varepsilon}=\frac{2r(6R^{2}\varepsilon^{2}+r^{2}(R^{2}-4\varepsilon^{2}))}{(4R^{2}\varepsilon^{2}+r^{2}(R^{2}-4\varepsilon^{2}))^{3/2}}. H ε = ( 4 R 2 ε 2 + r 2 ( R 2 − 4 ε 2 ) ) 3/2 2 r ( 6 R 2 ε 2 + r 2 ( R 2 − 4 ε 2 )) . In particular, although ∂ P R \partial P_{R} ∂ P R g ε g_{\varepsilon} g ε ε > 0 \varepsilon>0 ε > 0
We therefore construct a suitable perturbation Ω ε \Omega_\varepsilon Ω ε g ε g_\varepsilon g ε ∂ Ω \partial\Omega ∂ Ω approximation by prescribed mean curvature .
Theorem 1.2. Let Ω \Omega Ω C 2 C^2 C 2 H ( ∂ Ω ) H(\partial\Omega) H ( ∂ Ω ) C ( ∂ Ω ) C(\partial\Omega) C ( ∂ Ω ) c 1 , c 2 ∈ R c_1,c_2 \in\mathbb{R} c 1 , c 2 ∈ R H H H ∂ Ω \partial\Omega ∂ Ω c 1 ≤ H ≤ c 2 c_1 \le H \le c_2 c 1 ≤ H ≤ c 2 ∂ Ω \partial\Omega ∂ Ω ε > 0 \varepsilon> 0 ε > 0 Ω ε ⊂ R 3 \Omega_\varepsilon\subset\mathbb{R}^3 Ω ε ⊂ R 3 C 2 C^2 C 2 ∂ Ω ε → ∂ Ω \partial\Omega_\varepsilon\to\partial\Omega ∂ Ω ε → ∂ Ω ε → 0 \varepsilon\to0 ε → 0 η > 0 \eta> 0 η > 0 ε ˉ > 0 \bar{\varepsilon} > 0 ε ˉ > 0 g ε g_\varepsilon g ε ∂ Ω ε \partial\Omega_\varepsilon ∂ Ω ε c 1 − η ≤ H ε ≤ c 2 + η c_1-\eta\le H_\varepsilon\le c_2+\eta c 1 − η ≤ H ε ≤ c 2 + η ε < ε ˉ \varepsilon< \bar{\varepsilon} ε < ε ˉ
Theorem 1.2 covers assumption (A) of Theorem 1.1 . Axisymmetry of Ω \Omega Ω g ε g_\varepsilon g ε § 3 .
We are finally interested in understanding whether, in the distinguished setting given by assumption (A) of Theorem 1.1 , the horizontal mean curvature flow admits self-similar solutions. Here self-similarity is naturally understood with respect to the anisotropic dilations { δ λ } λ > 0 \{\delta_\lambda\}_{\lambda>0} { δ λ } λ > 0
δ λ ( x , y , z ) = ( λ x , λ y , λ 2 z ) , ( x , y , z ) ∈ R 3 , λ > 0. (8) \delta_\lambda(x,y,z)=(\lambda x,\lambda y,\lambda^2z), \qquad(x,y,z)\in\mathbb{R}^3,\quad\lambda>0. \tag*{(8)} δ λ ( x , y , z ) = ( λ x , λ y , λ 2 z ) , ( x , y , z ) ∈ R 3 , λ > 0. (8) Definition 1.3 (Self-similar solution). Let Ω ⊂ R 3 \Omega\subset\mathbb{R}^3 Ω ⊂ R 3 C 2 C^2 C 2 u ∈ C ( Ω ‾ ) u \in C(\overline{\Omega}) u ∈ C ( Ω ) Ω \Omega Ω T ∗ = max Ω ‾ u > 0 T^*=\max_{\overline{\Omega}}u>0 T ∗ = max Ω u > 0 u u u self-similar solution if there exists a continuous function
λ : [ 0 , T ∗ ) ⟶ ( 0 , 1 ] , λ ( 0 ) = 1 , λ ( t ) ⟶ t → T ∗ 0 \lambda:[0,T^*)\longrightarrow(0,1],\qquad\lambda(0)=1,\qquad\lambda(t)\mathop{\longrightarrow}_{t\to T^*}0 λ : [ 0 , T ∗ ) ⟶ ( 0 , 1 ] , λ ( 0 ) = 1 , λ ( t ) ⟶ t → T ∗ 0 such that for any t ∈ [ 0 , T ∗ ) t\in[0,T^*) t ∈ [ 0 , T ∗ )
Γ t = { p ∈ Ω ‾ : u ( p ) = t } , \Gamma_t=\{p\in\overline{\Omega}:u(p)=t\}, Γ t = { p ∈ Ω : u ( p ) = t } , satisfies
Γ t = δ λ ( t ) ( ∂ Ω ) . (9) \Gamma_t=\delta_{\lambda(t)}(\partial\Omega). \tag*{(9)} Γ t = δ λ ( t ) ( ∂ Ω ) . (9) In the Euclidean setting, the round sphere is simultaneously an isoperimetric set and a self-similarly shrinking solution to mean curvature flow. More generally, mean curvature flow is well known to have strong connections with the isoperimetry problem; see, for instance [33 ] for an account, and [25 , 26 , 1 ] for other applications of such connection. It is then natural to ask whether a similar picture holds in the Heisenberg group. The Pansu sphere is the natural candidate for such a role due to its isoperimetric properties [32 , 29 , 12 ] . In fact, characteristic points obstruct self-similar evolution through any right-Lipschitz solution.
Theorem 1.4. Let Ω ⊂ R 3 \Omega\subset\mathbb{R}^3 Ω ⊂ R 3 C 2 C^2 C 2 Ω \Omega Ω
The failure of the Pansu sphere in fulfilling natural isoperimetric-related properties was recently observed also in [21 ] .
At the core of Theorem 1.4 lies the following key observation: if u ∈ C 0 ( Ω ‾ ) u \in C^0(\overline{\Omega}) u ∈ C 0 ( Ω ) Γ t \Gamma_t Γ t instantaneous curvature blow-up and make it rigorous in Theorem 5.5 .
This blow-up can also be explicitly observed along the evolution of the Korányi sphere. This is the boundary of the set K R = { p ∈ R 3 : k R ( p ) < 0 } K_R = \{p \in\mathbb{R}^3 : k_R(p) < 0\} K R = { p ∈ R 3 : k R ( p ) < 0 } R > 0 R > 0 R > 0
k R ( x , y , z ) = ( x 2 + y 2 ) 2 + 16 z 2 − R 4 . k_R(x,y,z) = (x^2+y^2)^2 + 16z^2 - R^4. k R ( x , y , z ) = ( x 2 + y 2 ) 2 + 16 z 2 − R 4 . In terms of the radial distance r r r
H ∂ K R = 3 r R 2 , H_{\partial K_R} = \frac{3r}{R^2}, H ∂ K R = R 2 3 r , away from the characteristic set, which consists of the two poles. Although H ∂ K R > 0 H_{\partial K_R} > 0 H ∂ K R > 0 Theorem 1.1 does not apply to Ω = K R \Omega= K_R Ω = K R
Nevertheless, [20 ] , the authors show that the unique axisymmetric solution to
{ ∂ t w = Δ w − ∇ ∇ w ( ∇ w , ∇ w ) ∣ ∇ w ∣ 2 in R 3 × ( 0 , + ∞ ) , w ( ⋅ , 0 ) = k R in R 3 , (10) \begin{cases}
\partial_t w = \Delta w - \dfrac{\nabla\nabla w(\nabla w,\nabla w)}{|\nabla w|^2} & \text{in } \mathbb{R}^3 \times(0,+\infty),\\
w(\cdot,0) = k_R & \text{in } \mathbb{R}^3, \tag*{(10)}
\end{cases} ⎩ ⎨ ⎧ ∂ t w = Δ w − ∣∇ w ∣ 2 ∇∇ w ( ∇ w , ∇ w ) w ( ⋅ , 0 ) = k R in R 3 × ( 0 , + ∞ ) , in R 3 , (10) understood in the viscosity sense at points where ∇ w = 0 \nabla w = 0 ∇ w = 0
w ( p , t ) = ( x 2 + y 2 ) 2 + 12 t ( x 2 + y 2 ) + 16 z 2 + 12 t 2 − R 4 . (11) w(p,t) = (x^2+y^2)^2 + 12t(x^2+y^2) + 16z^2 + 12t^2 - R^4. \tag*{(11)} w ( p , t ) = ( x 2 + y 2 ) 2 + 12 t ( x 2 + y 2 ) + 16 z 2 + 12 t 2 − R 4 . (11) The horizontal mean curvature of the corresponding evolved level sets Σ t = { w ( ⋅ , t ) = 0 } \Sigma_t = \{w(\cdot,t) = 0\} Σ t = { w ( ⋅ , t ) = 0 } 0 ≤ t < T R ∗ 0 \le t < T_R^* 0 ≤ t < T R ∗ T R ∗ = R 2 12 T_R^* = \frac{R^2}{\sqrt{12}} T R ∗ = 12 R 2
H Σ t = ∂ t w ∣ ∇ w t ∣ = 12 ( x 2 + y 2 ) + 24 t ∣ ∇ w t ∣ . H_{\Sigma_t} = \frac{\partial_t w}{|\nabla w_t|} = \frac{12(x^2+y^2)+24t}{|\nabla w_t|}. H Σ t = ∣∇ w t ∣ ∂ t w = ∣∇ w t ∣ 12 ( x 2 + y 2 ) + 24 t . As a point of Σ t \Sigma_t Σ t H t H_t H t t > 0 t > 0 t > 0 t = 0 t = 0 t = 0 [9 , 20 ] , when the initial set is the Korányi sphere.
The explicit formula (1.11) reveals another interesting geometric aspect of this blow up. Indeed, setting λ t = R 4 − 12 t 2 \lambda_t = \sqrt{R^4 - 12t^2} λ t = R 4 − 12 t 2
δ 1 / λ t ( Σ t ) ⟶ Cyl R = { p ∈ R 3 : r ( p ) 2 = 1 R 2 12 } as t → T R ∗ \delta_{1/\lambda_t}(\Sigma_t) \longrightarrow\operatorname{Cyl}_R = \left\{p \in\mathbb{R}^3 : r(p)^2 = \frac{1}{R^2\sqrt{12}}\right\}
\qquad\text{as } t \to T_R^* δ 1/ λ t ( Σ t ) ⟶ Cyl R = { p ∈ R 3 : r ( p ) 2 = R 2 12 1 } as t → T R ∗ locally in Hausdorff distance. Thus, under this anisotropic rescaling, the asymptotic profile of the Korányi evolution is a vertical cylinder.
The instantaneous curvature blow-up, together with the cylindrical asymptotics of the Korányi evolution and the non-existence of compact right-Lipschitz self-similar solutions established above, leads us to conjecture that horizontal mean curvature flows starting from a suitable class of compact horizontally mean-convex hypersurfaces and becoming extinct at a single point do not approach a compact self-similar profile. Instead, their asymptotic geometry should be cylindrical. In this sense, vertical cylinders, which are shown to be self-similar for the general evolution in [9 ] , would provide the natural asymptotic models for compact mean-convex horizontal mean curvature flow evolutions in H \mathbb{H} H
Structure of the paper
Section 2 introduces the geometric analysis of the Heisenberg group needed in the paper: the horizontal notation for vectors, the right- and left-invariant sub-Riemannian distances, the horizontal gradient and the Laplacian of a function, and the curvature of hypersurfaces, together with some of their properties. It also collects the definitions of the Riemannian approximants and of viscosity solutions.
Section 3 contains the proof of Theorem 1.2. Its main building block is Proposition 3.10, which drives the approximation by prescribed mean curvature; this is then combined with a polynomial approximation, carried out in Section 3.3.
The proof of Theorem 1.1 is given in Section 4. It is based on the Riemannian approximation discussed above, at least in the axisymmetric case (Theorem 1.2), and relies on two further ingredients: stable gradient estimates for the solutions of the approximating Riemannian problems (Propositions 4.5 and 4.6), and the extraction of a limit viscosity solution (Proposition 4.8).
Section 5 shows that, whenever a horizontal mean curvature flow forms a foliation that is (slightly less than) C 2 C^2 C 2
Acknowledgments
The authors are members of INdAM–GNAMPA. The first and second authors are supported by the INdAM–GNAMPA projects “ANGELS” (CUP code E53C25002010001) and “GIANTS” (CUP code E53C23001740001). The second author is also supported by the STARS project “DEFORM” of the University of Padova (CUP code C93C25004860003). The third author is also affiliated with the Padova Neuroscience Center (PNC), University of Padova, and is supported by the French National Research Agency (ANR) through the project “SCOPE”.
The authors are grateful to Luca Capogna for his interest in this work and for helpful discussions, to Pierre Pansu for an inspiring discussion about the Koranyi evolution, and to Simone Verzellesi for pointing out an error in an argument in an earlier version of the paper.
AI Statement
The present paper is part of the Ph.D. project of the first author. All arguments presented herein were conceived and developed entirely by the authors. Large language models (ChatGPT and Claude) were used solely to assist with proof verification, the generation of TikZ figures, and linguistic editing.
The Heisenberg group
Horizontal distribution
The horizontal distribution D \mathcal{D} D R 3 \mathbb{R}^3 R 3 D \mathcal{D} D [ X , Y ] = Z [X,Y]=Z [ X , Y ] = Z Z = ∂ z Z=\partial_z Z = ∂ z
We explicit how horizontal vector fields change under rotations. For θ ∈ S 1 \theta\in\mathbb{S}^{1} θ ∈ S 1
R θ ( x , y , z ) = ( x cos θ − y sin θ , x sin θ + y cos θ , z ) (13) \mathcal{R}_{\theta}(x,y,z)=(x\cos\theta-y\sin\theta,x\sin\theta+y\cos\theta,z) \tag*{(13)} R θ ( x , y , z ) = ( x cos θ − y sin θ , x sin θ + y cos θ , z ) (13) denote the rotation about the z z z θ \theta θ u ∈ C 1 ( R 3 ) u \in C^{1}(\mathbb{R}^{3}) u ∈ C 1 ( R 3 ) θ ∈ S 1 \theta\in\mathbb{S}^{1} θ ∈ S 1
X ( u ∘ R θ ) = cos θ ( X u ) ∘ R θ + sin θ ( Y u ) ∘ R θ , X(u\circ\mathcal{R}_{\theta})=\cos\theta(Xu)\circ\mathcal{R}_{\theta}+\sin\theta(Yu)\circ\mathcal{R}_{\theta}, X ( u ∘ R θ ) = cos θ ( X u ) ∘ R θ + sin θ ( Y u ) ∘ R θ , Y ( u ∘ R θ ) = − sin θ ( X u ) ∘ R θ + cos θ ( Y u ) ∘ R θ . Y(u\circ\mathcal{R}_{\theta})=-\sin\theta(Xu)\circ\mathcal{R}_{\theta}+\cos\theta(Yu)\circ\mathcal{R}_{\theta}. Y ( u ∘ R θ ) = − sin θ ( X u ) ∘ R θ + cos θ ( Y u ) ∘ R θ . Moreover, the vertical field Z = ∂ z Z=\partial_z Z = ∂ z Z ( u ∘ R θ ) = ( Z u ) ∘ R θ Z(u\circ\mathcal{R}_{\theta})=(Zu)\circ\mathcal{R}_{\theta} Z ( u ∘ R θ ) = ( Z u ) ∘ R θ
Sub-Riemannian distance
A curve γ : [ 0 , 1 ] → R 3 \gamma:[0,1]\to\mathbb{R}^{3} γ : [ 0 , 1 ] → R 3 horizontal if it is absolutely continuous and γ ˙ ( t ) ∈ D γ ( t ) \dot{\gamma}(t)\in\mathcal{D}_{\gamma(t)} γ ˙ ( t ) ∈ D γ ( t ) t ∈ [ 0 , 1 ] t\in[0,1] t ∈ [ 0 , 1 ] [ 0 , 1 ] [0,1] [ 0 , 1 ] sub-Riemannian distance from p ∈ R 3 p\in\mathbb{R}^{3} p ∈ R 3 q ∈ R 3 q\in\mathbb{R}^{3} q ∈ R 3
d ( p , q ) = inf { ∫ 0 1 ∣ γ ˙ ( t ) ∣ d t | γ : [ 0 , 1 ] → R 3 is horizontal , γ ( 0 ) = p , γ ( 1 ) = q } . (14) d(p,q)=\inf\left\{\int_{0}^{1}|\dot{\gamma}(t)|\,dt\ \middle|\ \gamma:[0,1]\to\mathbb{R}^{3}\text{ is horizontal},\ \gamma(0)=p,\ \gamma(1)=q\right\}. \tag*{(14)} d ( p , q ) = inf { ∫ 0 1 ∣ γ ˙ ( t ) ∣ d t γ : [ 0 , 1 ] → R 3 is horizontal , γ ( 0 ) = p , γ ( 1 ) = q } . (14) Since D \mathcal{D} D d : R 3 × R 3 → [ 0 , + ∞ ] d:\mathbb{R}^{3}\times\mathbb{R}^{3}\to[0,+\infty] d : R 3 × R 3 → [ 0 , + ∞ ] R 3 \mathbb{R}^{3} R 3 [2 Theorem 3.31] ). By Remark 2.1, the distance d d d
left-translations invariant with respect to the group operation (2.1), i.e., d ( q ∗ p , q ∗ p ′ ) = d ( p , p ′ ) d(q*p,q*p')=d(p,p') d ( q ∗ p , q ∗ p ′ ) = d ( p , p ′ ) p , p ′ ∈ R 3 p,p'\in\mathbb{R}^{3} p , p ′ ∈ R 3
1-homogeneous with respect to the family of anisotropic dilations { δ λ } λ > 0 \{\delta_{\lambda}\}_{\lambda>0} { δ λ } λ > 0 d ( δ λ p , δ λ p ′ ) = λ d ( p , p ′ ) d(\delta_{\lambda}p,\delta_{\lambda}p')=\lambda d(p,p') d ( δ λ p , δ λ p ′ ) = λ d ( p , p ′ ) λ > 0 \lambda>0 λ > 0 p , p ′ ∈ R 3 p,p'\in\mathbb{R}^{3} p , p ′ ∈ R 3
Remark 2.2. The sub-Riemannian distance is not equivalent to the Euclidean one. The following estimates hold: for every compact set K ⊂ R 3 K\subset\mathbb{R}^{3} K ⊂ R 3 C K > 0 C_{K}>0 C K > 0
C K − 1 ∣ p − q ∣ R 3 ≤ d ( p , q ) ≤ C K ∣ p − q ∣ R 3 1 2 , ∀ p , q ∈ K , C_{K}^{-1}|p-q|_{\mathbb{R}^{3}}\le d(p,q)\le C_{K}|p-q|_{\mathbb{R}^{3}}^{\frac{1}{2}},\qquad\forall p,q\in K, C K − 1 ∣ p − q ∣ R 3 ≤ d ( p , q ) ≤ C K ∣ p − q ∣ R 3 2 1 , ∀ p , q ∈ K , see, e.g., [6 Proposition 5.15.1] .
Horizontal gradient, Horizontal Hessian and Heisenberg sub-Laplacian
Let u ∈ C 1 ( R 3 ) u\in C^{1}(\mathbb{R}^{3}) u ∈ C 1 ( R 3 ) horizontal gradient of u u u
∇ u = ( X u ) X + ( Y u ) Y ∈ D . (15) \nabla u=(Xu)X+(Yu)Y\in\mathcal{D}. \tag*{(15)} ∇ u = ( X u ) X + ( Y u ) Y ∈ D . (15) If u ∈ C 2 u\in C^{2} u ∈ C 2 horizontal Hessian of u u u D × D \mathcal{D}\times\mathcal{D} D × D { X , Y } \{X,Y\} { X , Y }
∇ ∇ u = ( X X u X Y u Y X u Y Y u ) . \nabla\nabla u=
\begin{pmatrix}
XXu & XYu\\
YXu & YYu
\end{pmatrix}. ∇∇ u = ( X X u Y X u X Y u Y Y u ) . The Heisenberg sub-Laplacian of u u u
Δ u = tr ( ∇ ∇ u ) = X 2 u + Y 2 u , \Delta u=\operatorname{tr}(\nabla\nabla u)=X^{2}u+Y^{2}u, Δ u = tr ( ∇∇ u ) = X 2 u + Y 2 u , and it is related to the horizontal gradient via the identity Δ u = div ( ∇ u ) \Delta u=\operatorname{div}(\nabla u) Δ u = div ( ∇ u )
Remark 2.3. By Remark 2.1, the horizontal gradient, the horizontal Hessian, and the Heisenberg sub-Laplacian are left-invariant with respect to the group law (2.1) and also invariant under rotations about the z z z Z = ∂ z Z=\partial_z Z = ∂ z z z z
Right-invariant structure
In this paper, right-invariant structure will also come into play. The Lie algebra of right-invariant vector fields with respect to the group law (12) is generated by X ~ \widetilde{X} X Y ~ \widetilde{Y} Y (5) and by Z = ∂ z Z = \partial_z Z = ∂ z [ X ~ , Y ~ ] = − Z [\widetilde{X},\widetilde{Y}] = -Z [ X , Y ] = − Z
[ X , X ~ ] = [ X , Y ~ ] = [ Y , X ~ ] = [ Y , Y ~ ] = 0. (16) [X,\widetilde{X}] = [X,\widetilde{Y}] = [Y,\widetilde{X}] = [Y,\widetilde{Y}] = 0. \tag*{(16)} [ X , X ] = [ X , Y ] = [ Y , X ] = [ Y , Y ] = 0. (16) We define the right-invariant distance d ~ \widetilde{d} d (14) with curves that are a.e. tangent to D ~ \widetilde{\mathcal{D}} D ~ \widetilde{\ } u u u
∇ ~ u = X ~ u X ~ + Y ~ u Y ~ . \widetilde{\nabla}u = \widetilde{X}u\,\widetilde{X} + \widetilde{Y}u\,\widetilde{Y}. ∇ u = X u X + Y u Y . Hypersurfaces in the Heisenberg group and mean curvature
The set of characteristic points of a C 1 C^1 C 1 S S S C ( S ) = { p ∈ S : D p ⊂ T p S } \mathcal{C}(S) = \{p \in S : \mathcal{D}_p \subset T_pS\} C ( S ) = { p ∈ S : D p ⊂ T p S } S S S u ∈ C k ( R 3 ) u \in C^k(\mathbb{R}^3) u ∈ C k ( R 3 ) k ≥ 1 k \ge1 k ≥ 1 S = { p ∈ R 3 : u ( p ) = 0 } S = \{p \in\mathbb{R}^3 : u(p) = 0\} S = { p ∈ R 3 : u ( p ) = 0 } ∇ E u ≠ 0 \nabla_Eu \ne0 ∇ E u = 0
C ( S ) = { p ∈ S ∣ X u ( p ) = Y u ( p ) = 0 } . \mathcal{C}(S) = \{p \in S \mid Xu(p) = Yu(p) = 0\}. C ( S ) = { p ∈ S ∣ X u ( p ) = Y u ( p ) = 0 } . Let Ω = { p ∈ R 3 ∣ u ( p ) > 0 } \Omega= \{p \in\mathbb{R}^3 \mid u(p) > 0\} Ω = { p ∈ R 3 ∣ u ( p ) > 0 } u ∈ C k ( R 3 ) u \in C^k(\mathbb{R}^3) u ∈ C k ( R 3 ) k ≥ 1 k \ge1 k ≥ 1 ∇ E u ( p ) ≠ 0 \nabla_Eu(p) \ne0 ∇ E u ( p ) = 0 p ∈ ∂ Ω p \in\partial\Omega p ∈ ∂ Ω Ω \Omega Ω u u u horizontal outer unit normal ν \nu ν Ω \Omega Ω
ν = − ∇ u ∣ ∇ u ∣ on ∂ Ω ∖ C ( ∂ Ω ) . \nu= -\frac{\nabla u}{|\nabla u|} \quad\text{on } \partial\Omega\setminus\mathcal{C}(\partial\Omega). ν = − ∣∇ u ∣ ∇ u on ∂ Ω ∖ C ( ∂ Ω ) . One can check that this definition does not depend on the choice of u u u [4 Theorem 1.1] .
If Ω \Omega Ω u ∈ C 2 ( R 3 ) u \in C^2(\mathbb{R}^3) u ∈ C 2 ( R 3 ) horizontal mean curvature of ∂ Ω \partial\Omega ∂ Ω p ∈ ∂ Ω ∖ C ( ∂ Ω ) p \in\partial\Omega\setminus\mathcal{C}(\partial\Omega) p ∈ ∂ Ω ∖ C ( ∂ Ω )
H = div ( ν ) = − 1 ∣ ∇ u ∣ ( Δ u − ∇ ∇ u ( ∇ u , ∇ u ) ∣ ∇ u ∣ 2 ) at p ∈ ∂ Ω ∖ C ( ∂ Ω ) . (17) H = \operatorname{div}(\nu) = -\frac{1}{|\nabla u|}\left(\Delta u - \frac{\nabla\nabla u(\nabla u,\nabla u)}{|\nabla u|^2}\right) \quad\text{at } p \in\partial\Omega\setminus\mathcal{C}(\partial\Omega). \tag*{(17)} H = div ( ν ) = − ∣∇ u ∣ 1 ( Δ u − ∣∇ u ∣ 2 ∇∇ u ( ∇ u , ∇ u ) ) at p ∈ ∂ Ω ∖ C ( ∂ Ω ) . (17) Definition 2.5 (Horizontal mean convexity and continuity of H H H Ω \Omega Ω C 2 C^2 C 2 H H H (17) at noncharacteristic point. We say that ∂ Ω \partial\Omega ∂ Ω strictly positive horizontal mean curvature if H > c > 0 H > c > 0 H > c > 0 p ∈ ∂ Ω ∖ C ( ∂ Ω ) p \in\partial\Omega\setminus\mathcal{C}(\partial\Omega) p ∈ ∂ Ω ∖ C ( ∂ Ω ) ∂ Ω \partial\Omega ∂ Ω horizontally mean convex .
Moreover, we say that H ∈ C 0 ( ∂ Ω ) H \in C^0(\partial\Omega) H ∈ C 0 ( ∂ Ω ) H H H ∂ Ω \partial\Omega ∂ Ω ∂ Ω \partial\Omega ∂ Ω H ∈ C 0 ( ∂ Ω ) H \in C^0(\partial\Omega) H ∈ C 0 ( ∂ Ω )
Riemannian approximation
For ε > 0 \varepsilon> 0 ε > 0 H ε = ( R 3 , g ε ) \mathbb{H}_\varepsilon= (\mathbb{R}^3,g_\varepsilon) H ε = ( R 3 , g ε ) g ε g_\varepsilon g ε (6) orthonormal. We denote by d ε d_\varepsilon d ε g ε g_\varepsilon g ε
Proposition 2.6. Let ε > 0 \varepsilon> 0 ε > 0 p , q ∈ R 3 p,q \in\mathbb{R}^3 p , q ∈ R 3 d ε ( p , q ) ≤ d ( p , q ) d_\varepsilon(p,q) \le d(p,q) d ε ( p , q ) ≤ d ( p , q ) d ~ ε ( p , q ) ≤ d ~ ( p , q ) \widetilde{d}_\varepsilon(p,q) \le\widetilde{d}(p,q) d ε ( p , q ) ≤ d ( p , q )
Proof. Recall that an absolutely continuous curve γ = ( γ x , γ y , γ z ) \gamma=(\gamma_x,\gamma_y,\gamma_z) γ = ( γ x , γ y , γ z ) [ 0 , 1 ] [0,1] [ 0 , 1 ] γ ˙ z = 1 2 ( γ x γ ˙ y − γ y γ ˙ x ) \dot{\gamma}_z=\frac{1}{2}(\gamma_x\dot{\gamma}_y-\gamma_y\dot{\gamma}_x) γ ˙ z = 2 1 ( γ x γ ˙ y − γ y γ ˙ x ) [ 0 , 1 ] [0,1] [ 0 , 1 ] γ \gamma γ
∣ γ ˙ ∣ ε = ∣ ( γ ˙ x , γ ˙ y , 0 ) ∣ R 3 = ∣ ( γ ˙ x , γ ˙ y ) ∣ R 2 = ∣ γ ˙ ∣ , |\dot{\gamma}|_{\varepsilon}=|(\dot{\gamma}_x,\dot{\gamma}_y,0)|_{\mathbb{R}^3}=|(\dot{\gamma}_x,\dot{\gamma}_y)|_{\mathbb{R}^2}=|\dot{\gamma}|, ∣ γ ˙ ∣ ε = ∣ ( γ ˙ x , γ ˙ y , 0 ) ∣ R 3 = ∣ ( γ ˙ x , γ ˙ y ) ∣ R 2 = ∣ γ ˙ ∣ , and the conclusion follows from the definition of the sub-Riemannian distance (14) . □
We use the subscript ε \varepsilon ε g ε g_\varepsilon g ε ∇ ε \nabla_\varepsilon ∇ ε ∇ ε ∇ ε \nabla_\varepsilon\nabla_\varepsilon ∇ ε ∇ ε Δ ε \Delta_\varepsilon Δ ε g ε g_\varepsilon g ε ∣ ∇ ε f ∣ 2 = g ε ( ∇ ε f , ∇ ε f ) |\nabla_\varepsilon f|^2=g_\varepsilon(\nabla_\varepsilon f,\nabla_\varepsilon f) ∣ ∇ ε f ∣ 2 = g ε ( ∇ ε f , ∇ ε f ) ( X f ) 2 + ( Y f ) 2 + ε 2 ( Z f ) 2 (Xf)^2+(Yf)^2+\varepsilon^2(Zf)^2 ( X f ) 2 + ( Y f ) 2 + ε 2 ( Z f ) 2 g ε g_\varepsilon g ε ∂ Ω \partial\Omega ∂ Ω Ω \Omega Ω u ∈ C 2 u\in C^2 u ∈ C 2 ∣ ∇ ε u ∣ ≠ 0 |\nabla_\varepsilon u|\ne0 ∣ ∇ ε u ∣ = 0 ∂ Ω \partial\Omega ∂ Ω
H ε = − div ( ∇ ε u ∣ ∇ ε u ∣ ε ) = − 1 ∣ ∇ ε u ∣ ε ( Δ ε u − ∇ ε ∇ ε u ( ∇ ε u , ∇ ε u ) ∣ ∇ ε u ∣ 2 ) at p ∈ ∂ Ω . H_\varepsilon=-\operatorname{div}\left(\frac{\nabla_\varepsilon u}{|\nabla_\varepsilon u|_\varepsilon}\right)=-\frac{1}{|\nabla_\varepsilon u|_\varepsilon}\left(\Delta_\varepsilon u-\frac{\nabla_\varepsilon\nabla_\varepsilon u(\nabla_\varepsilon u,\nabla_\varepsilon u)}{|\nabla_\varepsilon u|^2}\right)\quad\text{at }p\in\partial\Omega. H ε = − div ( ∣ ∇ ε u ∣ ε ∇ ε u ) = − ∣ ∇ ε u ∣ ε 1 ( Δ ε u − ∣ ∇ ε u ∣ 2 ∇ ε ∇ ε u ( ∇ ε u , ∇ ε u ) ) at p ∈ ∂ Ω. One can finally check that, if f ε → f f_\varepsilon\to f f ε → f C 2 C^2 C 2 ∇ ε f ε → ∇ f \nabla_\varepsilon f_\varepsilon\to\nabla f ∇ ε f ε → ∇ f ∇ ε ∇ ε f ε ( ∇ ε f ε , ∇ ε f ε ) → ∇ ∇ f ( ∇ f , ∇ f ) \nabla_\varepsilon\nabla_\varepsilon f_\varepsilon(\nabla_\varepsilon f_\varepsilon,\nabla_\varepsilon f_\varepsilon)\to\nabla\nabla f(\nabla f,\nabla f) ∇ ε ∇ ε f ε ( ∇ ε f ε , ∇ ε f ε ) → ∇∇ f ( ∇ f , ∇ f )
div ( ∇ ε f ε ∣ ∇ ε f ε ∣ ) → div ( ∇ f ∣ ∇ f ∣ ) \operatorname{div}\left(\frac{\nabla_\varepsilon f_\varepsilon}{|\nabla_\varepsilon f_\varepsilon|}\right)\to\operatorname{div}\left(\frac{\nabla f}{|\nabla f|}\right) div ( ∣ ∇ ε f ε ∣ ∇ ε f ε ) → div ( ∣∇ f ∣ ∇ f ) locally uniformly in { ∣ ∇ f ∣ ≠ 0 } \{|\nabla f|\ne0\} { ∣∇ f ∣ = 0 } Ω \Omega Ω C 2 C^2 C 2
H ε → H as ε → 0 locally uniformly in ∂ Ω ∖ C ( ∂ Ω ) . (18) H_\varepsilon\to H\quad\text{as }\varepsilon\to0\text{ locally uniformly in }\partial\Omega\setminus C(\partial\Omega). \tag*{(18)} H ε → H as ε → 0 locally uniformly in ∂ Ω ∖ C ( ∂ Ω ) . (18) This last observation is also recorded in [10 Corollary 4.21] . We are repeatedly using these simple observations without explicitly mentioning it. In full analogy one can consider Riemannian approximations for the sub-Riemannian structure given by the right-invariant distribution described above. Namely, one can define g ~ ε \widetilde{g}_\varepsilon g ε R 3 \mathbb{R}^3 R 3 X ~ , Y ~ \widetilde{X},\widetilde{Y} X , Y ε Z \varepsilon Z εZ ∇ ~ ε f \widetilde{\nabla}_\varepsilon f ∇ ε f g ~ ε \widetilde{g}_\varepsilon g ε f f f ∣ ∇ ~ ε f ∣ 2 = g ~ ε ( ∇ ~ ε f , ∇ ~ ε f ) |\widetilde{\nabla}_\varepsilon f|^2=\widetilde{g}_\varepsilon(\widetilde{\nabla}_\varepsilon f,\widetilde{\nabla}_\varepsilon f) ∣ ∇ ε f ∣ 2 = g ε ( ∇ ε f , ∇ ε f )
Remark 2.7. Let u ∈ C 1 ( R 3 ) u\in C^1(\mathbb{R}^3) u ∈ C 1 ( R 3 ) u ∘ R θ = u u\circ R_\theta=u u ∘ R θ = u θ ∈ S 1 \theta\in\mathbb{S}^1 θ ∈ S 1 R θ R_\theta R θ (13) . Then, we have
∣ ∇ u ∣ = ∣ ∇ ~ u ∣ , |\nabla u|=|\widetilde{\nabla}u|, ∣∇ u ∣ = ∣ ∇ u ∣ , and consequently, one also gets
∣ ∇ ε u ∣ = ∣ ∇ ~ ε u ∣ . (19) |\nabla_\varepsilon u|=|\widetilde{\nabla}_\varepsilon u|. \tag*{(19)} ∣ ∇ ε u ∣ = ∣ ∇ ε u ∣. (19) Viscosity solutions to (1.4) and (MCF)
The formal derivation described in the introduction leads us to the level-set formulation of the horizontal mean-convex mean curvature flow equation (1.4). To include characteristic initial data in the study of horizontal mean curvature flow evolutions we will study viscosity solutions to (MCF).
Definition 2.8. A function u ∈ C 0 ( Ω ) u\in C^0(\Omega) u ∈ C 0 ( Ω ) viscosity subsolution of (1.4) in Ω ⊂ R 3 \Omega\subset\mathbb{R}^3 Ω ⊂ R 3 p ∈ Ω p\in\Omega p ∈ Ω Φ ∈ C 2 ( Ω ) \Phi\in C^2(\Omega) Φ ∈ C 2 ( Ω ) u − Φ u-\Phi u − Φ p p p
{ 1 ≥ − Δ Φ + ∇ ∇ Φ ( ∇ Φ , ∇ Φ ) ∣ ∇ Φ ∣ 2 at p if ∇ Φ ( p ) ≠ 0 (20) \begin{cases}
1\ge-\Delta\Phi+\dfrac{\nabla\nabla\Phi(\nabla\Phi,\nabla\Phi)}{|\nabla\Phi|^2} & \text{at }p\\
\text{if }\nabla\Phi(p)\ne0 \tag*{(20)}
\end{cases} ⎩ ⎨ ⎧ 1 ≥ − ΔΦ + ∣∇Φ ∣ 2 ∇∇Φ ( ∇Φ , ∇Φ ) if ∇Φ ( p ) = 0 at p (20) and
{ 1 ≥ − Δ Φ + ∇ ∇ Φ ( η , η ) at p for some η ∈ D with ∣ η ∣ ≤ 1 , if ∇ Φ ( p ) = 0. (21) \begin{cases}
1 \ge-\Delta\Phi+ \nabla\nabla\Phi(\eta,\eta) & \text{at } p \\
\text{for some } \eta\in\mathcal{D} \text{ with } |\eta| \le1,\ \text{if } \nabla\Phi(p)=0. \tag*{(21)}
\end{cases} { 1 ≥ − ΔΦ + ∇∇Φ ( η , η ) for some η ∈ D with ∣ η ∣ ≤ 1 , if ∇Φ ( p ) = 0. at p (21) A function u ∈ C 0 ( Ω ) u \in C^{0}(\Omega) u ∈ C 0 ( Ω ) viscosity supersolution of (1.4) in Ω ⊂ R 3 \Omega\subset\mathbb{R}^{3} Ω ⊂ R 3 p 0 ∈ Ω p_{0} \in\Omega p 0 ∈ Ω Φ ∈ C 2 ( Ω ) \Phi\in C^{2}(\Omega) Φ ∈ C 2 ( Ω ) u − Φ u-\Phi u − Φ p p p
{ 1 ≤ − Δ Φ + ∇ ∇ Φ ( ∇ Φ , ∇ Φ ) ∣ ∇ Φ ∣ 2 at p if ∇ Φ ( p ) ≠ 0 \begin{cases}
1 \le-\Delta\Phi+ \dfrac{\nabla\nabla\Phi(\nabla\Phi,\nabla\Phi)}{|\nabla\Phi|^{2}} & \text{at } p \\
\text{if } \nabla\Phi(p)\ne0
\end{cases} ⎩ ⎨ ⎧ 1 ≤ − ΔΦ + ∣∇Φ ∣ 2 ∇∇Φ ( ∇Φ , ∇Φ ) if ∇Φ ( p ) = 0 at p and
{ 1 ≤ − Δ Φ + ∇ ∇ Φ ( η , η ) at p for some η ∈ D with ∣ η ∣ ≤ 1 , if ∇ Φ ( p ) = 0. (22) \begin{cases}
1 \le-\Delta\Phi+ \nabla\nabla\Phi(\eta,\eta) & \text{at } p \\
\text{for some } \eta\in\mathcal{D} \text{ with } |\eta| \le1,\ \text{if } \nabla\Phi(p)=0. \tag*{(22)}
\end{cases} { 1 ≤ − ΔΦ + ∇∇Φ ( η , η ) for some η ∈ D with ∣ η ∣ ≤ 1 , if ∇Φ ( p ) = 0. at p (22) A function u ∈ C 0 ( Ω ) u \in C^{0}(\Omega) u ∈ C 0 ( Ω ) viscosity solution to equation (1.4) if it is both a viscosity supersolution and a viscosity subsolution. A function u ∈ C 0 ( Ω ‾ ) u \in C^{0}(\overline{\Omega}) u ∈ C 0 ( Ω ) viscosity solution to (MCF) if it is a viscosity solution of (1.4) in Ω \Omega Ω
lim x → p u ( x ) = 0 ∀ p ∈ ∂ Ω . \lim_{x\to p}u(x)=0 \quad\forall p\in\partial\Omega. x → p lim u ( x ) = 0 ∀ p ∈ ∂ Ω. Approximating solutions
Our strategy to prove the existence of solutions to (MCF) (see Theorem 1.1) is based on an approximation procedure. We first apply Theorem 1.2 to approximate Ω \Omega Ω { Ω ε } ε > 0 \{\Omega_{\varepsilon}\}_{\varepsilon>0} { Ω ε } ε > 0 H ε \mathbb{H}_{\varepsilon} H ε Ω ε \Omega_{\varepsilon} Ω ε H ε \mathbb{H}_{\varepsilon} H ε
− Δ ε u ε + ∇ ε ∇ ε u ε ( ∇ ε u ε , ∇ ε u ε ) ∣ ∇ ε u ε ∣ 2 = 1. -\Delta_{\varepsilon}u_{\varepsilon}+
\frac{\nabla_{\varepsilon}\nabla_{\varepsilon}u_{\varepsilon}(\nabla_{\varepsilon}u_{\varepsilon},\nabla_{\varepsilon}u_{\varepsilon})}{|\nabla_{\varepsilon}u_{\varepsilon}|^{2}}=1. − Δ ε u ε + ∣ ∇ ε u ε ∣ 2 ∇ ε ∇ ε u ε ( ∇ ε u ε , ∇ ε u ε ) = 1. The resulting approximating problems correspond to the level set formulation for the mean curvature flow on Ω ε \Omega_{\varepsilon} Ω ε u ε ∈ C 0 ( Ω ε ‾ ) u_{\varepsilon}\in C^{0}(\overline{\Omega_{\varepsilon}}) u ε ∈ C 0 ( Ω ε )
{ − Δ ε u ε + ∇ ε ∇ ε u ε ( ∇ ε u ε , ∇ ε u ε ) ∣ ∇ ε u ε ∣ 2 = 1 in Ω ε , u ε = 0 on ∂ Ω ε . \begin{cases}
-\Delta_{\varepsilon}u_{\varepsilon}+
\dfrac{\nabla_{\varepsilon}\nabla_{\varepsilon}u_{\varepsilon}(\nabla_{\varepsilon}u_{\varepsilon},\nabla_{\varepsilon}u_{\varepsilon})}{|\nabla_{\varepsilon}u_{\varepsilon}|^{2}}=1 & \text{in } \Omega_{\varepsilon},\\
u_{\varepsilon}=0 & \text{on } \partial\Omega_{\varepsilon}.
\end{cases} ⎩ ⎨ ⎧ − Δ ε u ε + ∣ ∇ ε u ε ∣ 2 ∇ ε ∇ ε u ε ( ∇ ε u ε , ∇ ε u ε ) = 1 u ε = 0 in Ω ε , on ∂ Ω ε . Viscosity solutions to (MCFε _{\varepsilon} ε [34 ] as the limit as δ → 0 + \delta\to0^{+} δ → 0 + u ε δ u_{\varepsilon}^{\delta} u ε δ
{ − Δ ε u ε δ + ∇ ε ∇ ε u ε δ ( ∇ ε u ε δ , ∇ ε u ε δ ) ∣ ∇ ε u ε δ ∣ 2 + δ 2 = 1 in Ω ε , u ε δ = 0 on ∂ Ω ε . \begin{cases}
-\Delta_{\varepsilon}u_{\varepsilon}^{\delta}+
\dfrac{\nabla_{\varepsilon}\nabla_{\varepsilon}u_{\varepsilon}^{\delta}(\nabla_{\varepsilon}u_{\varepsilon}^{\delta},\nabla_{\varepsilon}u_{\varepsilon}^{\delta})}{|\nabla_{\varepsilon}u_{\varepsilon}^{\delta}|^{2}+\delta^{2}}=1 & \text{in } \Omega_{\varepsilon},\\
u_{\varepsilon}^{\delta}=0 & \text{on } \partial\Omega_{\varepsilon}.
\end{cases} ⎩ ⎨ ⎧ − Δ ε u ε δ + ∣ ∇ ε u ε δ ∣ 2 + δ 2 ∇ ε ∇ ε u ε δ ( ∇ ε u ε δ , ∇ ε u ε δ ) = 1 u ε δ = 0 in Ω ε , on ∂ Ω ε . To construct a solution of (MCF), we first obtain a family of solutions { u ε δ } ε , δ > 0 \{u_{\varepsilon}^{\delta}\}_{\varepsilon,\delta>0} { u ε δ } ε , δ > 0 ε δ _{\varepsilon}^{\delta} ε δ δ → 0 \delta\to0 δ → 0 ε → 0 \varepsilon\to0 ε → 0 u ε δ u_{\varepsilon}^{\delta} u ε δ C 0 ( Ω ‾ ) C^{0}(\overline{\Omega}) C 0 ( Ω )
This is the main technical step in which our assumptions are used. First, a Bochner-type identity based on [23 Proposition 3.4] allows to obtain right-invariant gradient estimates for u ε δ u_{\varepsilon}^{\delta} u ε δ D \mathcal{D} D D ~ \widetilde{\mathcal{D}} D ∇ \nabla ∇ d ~ \widetilde{d} d
Axisymmetry in the characteristic case is crucial for converting the right-invariant gradient estimate into a left-invariant one by Remark 2.7, allowing to exploit the boundary information H ε ∣ ∇ ε u ε ∣ = 1 H_\varepsilon\lvert\nabla_\varepsilon u_\varepsilon\rvert=1 H ε ∣ ∇ ε u ε ∣ = 1 ∂ Ω ε \partial\Omega_\varepsilon ∂ Ω ε g ε g_\varepsilon g ε ∂ Ω ε \partial\Omega_\varepsilon ∂ Ω ε Ω ‾ ε \overline{\Omega}_\varepsilon Ω ε ε \varepsilon ε
In the non-characteristic case we obtain an upper bound on the norm of right-invariant gradient, arguing similarly to the Euclidean case treated in [18 ] . Indeed, comparing u ε δ u_\varepsilon^\delta u ε δ d E d_E d E ∂ Ω ε \partial\Omega_\varepsilon ∂ Ω ε [19 Proposition 4.3] , one obtains a bound on ∥ ∇ ~ ε u ε δ ∥ L ∞ ( ∂ Ω ε ) \lVert\widetilde{\nabla}_\varepsilon u_\varepsilon^\delta\rVert_{L^\infty(\partial\Omega_\varepsilon)} ∥ ∇ ε u ε δ ∥ L ∞ ( ∂ Ω ε ) ε \varepsilon ε δ \delta δ d ~ \widetilde d d d d d
Remark 2.9. By Proposition 3.3, regular axisymmetric sets are topologically either spheres or tori. Theorem 1.2 deals only with the spherical case, since axisymmetric tori are non-characteristic by Corollary 3.5.
Consequently, by (18) , ∂ Ω \partial\Omega ∂ Ω g ε g_\varepsilon g ε ε > 0 \varepsilon>0 ε > 0 ε _\varepsilon ε [34 ] ).
Approximation by prescribed mean curvature and proof of theorem 1.2
In this section we collect some facts on axisymmetric sets and prove Theorem 1.2.
C2-regular connected axisymmetric sets
3.1. C 2 C^2 C 2 Axisymmetric connected sets are those generated by rotation of a C k C^k C k z z z
Definition 3.1. Let γ = ( γ 1 , γ 2 ) ∈ C k ( [ 0 , 1 ] , R + 2 ) \gamma=(\gamma_1,\gamma_2)\in C^k([0,1],\mathbb{R}_+^2) γ = ( γ 1 , γ 2 ) ∈ C k ([ 0 , 1 ] , R + 2 ) k ∈ N k\in\mathbb{N} k ∈ N R θ R_\theta R θ θ ∈ S 1 \theta\in S^1 θ ∈ S 1 Φ γ : [ 0 , 1 ] × S 1 → R 3 \Phi_\gamma:[0,1]\times S^1\to\mathbb{R}^3 Φ γ : [ 0 , 1 ] × S 1 → R 3
Φ γ ( τ , θ ) = R θ ( γ 1 ( τ ) , 0 , γ 2 ( τ ) ) , (23) \Phi_\gamma(\tau,\theta)=R_\theta(\gamma_1(\tau),0,\gamma_2(\tau)), \tag*{(23)} Φ γ ( τ , θ ) = R θ ( γ 1 ( τ ) , 0 , γ 2 ( τ )) , (23) We say that γ \gamma γ generating profile for a bounded, open, connected set Ω ⊂ R 3 \Omega\subset\mathbb{R}^3 Ω ⊂ R 3 C k C^k C k k ≥ 1 k\ge1 k ≥ 1 ∂ Ω = Φ γ ( [ 0 , 1 ] × S 1 ) \partial\Omega=\Phi_\gamma([0,1]\times S^1) ∂ Ω = Φ γ ([ 0 , 1 ] × S 1 ) Ω \Omega Ω C k C^k C k
Remark 3.2. Every C k C^k C k C k C^k C k { y = 0 , x ≥ 0 } \{y=0,\ x\ge0\} { y = 0 , x ≥ 0 }
In the following proposition we discuss the topology of C k C^k C k
Proposition 3.3. Let Ω ⊂ R 3 \Omega\subset\mathbb{R}^3 Ω ⊂ R 3 C 1 C^1 C 1 ∂ Ω \partial\Omega ∂ Ω z z z sphere , otherwise it is homeomorphic to a torus .
Remark 3.4. If ∂ Ω \partial\Omega ∂ Ω C 1 C^1 C 1 pinched torus , which is given by the rotation around the z z z
Proof. Let γ ∈ C 1 ( [ 0 , 1 ] ) \gamma\in C^1([0,1]) γ ∈ C 1 ([ 0 , 1 ]) Ω \Omega Ω γ \gamma γ z z z γ \gamma γ S 1 S^1 S 1 ∂ Ω \partial\Omega ∂ Ω S 1 × S 1 S^1\times S^1 S 1 × S 1
If γ \gamma γ z z z ∂ Ω \partial\Omega ∂ Ω ∂ Ω \partial\Omega ∂ Ω [ 0 , 1 ] × S 1 [0,1]\times S^1 [ 0 , 1 ] × S 1 { 0 } × S 1 \{0\}\times S^1 { 0 } × S 1 { 1 } × S 1 \{1\}\times S^1 { 1 } × S 1 S 2 S^2 S 2
Corollary 3.5. Let Ω ⊂ R 3 \Omega\subset\mathbb{R}^{3} Ω ⊂ R 3 C 1 C^{1} C 1
if ∂ Ω \partial\Omega ∂ Ω C ( ∂ Ω ) C(\partial\Omega) C ( ∂ Ω ) z z z
if ∂ Ω \partial\Omega ∂ Ω C ( ∂ Ω ) = ∅ C(\partial\Omega)=\varnothing C ( ∂ Ω ) = ∅
Proof. By Proposition 3.3, the statement follows once we prove that
C ( ∂ Ω ) = ∂ Ω ∩ { ( x , y , z ) ∈ R 3 ∣ x = 0 , y = 0 } . C(\partial\Omega)=\partial\Omega\cap\{(x,y,z)\in\mathbb{R}^{3}\mid x=0,\ y=0\}. C ( ∂ Ω ) = ∂ Ω ∩ {( x , y , z ) ∈ R 3 ∣ x = 0 , y = 0 } . In fact, if p = ( x , y , z ) ∈ ∂ Ω p=(x,y,z)\in\partial\Omega p = ( x , y , z ) ∈ ∂ Ω D ⊂ T p ∂ Ω D\subset T_{p}\partial\Omega D ⊂ T p ∂ Ω p p p ∂ Ω \partial\Omega ∂ Ω z = f ( x , y ) z=f(x,y) z = f ( x , y ) Ω \Omega Ω f f f r = r ( x , y ) r=r(x,y) r = r ( x , y ) ∣ ∇ ( z − f ( r ( x , y ) ) ) ∣ 2 = f ′ ( r ) 2 + r 2 4 = 0 \lvert\nabla(z-f(r(x,y)))\rvert^{2}=f'(r)^{2}+\frac{r^{2}}{4}=0 ∣ ∇ ( z − f ( r ( x , y ))) ∣ 2 = f ′ ( r ) 2 + 4 r 2 = 0 r = 0 r=0 r = 0 f ′ ( 0 ) = 0 f'(0)=0 f ′ ( 0 ) = 0
Remark 3.6 (Curvatures of axisymmetric z z z f ∈ C 2 ( I , R ) f\in C^{2}(I,\mathbb{R}) f ∈ C 2 ( I , R ) I ⊂ [ 0 , ∞ ) I\subset[0,\infty) I ⊂ [ 0 , ∞ ) ∂ Ω \partial\Omega ∂ Ω
u = z − f ∘ r ∈ C 2 ( R 3 ) , u=z-f\circ r\in C^{2}(\mathbb{R}^{3}), u = z − f ∘ r ∈ C 2 ( R 3 ) , then a direct computation shows that the horizontal mean curvature and the g ε g_{\varepsilon} g ε ∂ Ω \partial\Omega ∂ Ω H = φ ∘ r H=\varphi\circ r H = φ ∘ r H ε = φ ε ∘ r H_{\varepsilon}=\varphi_{\varepsilon}\circ r H ε = φ ε ∘ r
φ ( r ) = − σ r ∂ r ( r f ′ ( r ) f ′ ( r ) 2 + r 2 4 ) . (24) \varphi(r)=-\frac{\sigma}{r}\partial_{r}\left(\frac{rf'(r)}{\sqrt{f'(r)^{2}+\frac{r^{2}}{4}}}\right). \tag*{(24)} φ ( r ) = − r σ ∂ r f ′ ( r ) 2 + 4 r 2 r f ′ ( r ) . (24) φ ε ( r ) = − σ r ∂ r ( r f ′ ( r ) f ′ ( r ) 2 + r 2 4 + ε 2 ) . (25) \varphi_{\varepsilon}(r)=-\frac{\sigma}{r}\partial_{r}\left(\frac{rf'(r)}{\sqrt{f'(r)^{2}+\frac{r^{2}}{4}+\varepsilon^{2}}}\right). \tag*{(25)} φ ε ( r ) = − r σ ∂ r f ′ ( r ) 2 + 4 r 2 + ε 2 r f ′ ( r ) . (25) Here σ = 1 \sigma=1 σ = 1 Ω = { u < 0 } \Omega=\{u<0\} Ω = { u < 0 } σ = − 1 \sigma=-1 σ = − 1 Ω = { u > 0 } \Omega=\{u>0\} Ω = { u > 0 } φ \varphi φ r = 0 r=0 r = 0 φ ε \varphi_{\varepsilon} φ ε r = 0 r=0 r = 0 r → 0 + r\to0^{+} r → 0 +
Approximation by prescribed mean curvature.
Throughout this section, let Ω ⊂ R 3 \Omega\subset\mathbb{R}^{3} Ω ⊂ R 3 C 2 C^{2} C 2 H ( ∂ Ω ) H(\partial\Omega) H ( ∂ Ω ) C ( ∂ Ω ) C(\partial\Omega) C ( ∂ Ω ) φ \varphi φ ( 0 , R ] (0,R] ( 0 , R ] H = φ ∘ r H=\varphi\circ r H = φ ∘ r ∂ Ω ∖ C ( ∂ Ω ) \partial\Omega\setminus C(\partial\Omega) ∂ Ω ∖ C ( ∂ Ω ) H H H C ( ∂ Ω ) C(\partial\Omega) C ( ∂ Ω ) φ \varphi φ [ 0 , R ] [0,R] [ 0 , R ]
H = φ ∘ r on ∂ Ω . (26) H=\varphi\circ r\quad\text{on }\partial\Omega. \tag*{(26)} H = φ ∘ r on ∂ Ω. (26) The arguments of this section do not require any mean convexity assumption of Ω \Omega Ω
The case of a finite vertical set of the generating profile.
We first carry out the construction under the assumption that the generating profile has finitely many vertical points.
Definition 3.7. Given a C 1 C^{1} C 1 γ = ( γ 1 , γ 2 ) : [ 0 , 1 ] → R + 2 \gamma=(\gamma_{1},\gamma_{2}):[0,1]\to\mathbb{R}_{+}^{2} γ = ( γ 1 , γ 2 ) : [ 0 , 1 ] → R + 2 V \mathcal{V} V
V ( γ ) = { τ ∈ [ 0 , 1 ] ∣ γ 1 ′ ( τ ) = 0 } . \mathcal{V}(\gamma)=\{\tau\in[0,1]\mid\gamma_{1}'(\tau)=0\}. V ( γ ) = { τ ∈ [ 0 , 1 ] ∣ γ 1 ′ ( τ ) = 0 } . Accordingly, the vertical part of the boundary of the C 1 C^{1} C 1 Ω \Omega Ω γ \gamma γ
V ( ∂ Ω ) = { Φ γ ( τ , θ ) ∣ τ ∈ V ( γ ) , θ ∈ S 1 } , \mathcal{V}(\partial\Omega)=\{\Phi_{\gamma}(\tau,\theta)\mid\tau\in\mathcal{V}(\gamma),\ \theta\in\mathbb{S}^{1}\}, V ( ∂ Ω ) = { Φ γ ( τ , θ ) ∣ τ ∈ V ( γ ) , θ ∈ S 1 } , where Φ γ \Phi_{\gamma} Φ γ
Remark 3.8. The vertical part V ( ∂ Ω ) \mathcal{V}(\partial\Omega) V ( ∂ Ω ) γ 1 ′ \gamma_1' γ 1 ′ V ( ∂ Ω ) \mathcal{V}(\partial\Omega) V ( ∂ Ω ) ∂ Ω \partial\Omega ∂ Ω
The construction of { Ω ε } ε > 0 \{\Omega_\varepsilon\}_{\varepsilon>0} { Ω ε } ε > 0 ∂ Ω \partial\Omega ∂ Ω Figure 1 . Each resulting portion is a radial graph, whose horizontal mean curvature is given by (24) , with prescribed curvature φ \varphi φ (26) and a sign determined by the local graphical parametrization. These pieces will serve as local building blocks for the approximating boundaries, obtained by solving the corresponding prescribed g ε g_\varepsilon g ε (25) with φ ε = φ \varphi_\varepsilon=\varphi φ ε = φ
Figure 1. Decomposition of the generating profile into the graphical pieces Σ j \Sigma_j Σ j
Lemma 3.9 (Radial decomposition of the boundary). Let Ω \Omega Ω C 2 C^2 C 2 H ( ∂ Ω ) H(\partial\Omega) H ( ∂ Ω ) C ( ∂ Ω ) C(\partial\Omega) C ( ∂ Ω ) γ ∈ C 2 ( [ 0 , 1 ] , R + 2 ) \gamma\in C^2([0,1],\mathbb{R}_+^2) γ ∈ C 2 ([ 0 , 1 ] , R + 2 ) ∂ Ω \partial\Omega ∂ Ω # V ( γ ) < ∞ \#\mathcal{V}(\gamma)<\infty # V ( γ ) < ∞
γ 1 ( 0 ) = γ 1 ( 1 ) = 0 , γ 1 ( τ ) > 0 for every τ ∈ ( 0 , 1 ) , γ 2 ( 0 ) > γ 2 ( 1 ) . (27) \gamma_1(0)=\gamma_1(1)=0,\qquad\gamma_1(\tau)>0\quad\text{for every }\tau\in(0,1),\qquad\gamma_2(0)>\gamma_2(1). \tag*{(27)} γ 1 ( 0 ) = γ 1 ( 1 ) = 0 , γ 1 ( τ ) > 0 for every τ ∈ ( 0 , 1 ) , γ 2 ( 0 ) > γ 2 ( 1 ) . (27) Let 0 = α 0 < α 1 < ⋯ < α N = 1 0=\alpha_0<\alpha_1<\cdots<\alpha_N=1 0 = α 0 < α 1 < ⋯ < α N = 1 V ( γ ) ∪ { 0 , 1 } \mathcal{V}(\gamma)\cup\{0,1\} V ( γ ) ∪ { 0 , 1 }
I ~ j = ( α j − 1 , α j ) , a j = γ 1 ( α j ) , j = 0 , … , N . \widetilde I_j=(\alpha_{j-1},\alpha_j),\qquad a_j=\gamma_1(\alpha_j),\qquad j=0,\ldots,N. I j = ( α j − 1 , α j ) , a j = γ 1 ( α j ) , j = 0 , … , N . Then the following properties hold.
(i) Height functions. For every j = 1 , … , N j=1,\ldots,N j = 1 , … , N
γ 1 ∣ I ~ j : I ~ j ⟶ I j \gamma_1|_{\widetilde I_j}:\widetilde I_j\longrightarrow I_j γ 1 ∣ I j : I j ⟶ I j is a C 2 C^2 C 2
I j = γ 1 ( I ~ j ) = ( min { a j − 1 , a j } , max { a j − 1 , a j } ) . I_j=\gamma_1(\widetilde I_j)=\left(\min\{a_{j-1},a_j\},\max\{a_{j-1},a_j\}\right). I j = γ 1 ( I j ) = ( min { a j − 1 , a j } , max { a j − 1 , a j } ) . Letting τ j = ( γ 1 ∣ I ~ j ) − 1 \tau_j=(\gamma_1|_{\widetilde I_j})^{-1} τ j = ( γ 1 ∣ I j ) − 1
f j ( r ) = γ 2 ( τ j ( r ) ) , r ∈ I j (28) f_j(r)=\gamma_2(\tau_j(r)),\qquad r\in I_j \tag*{(28)} f j ( r ) = γ 2 ( τ j ( r )) , r ∈ I j (28) are C 2 ( I j ) C^2(I_j) C 2 ( I j )
Σ j = Φ γ ( I ~ j × S 1 ) = { ( x , y , z ) ∈ R 3 : r ( x , y ) ∈ I j , z = f j ( r ( x , y ) ) } \Sigma_j = \Phi_\gamma(\widetilde I_j \times S^1) = \{(x,y,z) \in\mathbb{R}^3 : r(x,y) \in I_j,\ z = f_j(r(x,y))\} Σ j = Φ γ ( I j × S 1 ) = {( x , y , z ) ∈ R 3 : r ( x , y ) ∈ I j , z = f j ( r ( x , y ))} are pairwise disjoint radial graphs satisfying
∂ Ω = ⋃ j = 1 N Σ j ‾ . \partial\Omega= \bigcup_{j=1}^{N} \overline{\Sigma_j}. ∂ Ω = j = 1 ⋃ N Σ j . Moreover, consecutive pieces meet along the corresponding vertical circle:
Σ j ‾ ∩ Σ j + 1 ‾ = Φ γ ( { a j } × S 1 ) , j = 1 , … , N − 1. \overline{\Sigma_j} \cap\overline{\Sigma_{j+1}} = \Phi_\gamma(\{a_j\} \times S^1), \qquad j=1,\ldots,N-1. Σ j ∩ Σ j + 1 = Φ γ ({ a j } × S 1 ) , j = 1 , … , N − 1. (ii) Poles and junctions. At the poles,
f 1 ∈ C 2 ( [ 0 , a 1 ) ) , f N ∈ C 2 ( [ 0 , a N − 1 ) ) , f_1 \in C^2([0,a_1)), \qquad f_N \in C^2([0,a_{N-1})), f 1 ∈ C 2 ([ 0 , a 1 )) , f N ∈ C 2 ([ 0 , a N − 1 )) , and
lim r → 0 + f 1 ′ ( r ) = lim r → 0 + f N ′ ( r ) = 0. \lim_{r\to0^+} f_1'(r) = \lim_{r\to0^+} f_N'(r) = 0. r → 0 + lim f 1 ′ ( r ) = r → 0 + lim f N ′ ( r ) = 0. For j = 1 , … , N j=1,\ldots,N j = 1 , … , N
σ j = sgn ( γ ′ ∣ I ~ j ) ∈ { − 1 , 1 } . \sigma_j = \operatorname{sgn}(\gamma'|_{\widetilde I_j}) \in\{-1,1\}. σ j = sgn ( γ ′ ∣ I j ) ∈ { − 1 , 1 } . At every junction point a j a_j a j j = 1 , … , N − 1 j=1,\ldots,N-1 j = 1 , … , N − 1
lim r → a j − σ j ∣ f j ′ ( r ) ∣ = lim r → a j σ j + 1 ∣ f j + 1 ′ ( r ) ∣ = + ∞ . (29) \lim_{r\to a_j^{-\sigma_j}} |f_j'(r)| = \lim_{r\to a_j^{\sigma_{j+1}}} |f_{j+1}'(r)| = +\infty. \tag*{(29)} r → a j − σ j lim ∣ f j ′ ( r ) ∣ = r → a j σ j + 1 lim ∣ f j + 1 ′ ( r ) ∣ = + ∞. (29) where r → a σ r\to a^\sigma r → a σ a a a σ \sigma σ a ∈ ∂ I j a\in\partial I_j a ∈ ∂ I j ρ > 0 \rho>0 ρ > 0 δ j ( a ) ∈ { − 1 , 1 } \delta_j(a)\in\{-1,1\} δ j ( a ) ∈ { − 1 , 1 }
δ j ( a ) f j ′ ( r ) > 0 for every r ∈ I j ∩ ( a − ρ , a + ρ ) . (30) \delta_j(a)f_j'(r)>0 \qquad\text{for every } r\in I_j\cap(a-\rho,a+\rho). \tag*{(30)} δ j ( a ) f j ′ ( r ) > 0 for every r ∈ I j ∩ ( a − ρ , a + ρ ) . (30) At a common endpoint a j a_j a j
σ j δ j ( a j ) = σ j + 1 δ j + 1 ( a j ) , j = 1 , … , N − 1. (31) \sigma_j\delta_j(a_j)=\sigma_{j+1}\delta_{j+1}(a_j), \qquad j=1,\ldots,N-1. \tag*{(31)} σ j δ j ( a j ) = σ j + 1 δ j + 1 ( a j ) , j = 1 , … , N − 1. (31) (iii) Equators. Let
R = max τ ∈ [ 0 , 1 ] γ 1 ( τ ) R=\max_{\tau\in[0,1]}\gamma_1(\tau) R = τ ∈ [ 0 , 1 ] max γ 1 ( τ ) and choose τ R ∈ ( 0 , 1 ) \tau_R\in(0,1) τ R ∈ ( 0 , 1 ) γ 1 ( τ R ) = R \gamma_1(\tau_R)=R γ 1 ( τ R ) = R τ R ∈ V ( γ ) \tau_R\in\mathcal{V}(\gamma) τ R ∈ V ( γ ) j R ∈ { 1 , … , N − 1 } j_R\in\{1,\ldots,N-1\} j R ∈ { 1 , … , N − 1 } τ R = α j R \tau_R=\alpha_{j_R} τ R = α j R
a j R = R , σ j R = 1 , σ j R + 1 = − 1 , a_{j_R}=R, \qquad\sigma_{j_R}=1, \qquad\sigma_{j_R+1}=-1, a j R = R , σ j R = 1 , σ j R + 1 = − 1 , and
lim r → R − f j R ′ ( r ) = − lim r → R − f j R + 1 ′ ( r ) = − ∞ . (32) \lim_{r\to R^-} f_{j_R}'(r)=-\lim_{r\to R^-} f_{j_R+1}'(r)=-\infty. \tag*{(32)} r → R − lim f j R ′ ( r ) = − r → R − lim f j R + 1 ′ ( r ) = − ∞. (32) (iv) Curvatures. For j = 1 , … , N j=1,\ldots,N j = 1 , … , N
φ j ( r ) = H ( r , 0 , f j ( r ) ) , r ∈ I j . \varphi_j(r)=H(r,0,f_j(r)), \qquad r\in I_j. φ j ( r ) = H ( r , 0 , f j ( r )) , r ∈ I j . Then φ j \varphi_j φ j I j ‾ \overline{I_j} I j
H ( p ) = φ j ( r ( p ) ) , p ∈ Σ j . H(p)=\varphi_j(r(p)), \qquad p\in\Sigma_j. H ( p ) = φ j ( r ( p )) , p ∈ Σ j . On I j I_j I j φ j \varphi_j φ j σ = σ j \sigma=\sigma_j σ = σ j H H H
lim r → a j − σ j φ j ( r ) = lim r → a j σ j + 1 φ j + 1 ( r ) , j = 1 , … , N − 1. (33) \lim_{r\to a_j^{-\sigma_j}}\varphi_j(r)=\lim_{r\to a_j^{\sigma_{j+1}}}\varphi_{j+1}(r), \qquad j=1,\ldots,N-1. \tag*{(33)} r → a j − σ j lim φ j ( r ) = r → a j σ j + 1 lim φ j + 1 ( r ) , j = 1 , … , N − 1. (33) Proof. Most of the assertions follow directly from the construction. We provide details only where needed. Up to reparametrization, condition (3.5) follows from the fact that ∂ Ω \partial\Omega ∂ Ω
(i) For every j = 1 , … , N j=1,\ldots,N j = 1 , … , N γ 1 ′ \gamma'_1 γ 1 ′ I ~ j \widetilde{I}_j I j γ 1 ∣ I ~ j \gamma_1|_{\widetilde{I}_j} γ 1 ∣ I j C 2 C^2 C 2 I j I_j I j ∂ Ω \partial\Omega ∂ Ω
(ii) The assertions at the poles follow from the C 2 C^2 C 2 ∂ Ω \partial\Omega ∂ Ω a 0 = a N = 0 a_0=a_N=0 a 0 = a N = 0
By (3.6), for every j = 1 , … , N j=1,\ldots,N j = 1 , … , N
f j ′ ( r ) = γ 2 ′ ( τ j ( r ) ) γ 1 ′ ( τ j ( r ) ) , r ∈ I j . (34) f'_j(r)=\frac{\gamma'_2(\tau_j(r))}{\gamma'_1(\tau_j(r))},\qquad r\in I_j. \tag*{(34)} f j ′ ( r ) = γ 1 ′ ( τ j ( r )) γ 2 ′ ( τ j ( r )) , r ∈ I j . (34) Since γ 1 ′ = 0 \gamma'_1=0 γ 1 ′ = 0 V ( γ ) \mathcal{V}(\gamma) V ( γ ) γ \gamma γ γ 2 ′ ≠ 0 \gamma'_2\ne0 γ 2 ′ = 0 f j ′ f'_j f j ′ I j I_j I j f j ′ f'_j f j ′ α j \alpha_j α j
σ j δ j ( a j ) = sgn γ 2 ′ ( α j ) = σ j + 1 δ j + 1 ( a j ) , \sigma_j\delta_j(a_j)=\operatorname{sgn}\gamma'_2(\alpha_j)=\sigma_{j+1}\delta_{j+1}(a_j), σ j δ j ( a j ) = sgn γ 2 ′ ( α j ) = σ j + 1 δ j + 1 ( a j ) , which proves (3.9).
(iii) Since γ 1 ( 0 ) = γ 1 ( 1 ) = 0 < R \gamma_1(0)=\gamma_1(1)=0<R γ 1 ( 0 ) = γ 1 ( 1 ) = 0 < R τ R ∈ ( 0 , 1 ) \tau_R\in(0,1) τ R ∈ ( 0 , 1 ) τ R \tau_R τ R γ 1 \gamma_1 γ 1 V ( γ ) \mathcal{V}(\gamma) V ( γ )
σ j R = 1 , σ j R + 1 = − 1. \sigma_{j_R}=1,\qquad\sigma_{j_R+1}=-1. σ j R = 1 , σ j R + 1 = − 1. Moreover, regularity gives γ 2 ′ ( α j R ) ≠ 0 \gamma'_2(\alpha_{j_R})\ne0 γ 2 ′ ( α j R ) = 0 γ 2 ′ ( α j R ) < 0 \gamma'_2(\alpha_{j_R})<0 γ 2 ′ ( α j R ) < 0 τ j R ( r ) → α j R \tau_{j_R}(r)\to\alpha_{j_R} τ j R ( r ) → α j R r → R − r\to R^- r → R −
lim r → R − f j R ′ ( r ) = lim r → R − γ 2 ′ ( τ j R ( r ) ) γ 1 ′ ( τ j R ( r ) ) = − ∞ . \lim_{r\to R^-}f'_{j_R}(r)=\lim_{r\to R^-}\frac{\gamma'_2(\tau_{j_R}(r))}{\gamma'_1(\tau_{j_R}(r))}=-\infty. r → R − lim f j R ′ ( r ) = r → R − lim γ 1 ′ ( τ j R ( r )) γ 2 ′ ( τ j R ( r )) = − ∞. Arguing in the same way for f j R + 1 f_{j_R+1} f j R + 1
(iv) The representation of H H H Σ j \Sigma_j Σ j φ j \varphi_j φ j I ‾ j \overline{I}_j I j H H H C ( ∂ Ω ) C(\partial\Omega) C ( ∂ Ω )
The next proposition contains the solution to the prescribed g ε g_\varepsilon g ε Ω ε \Omega_\varepsilon Ω ε
Proposition 3.10 (Prescribing g ε g_\varepsilon g ε Ω \Omega Ω C 2 C^2 C 2 H ( ∂ Ω ) H(\partial\Omega) H ( ∂ Ω ) C ( ∂ Ω ) C(\partial\Omega) C ( ∂ Ω ) γ ∈ C 2 ( [ 0 , 1 ] , R + 2 ) \gamma\in C^2([0,1],\mathbb{R}^2_+) γ ∈ C 2 ([ 0 , 1 ] , R + 2 ) ∂ Ω \partial\Omega ∂ Ω # V ( γ ) < ∞ \#\mathcal{V}(\gamma)<\infty # V ( γ ) < ∞ Σ 1 , … , Σ N \Sigma_1,\ldots,\Sigma_N Σ 1 , … , Σ N ε 0 > 0 \varepsilon_0>0 ε 0 > 0 ε ∈ ( 0 , ε 0 ) \varepsilon\in(0,\varepsilon_0) ε ∈ ( 0 , ε 0 ) C 2 C^2 C 2 Ω ε ⊂ R 3 \Omega_\varepsilon\subset\mathbb{R}^3 Ω ε ⊂ R 3 Σ ~ 1 , … , Σ ~ N \widetilde{\Sigma}_1,\ldots,\widetilde{\Sigma}_N Σ 1 , … , Σ N g ε g_\varepsilon g ε j = 1 , … , N j=1,\ldots,N j = 1 , … , N
H ε ( p ) = H ( p ′ ) for p ∈ Σ ~ j ∖ { r = 0 } , p ′ ∈ Σ j ∖ { r = 0 } such that r ( p ) = r ( p ′ ) . (35) H_\varepsilon(p)=H(p')
\qquad\text{for }p\in\widetilde{\Sigma}_j\setminus\{r=0\},\quad p'\in\Sigma_j\setminus\{r=0\}\text{ such that }r(p)=r(p'). \tag*{(35)} H ε ( p ) = H ( p ′ ) for p ∈ Σ j ∖ { r = 0 } , p ′ ∈ Σ j ∖ { r = 0 } such that r ( p ) = r ( p ′ ) . (35) In particular, H ε ∈ C 0 ( ∂ Ω ε ) H_\varepsilon\in C^0(\partial\Omega_\varepsilon) H ε ∈ C 0 ( ∂ Ω ε )
Moreover, as ε → 0 \varepsilon\to0 ε → 0 ∂ Ω ε \partial\Omega_\varepsilon ∂ Ω ε ∂ Ω \partial\Omega ∂ Ω d H d_H d H
We first prove a few auxiliary lemmas containing information on f f f φ \varphi φ (35) .
Lemma 3.11. Let σ ∈ { − 1 , 1 } \sigma\in\{-1,1\} σ ∈ { − 1 , 1 } 0 ≤ a < b 0 \leq a < b 0 ≤ a < b φ ∈ C ( [ a , b ] ) \varphi\in C([a,b]) φ ∈ C ([ a , b ]) f ∈ C ( [ a , b ] ) f \in C([a,b]) f ∈ C ([ a , b ])
a) f ∈ C 2 ( [ 0 , b ) ) f \in C^{2}([0,b)) f ∈ C 2 ([ 0 , b )) a = 0 a=0 a = 0 f ∈ C 2 ( ( a , b ) ) f \in C^{2}((a,b)) f ∈ C 2 (( a , b )) a > 0 a>0 a > 0
b) for r ∈ ( a , b ) r \in(a,b) r ∈ ( a , b )
− σ r d d r ( r f ′ ( r ) f ′ ( r ) 2 + r 2 4 ) = φ ( r ) ; (36) -\frac{\sigma}{r}\frac{\mathrm{d}}{\mathrm{d}r}\left(\frac{rf'(r)}{\sqrt{f'(r)^{2}+\frac{r^{2}}{4}}}\right)=\varphi(r); \tag*{(36)} − r σ d r d f ′ ( r ) 2 + 4 r 2 r f ′ ( r ) = φ ( r ) ; (36) c) there exist δ a , δ b ∈ { − 1 , 1 } \delta_a,\delta_b \in\{-1,1\} δ a , δ b ∈ { − 1 , 1 }
{ lim r → 0 + f ′ ( r ) = 0 , if a = 0 , lim r → a + δ a f ′ ( r ) = + ∞ , if a > 0 , lim r → b − δ b f ′ ( r ) = + ∞ . \begin{cases}
\lim_{r\to0^{+}}f'(r)=0, & \text{if }a=0,\\
\lim_{r\to a^{+}}\delta_a f'(r)=+\infty, & \text{if }a>0,
\end{cases}
\qquad
\lim_{r\to b^{-}}\delta_b f'(r)=+\infty. { lim r → 0 + f ′ ( r ) = 0 , lim r → a + δ a f ′ ( r ) = + ∞ , if a = 0 , if a > 0 , r → b − lim δ b f ′ ( r ) = + ∞. Then the function ψ : [ a , b ] → R \psi:[a,b]\to\mathbb{R} ψ : [ a , b ] → R
ψ ( r ) = ∫ a r t φ ( t ) d t , a ≤ r ≤ b , (37) \psi(r)=\int_{a}^{r}t\varphi(t)\,\mathrm{d}t,\qquad a\leq r\leq b, \tag*{(37)} ψ ( r ) = ∫ a r tφ ( t ) d t , a ≤ r ≤ b , (37) satisfies the following:
i. ψ ∈ C 1 ( ( a , b ) ) ∩ C 0 ( [ a , b ] ) \psi\in C^{1}((a,b))\cap C^{0}([a,b]) ψ ∈ C 1 (( a , b )) ∩ C 0 ([ a , b ])
ii. r 2 − ( ψ ( r ) − a δ a σ ) 2 ≥ 0 r^{2}-(\psi(r)-a\delta_a\sigma)^{2}\geq0 r 2 − ( ψ ( r ) − a δ a σ ) 2 ≥ 0 [ a , b ] [a,b] [ a , b ]
r 2 − ( ψ ( r ) − a δ a σ ) 2 = 0 ⟺ r ∈ { a , b } ; (38) r^{2}-(\psi(r)-a\delta_a\sigma)^{2}=0\Longleftrightarrow r\in\{a,b\}; \tag*{(38)} r 2 − ( ψ ( r ) − a δ a σ ) 2 = 0 ⟺ r ∈ { a , b } ; (38) iii. Let Ψ : ( a , b ) → R \Psi:(a,b)\to\mathbb{R} Ψ : ( a , b ) → R
Ψ ( r ) = ∣ ψ ( r ) − a δ a σ ∣ r 2 r 2 − ( ψ ( r ) − a δ a σ ) 2 . (39) \Psi(r)=\frac{\left|\psi(r)-a\delta_a\sigma\right|\frac{r}{2}}{\sqrt{r^{2}-(\psi(r)-a\delta_a\sigma)^{2}}}. \tag*{(39)} Ψ ( r ) = r 2 − ( ψ ( r ) − a δ a σ ) 2 ∣ ψ ( r ) − a δ a σ ∣ 2 r . (39) Then Ψ ∈ L 1 ( [ a , b ] ) \Psi\in L^{1}([a,b]) Ψ ∈ L 1 ([ a , b ]) r ∈ ( a , b ) r\in(a,b) r ∈ ( a , b )
f ′ ( r ) = δ ( r ) Ψ ( r ) . (40) f'(r)=\delta(r)\Psi(r). \tag*{(40)} f ′ ( r ) = δ ( r ) Ψ ( r ) . (40) where δ : [ a , b ] → R \delta:[a,b]\to\mathbb{R} δ : [ a , b ] → R
δ ( r ) = { sgn ( f ′ ( r ) ) , if f ′ ( r ) ≠ 0 , 0 , if f ′ ( r ) = 0. (41) \delta(r)=
\begin{cases}
\operatorname{sgn}(f'(r)), & \text{if }f'(r)\ne0,\\
0, & \text{if }f'(r)=0.
\end{cases} \tag*{(41)} δ ( r ) = { sgn ( f ′ ( r )) , 0 , if f ′ ( r ) = 0 , if f ′ ( r ) = 0. (41) Proof. Since φ \varphi φ ψ ∈ C 1 ( ( a , b ) ) ∩ C 0 ( [ a , b ] ) \psi\in C^{1}((a,b))\cap C^{0}([a,b]) ψ ∈ C 1 (( a , b )) ∩ C 0 ([ a , b ])
We prove ii). Integrating (36) on ( a , r ) (a,r) ( a , r ) r ∈ ( a , b ) r\in(a,b) r ∈ ( a , b )
ψ ( r ) = − σ r f ′ ( r ) f ′ ( r ) 2 + r 2 4 + σ C a , a < r < b , \psi(r)=-\sigma\frac{rf'(r)}{\sqrt{f'(r)^{2}+\frac{r^{2}}{4}}}+\sigma C_a,\qquad a<r<b, ψ ( r ) = − σ f ′ ( r ) 2 + 4 r 2 r f ′ ( r ) + σ C a , a < r < b , where
C a = lim r → a + r f ′ ( r ) f ′ ( r ) 2 + r 2 4 . C_a=\lim_{r\to a^{+}}\frac{rf'(r)}{\sqrt{f'(r)^{2}+\frac{r^{2}}{4}}}. C a = r → a + lim f ′ ( r ) 2 + 4 r 2 r f ′ ( r ) . If a = 0 a=0 a = 0
C 0 = lim r → 0 + r ( f ′ ′ ( 0 ) r + o ( r ) ) ( f ′ ′ ( 0 ) r + o ( r ) ) 2 + r 2 4 = lim r → 0 + f ′ ′ ( 0 ) r 2 + o ( r 2 ) r f ′ ′ ( 0 ) 2 + 1 4 + o ( 1 ) = 0. C_0=\lim_{r\to0^{+}}\frac{r(f''(0)r+o(r))}{\sqrt{(f''(0)r+o(r))^{2}+\frac{r^{2}}{4}}}
=\lim_{r\to0^{+}}\frac{f''(0)r^{2}+o(r^{2})}{r\sqrt{f''(0)^{2}+\frac{1}{4}+o(1)}}=0. C 0 = r → 0 + lim ( f ′′ ( 0 ) r + o ( r ) ) 2 + 4 r 2 r ( f ′′ ( 0 ) r + o ( r )) = r → 0 + lim r f ′′ ( 0 ) 2 + 4 1 + o ( 1 ) f ′′ ( 0 ) r 2 + o ( r 2 ) = 0. If a > 0 a>0 a > 0
C a = lim r → a + r δ a 1 + r 2 4 f ′ ( r ) 2 = a δ a . (42) C_a=\lim_{r\to a^{+}}\frac{r\delta_a}{\sqrt{1+\frac{r^{2}}{4f'(r)^{2}}}}=a\delta_a. \tag*{(42)} C a = r → a + lim 1 + 4 f ′ ( r ) 2 r 2 r δ a = a δ a . (42) We therefore obtain
ψ ( r ) = − σ r f ′ ( r ) f ′ ( r ) 2 + r 2 4 + σ a δ a , r ∈ ( a , b ) , \psi(r)=-\sigma\frac{rf'(r)}{\sqrt{f'(r)^2+\frac{r^2}{4}}}+\sigma a\delta_a,\qquad r\in(a,b), ψ ( r ) = − σ f ′ ( r ) 2 + 4 r 2 r f ′ ( r ) + σ a δ a , r ∈ ( a , b ) , that can be rewritten as
f ′ 2 ( r ) [ r 2 − ( ψ ( r ) − a δ a σ ) 2 ] = ( ψ ( r ) − a δ a σ ) 2 r 2 4 , r ∈ ( a , b ) . (43) f'^{2}(r)\left[r^2-\left(\psi(r)-a\delta_a\sigma\right)^2\right]=\left(\psi(r)-a\delta_a\sigma\right)^2\frac{r^2}{4},\qquad r\in(a,b). \tag*{(43)} f ′ 2 ( r ) [ r 2 − ( ψ ( r ) − a δ a σ ) 2 ] = ( ψ ( r ) − a δ a σ ) 2 4 r 2 , r ∈ ( a , b ) . (43) Since the right-hand side is nonnegative we have that
r 2 − ( ψ ( r ) − a δ a σ ) 2 ≥ 0 for every r ∈ ( a , b ) . r^2-\left(\psi(r)-a\delta_a\sigma\right)^2\geq0\qquad\text{for every }r\in(a,b). r 2 − ( ψ ( r ) − a δ a σ ) 2 ≥ 0 for every r ∈ ( a , b ) . We show that equality holds at the endpoints. At r = a r=a r = a ψ \psi ψ r → b − r\to b^- r → b −
lim r → b − [ r 2 − ( ψ ( r ) − a δ a σ ) 2 ] = lim r → b − ( ψ ( r ) − a δ a σ ) 2 r 2 4 f ′ 2 ( r ) = 0 , \lim_{r\to b^-}\left[r^2-\left(\psi(r)-a\delta_a\sigma\right)^2\right]=\lim_{r\to b^-}\frac{\left(\psi(r)-a\delta_a\sigma\right)^2r^2}{4f'^{2}(r)}=0, r → b − lim [ r 2 − ( ψ ( r ) − a δ a σ ) 2 ] = r → b − lim 4 f ′ 2 ( r ) ( ψ ( r ) − a δ a σ ) 2 r 2 = 0 , because ( ψ ( r ) − a δ a σ ) 2 r 2 \left(\psi(r)-a\delta_a\sigma\right)^2r^2 ( ψ ( r ) − a δ a σ ) 2 r 2 [ a , b ] [a,b] [ a , b ]
On the other hand, if r ∈ ( a , b ) r\in(a,b) r ∈ ( a , b )
r 2 = ( ψ ( r ) − a δ a σ ) 2 (44) r^2=\left(\psi(r)-a\delta_a\sigma\right)^2 \tag*{(44)} r 2 = ( ψ ( r ) − a δ a σ ) 2 (44) equation (3.20) yields r 2 4 ( ψ ( r ) − a δ a σ ) 2 = 0 \frac{r^2}{4}\left(\psi(r)-a\delta_a\sigma\right)^2=0 4 r 2 ( ψ ( r ) − a δ a σ ) 2 = 0 (44) in the latter we thus obtain r = 0 r=0 r = 0
We conclude by proving iii). Observe that Ψ \Psi Ψ ( a , b ) (a,b) ( a , b ) (38) . By (3.20) and ii) we therefore deduce that ∣ f ′ ( r ) ∣ = Ψ ( r ) |f'(r)|=\Psi(r) ∣ f ′ ( r ) ∣ = Ψ ( r ) (40) . We deduce that for any c , d ∈ ( a , b ) c,d\in(a,b) c , d ∈ ( a , b ) c < d c<d c < d Ψ ∈ L 1 ( [ c , d ] ) \Psi\in L^1([c,d]) Ψ ∈ L 1 ([ c , d ]) f ′ ∈ C 0 ( [ c , d ] ) f'\in C^0([c,d]) f ′ ∈ C 0 ([ c , d ])
If a = 0 a=0 a = 0 f ′ ∈ C 0 ( [ 0 , b ) ) f'\in C^0([0,b)) f ′ ∈ C 0 ([ 0 , b )) Ψ ∈ L 1 ( [ 0 , d ] ) \Psi\in L^1([0,d]) Ψ ∈ L 1 ([ 0 , d ]) d ∈ ( 0 , b ) d\in(0,b) d ∈ ( 0 , b )
If a > 0 a>0 a > 0 r a > a r_a>a r a > a r ∈ ( a , r a ) r\in(a,r_a) r ∈ ( a , r a ) ∣ f ′ ( r ) ∣ = δ a f ′ ( r ) |f'(r)|=\delta_a f'(r) ∣ f ′ ( r ) ∣ = δ a f ′ ( r ) r ∈ ( a , r a ) r\in(a,r_a) r ∈ ( a , r a )
lim η → 0 + ∫ a + η r ∣ Ψ ( s ) ∣ d s = lim η → 0 + ∫ a + η r δ a f ′ ( s ) d s = lim η → 0 + δ a ( f ( r ) − f ( a + η ) ) ∈ R , \lim_{\eta\to0^+}\int_{a+\eta}^{r}|\Psi(s)|\,ds=\lim_{\eta\to0^+}\int_{a+\eta}^{r}\delta_a f'(s)\,ds=\lim_{\eta\to0^+}\delta_a\left(f(r)-f(a+\eta)\right)\in\mathbb{R}, η → 0 + lim ∫ a + η r ∣Ψ ( s ) ∣ d s = η → 0 + lim ∫ a + η r δ a f ′ ( s ) d s = η → 0 + lim δ a ( f ( r ) − f ( a + η ) ) ∈ R , thus proving Ψ ∈ L 1 ( [ a , r ] ) \Psi\in L^1([a,r]) Ψ ∈ L 1 ([ a , r ])
The same argument proves integrability of ∣ Ψ ∣ |\Psi| ∣Ψ∣ [ r , b ] [r,b] [ r , b ]
Remark 3.12. An explicit formula for f f f ψ \psi ψ (40) . For r ∈ [ a , b ] r\in[a,b] r ∈ [ a , b ]
f ( r ) = f ( b ) − ∫ r b δ ( t ) Ψ ( t ) d t = f ( a ) + ∫ a r δ ( t ) Ψ ( t ) d t . f(r)=f(b)-\int_r^b\delta(t)\Psi(t)\,dt=f(a)+\int_a^r\delta(t)\Psi(t)\,dt. f ( r ) = f ( b ) − ∫ r b δ ( t ) Ψ ( t ) d t = f ( a ) + ∫ a r δ ( t ) Ψ ( t ) d t . Lemma 3.13. Let σ ∈ { − 1 , 1 } \sigma\in\{-1,1\} σ ∈ { − 1 , 1 } b > 0 b>0 b > 0 φ ∈ C ( [ 0 , b ] ) \varphi\in C([0,b]) φ ∈ C ([ 0 , b ]) f ∈ C ( [ 0 , b ] ) ∩ C 2 ( [ 0 , b ) ) f\in C([0,b])\cap C^2([0,b)) f ∈ C ([ 0 , b ]) ∩ C 2 ([ 0 , b )) (36) is satisfied on ( 0 , b ) (0,b) ( 0 , b )
lim r → 0 + f ′ ( r ) = 0 \lim_{r\to0^+}f'(r)=0 r → 0 + lim f ′ ( r ) = 0 Then as r → 0 + r\to0^+ r → 0 +
f ′ ( r ) = − σ φ ( 0 ) 4 r 2 + o ( r 2 ) . (45) f'(r)=-\frac{\sigma\varphi(0)}{4}r^2+o(r^2). \tag*{(45)} f ′ ( r ) = − 4 σ φ ( 0 ) r 2 + o ( r 2 ) . (45) Proof of Lemma 3.13. Since f ∈ C 2 ( [ 0 , b ) ) f\in C^2([0,b)) f ∈ C 2 ([ 0 , b ))
lim r → 0 + r f ′ ( r ) f ′ ( r ) 2 + r 2 4 = lim r → 0 + r 2 f ′ ′ ( 0 ) r f ′ ′ ( 0 ) 2 + 1 4 + o ( 1 ) = 0. \lim_{r\to0^+}\frac{rf'(r)}{\sqrt{f'(r)^2+\frac{r^2}{4}}}=\lim_{r\to0^+}\frac{r^2f''(0)}{r\sqrt{f''(0)^2+\frac{1}{4}+o(1)}}=0. r → 0 + lim f ′ ( r ) 2 + 4 r 2 r f ′ ( r ) = r → 0 + lim r f ′′ ( 0 ) 2 + 4 1 + o ( 1 ) r 2 f ′′ ( 0 ) = 0. Using also (3.14), this implies that for r → 0 + r \to0^{+} r → 0 +
r f ′ ( r ) f ′ ( r ) 2 + r 2 4 = ∫ 0 r d d t ( t f ′ ( t ) f ′ ( t ) 2 + t 2 4 ) d t = ∫ 0 r ( − σ φ ( 0 ) t + o ( t ) ) d t , \frac{r f'(r)}{\sqrt{f'(r)^2+\frac{r^2}{4}}}=\int_{0}^{r}\frac{d}{dt}\left(\frac{t f'(t)}{\sqrt{f'(t)^2+\frac{t^2}{4}}}\right)\,dt=\int_{0}^{r}(-\sigma\varphi(0)t+o(t))\,dt, f ′ ( r ) 2 + 4 r 2 r f ′ ( r ) = ∫ 0 r d t d f ′ ( t ) 2 + 4 t 2 t f ′ ( t ) d t = ∫ 0 r ( − σ φ ( 0 ) t + o ( t )) d t , where in the last equality we used the continuity of φ \varphi φ 0 0 0 r → 0 + r \to0^{+} r → 0 +
f ′ ( r ) f ′ ( r ) 2 + r 2 4 = − σ φ ( 0 ) r 2 + o ( r ) , \frac{f'(r)}{\sqrt{f'(r)^2+\frac{r^2}{4}}}=-\sigma\varphi(0)\frac{r}{2}+o(r), f ′ ( r ) 2 + 4 r 2 f ′ ( r ) = − σ φ ( 0 ) 2 r + o ( r ) , If φ ( 0 ) ≠ 0 \varphi(0)\ne0 φ ( 0 ) = 0 δ ( r ) = sgn ( − σ φ ( 0 ) ) \delta(r)=\operatorname{sgn}(-\sigma\varphi(0)) δ ( r ) = sgn ( − σ φ ( 0 ))
f ′ ( r ) = r 2 ( − σ φ ( 0 ) r 2 + o ( r ) ) 1 − ( σ φ ( 0 ) r 2 + o ( r ) ) 2 = − σ φ ( 0 ) r 2 4 + o ( r 2 ) . f'(r)=\frac{\frac{r}{2}\left(-\sigma\varphi(0)\frac{r}{2}+o(r)\right)}{\sqrt{1-\left(\sigma\varphi(0)\frac{r}{2}+o(r)\right)^2}}=-\sigma\varphi(0)\frac{r^2}{4}+o(r^2). f ′ ( r ) = 1 − ( σ φ ( 0 ) 2 r + o ( r ) ) 2 2 r ( − σ φ ( 0 ) 2 r + o ( r ) ) = − σ φ ( 0 ) 4 r 2 + o ( r 2 ) . The computation directly extends to the case φ ( 0 ) = 0 \varphi(0)=0 φ ( 0 ) = 0
In the next proposition, we solve the local boundary value problems for the prescribed g ε g_{\varepsilon} g ε (25) on intervals [ a , b ] ⊂ [ 0 , R ] [a,b]\subset[0,R] [ a , b ] ⊂ [ 0 , R ] φ ε = φ \varphi_{\varepsilon}=\varphi φ ε = φ ∂ Ω \partial\Omega ∂ Ω a a a b b b
Proposition 3.14. Let σ ∈ { − 1 , 1 } \sigma\in\{-1,1\} σ ∈ { − 1 , 1 } 0 ≤ a < b 0\le a<b 0 ≤ a < b φ ∈ C ( [ a , b ] ) \varphi\in C([a,b]) φ ∈ C ([ a , b ]) f ∈ C ( [ a , b ] ) f\in C([a,b]) f ∈ C ([ a , b ])
a) f ∈ C 2 ( [ 0 , b ) ) f\in C^2([0,b)) f ∈ C 2 ([ 0 , b )) a = 0 a=0 a = 0 f ∈ C 2 ( ( a , b ) ) f\in C^2((a,b)) f ∈ C 2 (( a , b )) a > 0 a>0 a > 0
b) for r ∈ ( a , b ) r\in(a,b) r ∈ ( a , b )
− σ r d d r ( r f ′ ( r ) f ′ ( r ) 2 + r 2 4 ) = φ ( r ) ; (46) -\frac{\sigma}{r}\frac{d}{dr}\left(\frac{r f'(r)}{\sqrt{f'(r)^2+\frac{r^2}{4}}}\right)=\varphi(r); \tag*{(46)} − r σ d r d f ′ ( r ) 2 + 4 r 2 r f ′ ( r ) = φ ( r ) ; (46) c) there exist δ a , δ b ∈ { − 1 , 1 } \delta_a,\delta_b\in\{-1,1\} δ a , δ b ∈ { − 1 , 1 }
\begin{cases}
\lim_{r\to0^+}f'(r)=0, & \text{if }a=0,\\[6pt]
\lim_{r\to a^+}\delta_a f'(r)=+\infty, & \text{if }a>0,
\end{cases}
\qquad
\lim_{r\to b^-}\delta_b f'(r)=+\infty. \tag*{(47)}\end{cases}
Let z a , z b ∈ R z_a,z_b\in\mathbb{R} z a , z b ∈ R ε > 0 \varepsilon>0 ε > 0 g ε g_{\varepsilon} g ε ( a , b ) (a,b) ( a , b )
− σ r ∂ r ( r f ε ′ ( r ) f ε ′ ( r ) 2 + r 2 4 + ε 2 ) = φ , (48) -\frac{\sigma}{r}\partial_r\left(\frac{r f'_{\varepsilon}(r)}{\sqrt{f'_{\varepsilon}(r)^2+\frac{r^2}{4}+\varepsilon^2}}\right)=\varphi, \tag*{(48)} − r σ ∂ r f ε ′ ( r ) 2 + 4 r 2 + ε 2 r f ε ′ ( r ) = φ , (48) together with one of the following boundary conditions
f ε ′ ( 0 ) = 0 , lim r → b − δ b f ε ′ ( r ) = + ∞ , f ε ( b ) = z b with a = 0 ; f'_{\varepsilon}(0)=0,\qquad\lim_{r\to b^-}\delta_b f'_{\varepsilon}(r)=+\infty,\qquad f_{\varepsilon}(b)=z_b\quad\text{with }a=0; f ε ′ ( 0 ) = 0 , r → b − lim δ b f ε ′ ( r ) = + ∞ , f ε ( b ) = z b with a = 0 ; (BC0 _0 0
lim r → a + δ a f ε ′ ( r ) = + ∞ , lim r → b − δ b f ε ′ ( r ) = + ∞ , f ε ( a ) = z a with a > 0 ; \lim_{r\to a^+}\delta_a f'_{\varepsilon}(r)=+\infty,\qquad\lim_{r\to b^-}\delta_b f'_{\varepsilon}(r)=+\infty,\qquad f_{\varepsilon}(a)=z_a\quad\text{with }a>0; r → a + lim δ a f ε ′ ( r ) = + ∞ , r → b − lim δ b f ε ′ ( r ) = + ∞ , f ε ( a ) = z a with a > 0 ; (BCa _a a
lim r → a + δ a f ε ′ ( r ) = + ∞ , lim r → b − δ b f ε ′ ( r ) = + ∞ , f ε ( b ) = z b with a > 0. \lim_{r\to a^+}\delta_a f'_{\varepsilon}(r)=+\infty,\qquad\lim_{r\to b^-}\delta_b f'_{\varepsilon}(r)=+\infty,\qquad f_{\varepsilon}(b)=z_b\quad\text{with }a>0. r → a + lim δ a f ε ′ ( r ) = + ∞ , r → b − lim δ b f ε ′ ( r ) = + ∞ , f ε ( b ) = z b with a > 0. (BCb _b b
Then the corresponding boundary value problem admits a unique solution f ε ∈ C 2 ( ( a , b ) ) ∩ C 0 ( [ a , b ] ) f_{\varepsilon}\in C^2((a,b))\cap C^0([a,b]) f ε ∈ C 2 (( a , b )) ∩ C 0 ([ a , b ]) a > 0 a>0 a > 0 f ε ∈ C 2 ( [ 0 , b ) ) ∩ C 0 ( [ 0 , b ] ) f_{\varepsilon}\in C^2([0,b))\cap C^0([0,b]) f ε ∈ C 2 ([ 0 , b )) ∩ C 0 ([ 0 , b ]) a = 0 a=0 a = 0
f ε ( r ) = z b − ∫ r b f ′ ( t ) 1 + 4 ε 2 t 2 d t (49) f_{\varepsilon}(r)=z_b-\int_{r}^{b}f'(t)\sqrt{1+\frac{4\varepsilon^2}{t^2}}\,dt \tag*{(49)} f ε ( r ) = z b − ∫ r b f ′ ( t ) 1 + t 2 4 ε 2 d t (49) in cases ( B C 0 ) (\mathrm{BC}_0) ( BC 0 ) ( B C b ) (\mathrm{BC}_b) ( BC b )
f ε ( r ) = z a + ∫ a r f ′ ( t ) 1 + 4 ε 2 t 2 d t (50) f_\varepsilon(r)=z_a+\int_a^r f'(t)\sqrt{1+\frac{4\varepsilon^2}{t^2}}\,dt \tag*{(50)} f ε ( r ) = z a + ∫ a r f ′ ( t ) 1 + t 2 4 ε 2 d t (50) in case ( B C a ) (\mathrm{BC}_a) ( BC a )
Moreover, if a = 0 a=0 a = 0 r → 0 + r\to0^+ r → 0 +
f ε ′ ( r ) = − σ φ ( 0 ) ε 2 r + o ( r ) . (51) f_\varepsilon'(r)=-\frac{\sigma\varphi(0)\varepsilon}{2}r+o(r). \tag*{(51)} f ε ′ ( r ) = − 2 σ φ ( 0 ) ε r + o ( r ) . (51) Finally, as ε → 0 + \varepsilon\to0^+ ε → 0 + f ε f_\varepsilon f ε [ a , b ] [a,b] [ a , b ] f a = f − f ( a ) + z a f_a=f-f(a)+z_a f a = f − f ( a ) + z a ( B C a ) (\mathrm{BC}_a) ( BC a ) f b = f − f ( b ) + z b f_b=f-f(b)+z_b f b = f − f ( b ) + z b ( B C 0 ) (\mathrm{BC}_0) ( BC 0 ) ( B C b ) (\mathrm{BC}_b) ( BC b )
Proof. Step I: Existence of a solution. Let ψ : [ a , b ] → R \psi:[a,b]\to\mathbb{R} ψ : [ a , b ] → R Ψ : ( a , b ) → R \Psi:(a,b)\to\mathbb{R} Ψ : ( a , b ) → R ε > 0 \varepsilon>0 ε > 0
Ψ ε ( r ) = 2 Ψ ( r ) r r 2 4 + ε 2 , r ∈ ( a , b ) . \Psi_\varepsilon(r)=\frac{2\Psi(r)}{r}\sqrt{\frac{r^2}{4}+\varepsilon^2},\qquad r\in(a,b). Ψ ε ( r ) = r 2Ψ ( r ) 4 r 2 + ε 2 , r ∈ ( a , b ) . We first prove that
Ψ ε ∈ L 1 ( [ a , b ] ) . (52) \Psi_\varepsilon\in L^1([a,b]). \tag*{(52)} Ψ ε ∈ L 1 ([ a , b ]) . (52) Fix c ∈ ( a , b ) c\in(a,b) c ∈ ( a , b )
1 r r 2 4 + ε 2 ≤ 1 c b 2 4 + ε 2 for r ∈ [ c , b ] , \frac{1}{r}\sqrt{\frac{r^2}{4}+\varepsilon^2}\leq\frac{1}{c}\sqrt{\frac{b^2}{4}+\varepsilon^2}\qquad\text{for }r\in[c,b], r 1 4 r 2 + ε 2 ≤ c 1 4 b 2 + ε 2 for r ∈ [ c , b ] , and since Ψ ∈ L 1 ( [ a , b ] ) \Psi\in L^1([a,b]) Ψ ∈ L 1 ([ a , b ]) Ψ ε ∈ L 1 ( [ c , b ] ) \Psi_\varepsilon\in L^1([c,b]) Ψ ε ∈ L 1 ([ c , b ])
It remains to consider the behavior near a a a a > 0 a>0 a > 0 [ a , c ] [a,c] [ a , c ] Ψ ε ∈ L 1 ( [ a , c ] ) \Psi_\varepsilon\in L^1([a,c]) Ψ ε ∈ L 1 ([ a , c ]) a = 0 a=0 a = 0
lim r → 0 + ψ ( r ) r = lim r → 0 + 1 r ∫ 0 r t φ ( t ) d t = 0 , \lim_{r\to0^+}\frac{\psi(r)}{r}=\lim_{r\to0^+}\frac{1}{r}\int_0^r t\varphi(t)\,dt=0, r → 0 + lim r ψ ( r ) = r → 0 + lim r 1 ∫ 0 r tφ ( t ) d t = 0 , because φ \varphi φ 0 0 0
lim r → 0 + 2 Ψ ( r ) r = lim r → 0 + ∣ ψ ( r ) ∣ r 2 − ψ ( r ) 2 = lim r → 0 + ∣ ψ ( r ) ∣ r 1 1 − ( ψ ( r ) r ) 2 = 0. (53) \lim_{r\to0^+}\frac{2\Psi(r)}{r}
=\lim_{r\to0^+}\frac{|\psi(r)|}{\sqrt{r^2-\psi(r)^2}}
=\lim_{r\to0^+}\frac{|\psi(r)|}{r}\frac{1}{\sqrt{1-\left(\frac{\psi(r)}{r}\right)^2}}
=0. \tag*{(53)} r → 0 + lim r 2Ψ ( r ) = r → 0 + lim r 2 − ψ ( r ) 2 ∣ ψ ( r ) ∣ = r → 0 + lim r ∣ ψ ( r ) ∣ 1 − ( r ψ ( r ) ) 2 1 = 0. (53) Thus Ψ ε \Psi_\varepsilon Ψ ε r = 0 r=0 r = 0 Ψ ε ( 0 ) = 0 \Psi_\varepsilon(0)=0 Ψ ε ( 0 ) = 0 Ψ ε ∈ L 1 ( [ 0 , c ] ) \Psi_\varepsilon\in L^1([0,c]) Ψ ε ∈ L 1 ([ 0 , c ])
Let δ : [ a , b ] → { − 1 , 0 , 1 } \delta:[a,b]\to\{-1,0,1\} δ : [ a , b ] → { − 1 , 0 , 1 } ε > 0 \varepsilon>0 ε > 0 f ε : [ a , b ] → R f_\varepsilon:[a,b]\to\mathbb{R} f ε : [ a , b ] → R
f ε ( r ) = z b − ∫ r b δ ( t ) Ψ ε ( t ) d t f_\varepsilon(r)=z_b-\int_r^b\delta(t)\Psi_\varepsilon(t)\,dt f ε ( r ) = z b − ∫ r b δ ( t ) Ψ ε ( t ) d t in cases ( B C 0 ) (\mathrm{BC}_0) ( BC 0 ) ( B C b ) (\mathrm{BC}_b) ( BC b )
f ε ( r ) = z a + ∫ a r δ ( t ) Ψ ε ( t ) d t f_\varepsilon(r)=z_a+\int_a^r\delta(t)\Psi_\varepsilon(t)\,dt f ε ( r ) = z a + ∫ a r δ ( t ) Ψ ε ( t ) d t in case ( B C a ) (\mathrm{BC}_a) ( BC a ) f ε f_\varepsilon f ε [ a , b ] [a,b] [ a , b ] δ Ψ ∈ C 1 ( ( a , b ) ) \delta\Psi\in C^1((a,b)) δ Ψ ∈ C 1 (( a , b )) δ Ψ ε ∈ C 1 ( ( a , b ) ) \delta\Psi_\varepsilon\in C^1((a,b)) δ Ψ ε ∈ C 1 (( a , b )) f ε ∈ C 2 ( ( a , b ) ) f_\varepsilon\in C^2((a,b)) f ε ∈ C 2 (( a , b ))
f ε ′ ( r ) = δ ( r ) Ψ ε ( r ) , r ∈ ( a , b ) . (54) f_\varepsilon'(r)=\delta(r)\Psi_\varepsilon(r),\qquad r\in(a,b). \tag*{(54)} f ε ′ ( r ) = δ ( r ) Ψ ε ( r ) , r ∈ ( a , b ) . (54) If a = 0 a=0 a = 0 f ε f_\varepsilon f ε C 2 C^2 C 2
f ε ′ ( r ) = f ′ ( r ) 1 + 4 ε 2 r 2 . f_\varepsilon'(r)=f'(r)\sqrt{1+\frac{4\varepsilon^2}{r^2}}. f ε ′ ( r ) = f ′ ( r ) 1 + r 2 4 ε 2 . By (3.22), we therefore obtain (3.28), that implies that
lim r → 0 + f ε ′ ( r ) = 0 , f ε ′ ′ ( 0 ) = lim r → 0 + f ε ′ ( r ) r = − σ φ ( 0 ) ε 2 , (55) \lim_{r\to0^+}f_\varepsilon'(r)=0,\qquad f_\varepsilon''(0)=\lim_{r\to0^+}\frac{f_\varepsilon'(r)}{r}=-\frac{\sigma\varphi(0)\varepsilon}{2}, \tag*{(55)} r → 0 + lim f ε ′ ( r ) = 0 , f ε ′′ ( 0 ) = r → 0 + lim r f ε ′ ( r ) = − 2 σ φ ( 0 ) ε , (55) hence yielding C 2 C^2 C 2
f ε ′ ′ ( r ) = − 1 r 2 4 + ε 2 { σ φ ( r ) ( f ε ′ ( r ) 2 + r 2 4 + ε 2 ) 3 / 2 + f ε ′ ( r ) 3 r + ε 2 f ε ′ ( r ) r } , f_\varepsilon''(r)=-\frac{1}{\frac{r^2}{4}+\varepsilon^2}\left\{\sigma\varphi(r)\left(f_\varepsilon'(r)^2+\frac{r^2}{4}+\varepsilon^2\right)^{3/2}+\frac{f_\varepsilon'(r)^3}{r}+\varepsilon^2\frac{f_\varepsilon'(r)}{r}\right\}, f ε ′′ ( r ) = − 4 r 2 + ε 2 1 { σ φ ( r ) ( f ε ′ ( r ) 2 + 4 r 2 + ε 2 ) 3/2 + r f ε ′ ( r ) 3 + ε 2 r f ε ′ ( r ) } , and using (3.32), we obtain
lim r → 0 + f ε ′ ′ ( r ) = − 1 ε 2 ( σ φ ( 0 ) ε 3 − σ φ ( 0 ) ε 3 2 ) = − σ φ ( 0 ) ε 2 = f ε ′ ′ ( 0 ) . \lim_{r\to0^+}f_\varepsilon''(r)=-\frac{1}{\varepsilon^2}\left(\sigma\varphi(0)\varepsilon^3-\frac{\sigma\varphi(0)\varepsilon^3}{2}\right)=-\frac{\sigma\varphi(0)\varepsilon}{2}=f_\varepsilon''(0). r → 0 + lim f ε ′′ ( r ) = − ε 2 1 ( σ φ ( 0 ) ε 3 − 2 σ φ ( 0 ) ε 3 ) = − 2 σ φ ( 0 ) ε = f ε ′′ ( 0 ) . Therefore f ε ∈ C 2 ( [ 0 , b ) ) f_\varepsilon\in C^2([0,b)) f ε ∈ C 2 ([ 0 , b ))
We show that f ε f_\varepsilon f ε Ψ ε \Psi_\varepsilon Ψ ε
f ε ′ ( r ) 2 + r 2 4 + ε 2 = r 2 ( r 2 4 + ε 2 ) r 2 − ( ψ ( r ) − a δ a σ ) 2 , r ∈ ( a , b ) . f_\varepsilon'(r)^2+\frac{r^2}{4}+\varepsilon^2=\frac{r^2\left(\frac{r^2}{4}+\varepsilon^2\right)}{r^2-\left(\psi(r)-a\delta_a\sigma\right)^2},\qquad r\in(a,b). f ε ′ ( r ) 2 + 4 r 2 + ε 2 = r 2 − ( ψ ( r ) − a δ a σ ) 2 r 2 ( 4 r 2 + ε 2 ) , r ∈ ( a , b ) . In this computation, the case δ ( r ) = 0 \delta(r)=0 δ ( r ) = 0 ψ ( r ) − a δ a σ = 0 \psi(r)-a\delta_a\sigma=0 ψ ( r ) − a δ a σ = 0
r f ε ′ ( r ) f ε ′ ( r ) 2 + r 2 4 + ε 2 = δ ( r ) ∣ ψ ( r ) − a δ a σ ∣ , r ∈ ( a , b ) . \frac{r f_\varepsilon'(r)}{\sqrt{f_\varepsilon'(r)^2+\frac{r^2}{4}+\varepsilon^2}}=\delta(r)\left|\psi(r)-a\delta_a\sigma\right|,\qquad r\in(a,b). f ε ′ ( r ) 2 + 4 r 2 + ε 2 r f ε ′ ( r ) = δ ( r ) ∣ ψ ( r ) − a δ a σ ∣ , r ∈ ( a , b ) . By (3.18), the right-hand side agrees with the corresponding quantity for the limiting profile, namely
r f ′ ( r ) f ′ ( r ) 2 + r 2 4 = δ ( r ) ∣ ψ ( r ) − a δ a σ ∣ , r ∈ ( a , b ) . \frac{r f'(r)}{\sqrt{f'(r)^2+\frac{r^2}{4}}}=\delta(r)\left|\psi(r)-a\delta_a\sigma\right|,\qquad r\in(a,b). f ′ ( r ) 2 + 4 r 2 r f ′ ( r ) = δ ( r ) ∣ ψ ( r ) − a δ a σ ∣ , r ∈ ( a , b ) . Hence, using (3.23), we get
− σ r d d r ( r f ε ′ ( r ) f ε ′ ( r ) 2 + r 2 4 + ε 2 ) = − σ r d d r ( r f ′ ( r ) f ′ ( r ) 2 + r 2 4 ) = φ ( r ) , r ∈ ( a , b ) . -\frac{\sigma}{r}\frac{\mathrm{d}}{\mathrm{d}r}\left(\frac{r f_\varepsilon'(r)}{\sqrt{f_\varepsilon'(r)^2+\frac{r^2}{4}+\varepsilon^2}}\right)=-\frac{\sigma}{r}\frac{\mathrm{d}}{\mathrm{d}r}\left(\frac{r f'(r)}{\sqrt{f'(r)^2+\frac{r^2}{4}}}\right)=\varphi(r),\qquad r\in(a,b). − r σ d r d f ε ′ ( r ) 2 + 4 r 2 + ε 2 r f ε ′ ( r ) = − r σ d r d f ′ ( r ) 2 + 4 r 2 r f ′ ( r ) = φ ( r ) , r ∈ ( a , b ) . Thus f ε f_\varepsilon f ε
We finally check the boundary conditions. All derivatives are taken in the open interval ( a , b ) (a,b) ( a , b ) f ε ′ f_\varepsilon' f ε ′ 0 _0 0 a = 0 a=0 a = 0 b b b δ ( r ) = δ b \delta(r)=\delta_b δ ( r ) = δ b r r r b b b ψ \psi ψ b b b
lim r → b − δ b f ε ′ ( r ) = lim r → b − ∣ ψ ( r ) ∣ r 2 − ψ ( r ) 2 r 2 4 + ε 2 = + ∞ . \lim_{r\to b^-}\delta_b f_\varepsilon'(r)=\lim_{r\to b^-}\frac{|\psi(r)|}{\sqrt{r^2-\psi(r)^2}}\sqrt{\frac{r^2}{4}+\varepsilon^2}=+\infty. r → b − lim δ b f ε ′ ( r ) = r → b − lim r 2 − ψ ( r ) 2 ∣ ψ ( r ) ∣ 4 r 2 + ε 2 = + ∞. The prescribed value f ε ( b ) = z b f_\varepsilon(b)=z_b f ε ( b ) = z b f ε f_\varepsilon f ε
The same argument gives the boundary condition at b b b b _b b ψ ( r ) \psi(r) ψ ( r ) ψ ( r ) − a δ a σ \psi(r)-a\delta_a\sigma ψ ( r ) − a δ a σ δ ( r ) = δ a \delta(r)=\delta_a δ ( r ) = δ a r r r a a a a a a a _a a b _b b f ε ( a ) = z a f_\varepsilon(a)=z_a f ε ( a ) = z a a _a a f ε ( b ) = z b f_\varepsilon(b)=z_b f ε ( b ) = z b b _b b f ε f_\varepsilon f ε
Step II: Uniqueness of the solution. We prove uniqueness. Let h h h
W h ( r ) = r h ′ ( r ) h ′ ( r ) 2 + r 2 4 + ε 2 , r ∈ ( a , b ) . W_h(r)=\frac{rh'(r)}{\sqrt{h'(r)^2+\frac{r^2}{4}+\varepsilon^2}},\qquad r\in(a,b). W h ( r ) = h ′ ( r ) 2 + 4 r 2 + ε 2 r h ′ ( r ) , r ∈ ( a , b ) . Then the ODE (48) rewrites as W h ′ ( r ) = − σ r φ ( r ) W_h'(r)=-\sigma r\varphi(r) W h ′ ( r ) = − σ r φ ( r ) r ∈ ( a , b ) r\in(a,b) r ∈ ( a , b )
W h ( r ) = C a − σ ∫ a r t φ ( t ) d t . W_h(r)=C_a-\sigma\int_a^r t\varphi(t)\,\mathrm{d}t. W h ( r ) = C a − σ ∫ a r tφ ( t ) d t . where
C a = lim r → a + r h ′ ( r ) h ′ ( r ) 2 + r 2 4 + ε 2 = a δ a . C_a=\lim_{r\to a^+}\frac{rh'(r)}{\sqrt{h'(r)^2+\frac{r^2}{4}+\varepsilon^2}}=a\delta_a. C a = r → a + lim h ′ ( r ) 2 + 4 r 2 + ε 2 r h ′ ( r ) = a δ a . Here, the computation of C a C_a C a h h h
W h ( r ) = W f ε ( r ) for any r ∈ ( a , b ) . W_h(r)=W_{f_\varepsilon}(r)\qquad\text{for any }r\in(a,b). W h ( r ) = W f ε ( r ) for any r ∈ ( a , b ) . It remains to observe that, for each fixed r ∈ ( a , b ) r\in(a,b) r ∈ ( a , b )
p ↦ r p p 2 + r 2 4 + ε 2 p\mapsto\frac{rp}{\sqrt{p^2+\frac{r^2}{4}+\varepsilon^2}} p ↦ p 2 + 4 r 2 + ε 2 r p is strictly increasing from R \mathbb{R} R ( − r , r ) (-r,r) ( − r , r ) h ′ ( r ) = f ε ′ ( r ) h'(r)=f_\varepsilon'(r) h ′ ( r ) = f ε ′ ( r ) r ∈ ( a , b ) r\in(a,b) r ∈ ( a , b ) h h h f ε f_\varepsilon f ε h ( a ) = z a h(a)=z_a h ( a ) = z a h ( b ) = z b h(b)=z_b h ( b ) = z b
Step III: Uniform convergence as ε → 0 + \varepsilon\to0^+ ε → 0 + We prove the uniform convergence in the case of ( B C 0 ) (\mathrm{BC}_0) ( BC 0 ) ( B C b ) (\mathrm{BC}_b) ( BC b ) ( B C a ) (\mathrm{BC}_a) ( BC a )
We first claim that
lim ε → 0 + ∫ a b ∣ Ψ ( t ) ∣ ( 2 t t 2 4 + ε 2 − 1 ) d t = 0. (56) \lim_{\varepsilon\to0^+}\int_a^b|\Psi(t)|\left(\frac{2}{t}\sqrt{\frac{t^2}{4}+\varepsilon^2}-1\right)\,\mathrm{d}t=0. \tag*{(56)} ε → 0 + lim ∫ a b ∣Ψ ( t ) ∣ ( t 2 4 t 2 + ε 2 − 1 ) d t = 0. (56) Indeed, this implies
f ε → f − f ( b ) + z b uniformly on [ a , b ] , f_\varepsilon\to f-f(b)+z_b\qquad\text{uniformly on }[a,b], f ε → f − f ( b ) + z b uniformly on [ a , b ] , since
sup r ∈ [ a , b ] ∣ f ε ( r ) − f ( r ) + f ( b ) − z b ∣ = sup r ∈ [ a , b ] ∣ ∫ r b δ ( t ) ( Ψ ε ( t ) − Ψ ( t ) ) d t ∣ ≤ ∫ a b ∣ Ψ ( t ) ∣ ( 2 t t 2 4 + ε 2 − 1 ) d t . \begin{aligned}
\sup_{r\in[a,b]}|f_\varepsilon(r)-f(r)+f(b)-z_b|
=\sup_{r\in[a,b]}\left|\int_r^b\delta(t)(\Psi_\varepsilon(t)-\Psi(t))\,\mathrm{d}t\right| \\
&\leq\int_a^b|\Psi(t)|\left(\frac{2}{t}\sqrt{\frac{t^2}{4}+\varepsilon^2}-1\right)\,\mathrm{d}t.
\end{aligned} r ∈ [ a , b ] sup ∣ f ε ( r ) − f ( r ) + f ( b ) − z b ∣ = r ∈ [ a , b ] sup ∫ r b δ ( t ) ( Ψ ε ( t ) − Ψ ( t )) d t ≤ ∫ a b ∣Ψ ( t ) ∣ ( t 2 4 t 2 + ε 2 − 1 ) d t . It remains to prove (56) . For every fixed t ∈ ( a , b ] t\in(a,b] t ∈ ( a , b ]
2 t t 2 4 + ε 2 − 1 ⟶ 0 as ε → 0 + . \frac{2}{t}\sqrt{\frac{t^2}{4}+\varepsilon^2}-1\longrightarrow0\qquad\text{as }\varepsilon\to0^+. t 2 4 t 2 + ε 2 − 1 ⟶ 0 as ε → 0 + . Thus, (3.33) follows by dominated convergence once we find an L 1 L^1 L 1 ε \varepsilon ε ε \varepsilon ε
If a > 0 a>0 a > 0 0 < ε ≤ 1 0<\varepsilon\leq1 0 < ε ≤ 1 t ∈ [ a , b ] t\in[a,b] t ∈ [ a , b ]
0 ≤ 2 t t 2 4 + ε 2 − 1 ≤ 2 a b 2 4 + 1 − 1. 0\leq\frac{2}{t}\sqrt{\frac{t^2}{4}+\varepsilon^2}-1\leq\frac{2}{a}\sqrt{\frac{b^2}{4}+1}-1. 0 ≤ t 2 4 t 2 + ε 2 − 1 ≤ a 2 4 b 2 + 1 − 1. Therefore the integrand is bounded by a constant multiple of ∣ Ψ ∣ |\Psi| ∣Ψ∣ Ψ ∈ L 1 ( [ a , b ] ) \Psi\in L^1([a,b]) Ψ ∈ L 1 ([ a , b ]) Lemma 3.11 , statement iii), the conclusion follows by dominated convergence.
Let now a = 0 a=0 a = 0 (53) , the map t ↦ Ψ ( t ) / t t\mapsto\Psi(t)/t t ↦ Ψ ( t ) / t t = 0 t=0 t = 0
Ψ ( t ) t ∈ L 1 ( [ 0 , b ] ) . (57) \frac{\Psi(t)}{t}\in L^1([0,b]). \tag*{(57)} t Ψ ( t ) ∈ L 1 ([ 0 , b ]) . (57) For 0 < ε ≤ 1 0<\varepsilon\le1 0 < ε ≤ 1
0 ≤ 2 t ( t 2 4 + ε 2 − 1 ) = 2 t ( t 2 4 + ε 2 − t 2 ) ≤ 2 t ( t 2 4 + 1 − t 2 ) . 0\le\frac{2}{t}\left(\sqrt{\frac{t^2}{4}+\varepsilon^2}-1\right)=\frac{2}{t}\left(\sqrt{\frac{t^2}{4}+\varepsilon^2}-\frac{t}{2}\right)\le\frac{2}{t}\left(\sqrt{\frac{t^2}{4}+1}-\frac{t}{2}\right). 0 ≤ t 2 ( 4 t 2 + ε 2 − 1 ) = t 2 ( 4 t 2 + ε 2 − 2 t ) ≤ t 2 ( 4 t 2 + 1 − 2 t ) . Moreover,
t 2 4 + 1 − t 2 = 1 t 2 4 + 1 + t 2 ≤ 1. \sqrt{\frac{t^2}{4}+1}-\frac{t}{2}=\frac{1}{\sqrt{\frac{t^2}{4}+1+\frac{t}{2}}}\le1. 4 t 2 + 1 − 2 t = 4 t 2 + 1 + 2 t 1 ≤ 1. Hence
∣ Ψ ( t ) ∣ ( 2 t t 2 4 + ε 2 − 1 ) ≤ 2 ∣ Ψ ( t ) t ∣ . |\Psi(t)|\left(\frac{2}{t}\sqrt{\frac{t^2}{4}+\varepsilon^2}-1\right)\le2\left|\frac{\Psi(t)}{t}\right|. ∣Ψ ( t ) ∣ ( t 2 4 t 2 + ε 2 − 1 ) ≤ 2 t Ψ ( t ) . Therefore by (57) , dominated convergence gives (56) with a = 0 a=0 a = 0 □ \square □
Proof of Proposition 3.10. Step I: Construction of the approximants Σ j ε \Sigma_j^\varepsilon Σ j ε For ε > 0 \varepsilon>0 ε > 0 Σ 1 ε , … , Σ N ε \Sigma_1^\varepsilon,\ldots,\Sigma_N^\varepsilon Σ 1 ε , … , Σ N ε z z z { z = f j ε } \{z=f_j^\varepsilon\} { z = f j ε } f j ε ∈ C 2 ( I j ) ∩ C 0 ( I j ‾ ) f_j^\varepsilon\in C^2(I_j)\cap C^0(\overline{I_j}) f j ε ∈ C 2 ( I j ) ∩ C 0 ( I j ) (48) on I j I_j I j σ = σ j \sigma=\sigma_j σ = σ j j = 1 , … , N j=1,\ldots,N j = 1 , … , N
We start from an equator of Ω \Omega Ω j = j R j=j_R j = j R f j R ε ∈ C 2 ( ( a j R − 1 , R ) ) ∩ C 0 ( [ a j R − 1 , R ] ) f_{j_R}^\varepsilon\in C^2((a_{j_R-1},R))\cap C^0([a_{j_R-1},R]) f j R ε ∈ C 2 (( a j R − 1 , R )) ∩ C 0 ([ a j R − 1 , R ]) (48) with σ = σ j R = 1 \sigma=\sigma_{j_R}=1 σ = σ j R = 1 a j R − 1 = 0 a_{j_R-1}=0 a j R − 1 = 0 a j R − 1 > 0 a_{j_R-1}>0 a j R − 1 > 0
If j R = 1 j_R=1 j R = 1 a j R − 1 = 0 a_{j_R-1}=0 a j R − 1 = 0 ( B C 0 ) (\mathrm{BC}_0) ( BC 0 ) z b = f j R ( R ) z_b=f_{j_R}(R) z b = f j R ( R ) δ b = − 1 \delta_b=-1 δ b = − 1
If j R ≠ 1 j_R\ne1 j R = 1 a j R − 1 > 0 a_{j_R-1}>0 a j R − 1 > 0 ( B C b ) (\mathrm{BC}_b) ( BC b ) f j R ε f_{j_R}^\varepsilon f j R ε r = a j R − 1 r=a_{j_R-1} r = a j R − 1 r = R r=R r = R f j R f_{j_R} f j R
a = a j R − 1 , b = a j R = R , z b = f j R ( R ) , δ a = δ j R ( a j R − 1 ) , δ b = − 1 , a=a_{j_R-1},\qquad b=a_{j_R}=R,\qquad z_b=f_{j_R}(R),\qquad\delta_a=\delta_{j_R}(a_{j_R-1}),\qquad\delta_b=-1, a = a j R − 1 , b = a j R = R , z b = f j R ( R ) , δ a = δ j R ( a j R − 1 ) , δ b = − 1 , as follows from (32) . Proposition 3.14, then yields the explicit formula (49) for the solution f j R ε f_{j_R}^\varepsilon f j R ε
We proceed recursively for j = j R − 1 , … , 2 j=j_R-1,\ldots,2 j = j R − 1 , … , 2 f j + 1 ε f_{j+1}^\varepsilon f j + 1 ε f j ε f_j^\varepsilon f j ε (48) on I j I_j I j σ = σ j \sigma=\sigma_j σ = σ j f j + 1 ε f_{j+1}^\varepsilon f j + 1 ε a j a_j a j f j f_j f j σ j = 1 \sigma_j=1 σ j = 1 I j = ( a j − 1 , a j ) I_j=(a_{j-1},a_j) I j = ( a j − 1 , a j ) ( B C b ) (\mathrm{BC}_b) ( BC b )
a = a j − 1 , b = a j , z b = f j + 1 ε ( a j ) , δ a = δ j ( a j − 1 ) , δ b = δ j ( a j ) . a=a_{j-1},\qquad b=a_j,\qquad z_b=f_{j+1}^\varepsilon(a_j),\qquad\delta_a=\delta_j(a_{j-1}),\qquad\delta_b=\delta_j(a_j). a = a j − 1 , b = a j , z b = f j + 1 ε ( a j ) , δ a = δ j ( a j − 1 ) , δ b = δ j ( a j ) . If instead σ j = − 1 \sigma_j=-1 σ j = − 1 I j = ( a j , a j − 1 ) I_j=(a_j,a_{j-1}) I j = ( a j , a j − 1 ) ( B C a ) (\mathrm{BC}_a) ( BC a )
a = a j , b = a j − 1 , z a = f j + 1 ε ( a j ) , δ a = δ j ( a j ) , δ b = δ j ( a j − 1 ) . a=a_j,\qquad b=a_{j-1},\qquad z_a=f_{j+1}^\varepsilon(a_j),\qquad\delta_a=\delta_j(a_j),\qquad\delta_b=\delta_j(a_{j-1}). a = a j , b = a j − 1 , z a = f j + 1 ε ( a j ) , δ a = δ j ( a j ) , δ b = δ j ( a j − 1 ) . Proceeding recursively, we eventually reach I 1 = ( 0 , a 1 ) I_1=(0,a_1) I 1 = ( 0 , a 1 ) (48) with σ = σ 1 \sigma=\sigma_1 σ = σ 1 ( B C 0 ) (\mathrm{BC}_0) ( BC 0 ) b = a 1 b=a_1 b = a 1 δ b = sgn ( γ 2 ′ ( α 1 ) ) \delta_b=\operatorname{sgn}(\gamma_2'(\alpha_1)) δ b = sgn ( γ 2 ′ ( α 1 ))
Arguing in the same way on the remaining intervals I j I_j I j j = j R + 1 , … , N j=j_R+1,\ldots,N j = j R + 1 , … , N f j ε f_j^\varepsilon f j ε j = 1 , … , N j=1,\ldots,N j = 1 , … , N a j a_j a j j = 1 , … , N j=1,\ldots,N j = 1 , … , N
Σ j ε = { z = f j ε ( r ( x , y ) ) , r ( x , y ) ∈ I j } . \Sigma_j^\varepsilon=\{z=f_j^\varepsilon(r(x,y)),\ r(x,y)\in I_j\}. Σ j ε = { z = f j ε ( r ( x , y )) , r ( x , y ) ∈ I j } . Step II: Uniform convergence and absence of self intersections among the Σ j ε \Sigma_j^\varepsilon Σ j ε We first claim that for any j = 1 , … , N j=1,\ldots,N j = 1 , … , N f j ε f_j^\varepsilon f j ε f j f_j f j I j ‾ \overline{I_j} I j ε → 0 + \varepsilon\to0^+ ε → 0 + j = j R j=j_R j = j R b = R b=R b = R z b = f j R ( a j R ) z_b=f_{j_R}(a_{j_R}) z b = f j R ( a j R ) j j j j R − 1 j_R-1 j R − 1 j j j j R + 1 j_R+1 j R + 1 N N N
Assume that f j ε f_j^\varepsilon f j ε f j f_j f j I j ‾ \overline{I_j} I j ε → 0 + \varepsilon\to0^+ ε → 0 + I ‾ j − 1 \overline{I}_{j-1} I j − 1 σ j − 1 = 1 \sigma_{j-1}=1 σ j − 1 = 1 I ‾ j − 1 = [ a j − 2 , a j − 1 ] \overline{I}_{j-1}=[a_{j-2},a_{j-1}] I j − 1 = [ a j − 2 , a j − 1 ]
sup I ‾ j − 1 ∣ f j − 1 ε − f j − 1 ∣ = sup r ∈ I ‾ j − 1 ∣ f j ε ( a j − 1 ) − f j ( a j − 1 ) + ∫ r a j − 1 f j − 1 ε ′ ( t ) ( 1 − 1 + 4 ε 2 t 2 ) d t ∣ ≤ ∣ f j ε ( a j − 1 ) − f j ( a j − 1 ) ∣ + sup r ∈ I ‾ j − 1 ∫ r a j − 1 ∣ Ψ j − 1 ( t ) ∣ ( 1 + 4 ε 2 t 2 − 1 ) d t ≤ ∣ f j ε ( a j − 1 ) − f j ( a j − 1 ) ∣ + ∫ I j − 1 ∣ Ψ j − 1 ( t ) ∣ ( 1 + 4 ε 2 t 2 − 1 ) d t , (58) \begin{aligned}
\sup_{\overline{I}_{j-1}}|f_{j-1}^\varepsilon-f_{j-1}|&=\sup_{r\in\overline{I}_{j-1}}\left|f_j^\varepsilon(a_{j-1})-f_j(a_{j-1})+\int_r^{a_{j-1}}f_{j-1}^{\varepsilon\prime}(t)\left(1-\sqrt{1+\frac{4\varepsilon^2}{t^2}}\right)\,dt\right|\\
&\leq|f_j^\varepsilon(a_{j-1})-f_j(a_{j-1})|+\sup_{r\in\overline{I}_{j-1}}\int_r^{a_{j-1}}|\Psi_{j-1}(t)|\left(\sqrt{1+\frac{4\varepsilon^2}{t^2}}-1\right)\,dt \\
&\leq|f_j^\varepsilon(a_{j-1})-f_j(a_{j-1})|+\int_{I_{j-1}}|\Psi_{j-1}(t)|\left(\sqrt{1+\frac{4\varepsilon^2}{t^2}}-1\right)\,dt, \tag*{(58)}
\end{aligned} I j − 1 sup ∣ f j − 1 ε − f j − 1 ∣ = r ∈ I j − 1 sup f j ε ( a j − 1 ) − f j ( a j − 1 ) + ∫ r a j − 1 f j − 1 ε ′ ( t ) ( 1 − 1 + t 2 4 ε 2 ) d t ≤ ∣ f j ε ( a j − 1 ) − f j ( a j − 1 ) ∣ + r ∈ I j − 1 sup ∫ r a j − 1 ∣ Ψ j − 1 ( t ) ∣ ( 1 + t 2 4 ε 2 − 1 ) d t ≤ ∣ f j ε ( a j − 1 ) − f j ( a j − 1 ) ∣ + ∫ I j − 1 ∣ Ψ j − 1 ( t ) ∣ ( 1 + t 2 4 ε 2 − 1 ) d t , (58) where Ψ j − 1 ∈ L 1 ( I j − 1 ) \Psi_{j-1}\in L^1(I_{j-1}) Ψ j − 1 ∈ L 1 ( I j − 1 ) a = a j − 2 a=a_{j-2} a = a j − 2 b = a j − 1 b=a_{j-1} b = a j − 1 σ = σ j − 1 \sigma=\sigma_{j-1} σ = σ j − 1 σ j − 1 = − 1 \sigma_{j-1}=-1 σ j − 1 = − 1
We now show that, for ε > 0 \varepsilon>0 ε > 0 j ≠ h j\ne h j = h I j ∩ I h = ∅ I_j\cap I_h=\varnothing I j ∩ I h = ∅ Σ j ε ∩ Σ h ε = ∅ \Sigma_j^\varepsilon\cap\Sigma_h^\varepsilon=\varnothing Σ j ε ∩ Σ h ε = ∅
I j ∩ I h ≠ ∅ . I_j\cap I_h\ne\varnothing. I j ∩ I h = ∅ . We first treat adjacent profiles near the junction radii, namely h = j + 1 h=j+1 h = j + 1 sgn ( f j ε ′ ) = sgn ( f j ′ ) \operatorname{sgn}(f_j^{\varepsilon\prime})=\operatorname{sgn}(f_j') sgn ( f j ε ′ ) = sgn ( f j ′ ) ε > 0 \varepsilon>0 ε > 0 ρ = min j ρ j > 0 \rho=\min_j\rho_j>0 ρ = min j ρ j > 0 ε > 0 \varepsilon>0 ε > 0
δ j ( a j ) f j ε ′ > 0 on I j ∩ ( a j − ρ , a j + ρ ) , δ j + 1 ( a j ) f j + 1 ε ′ > 0 on I j + 1 ∩ ( a j − ρ , a j + ρ ) . (59) \begin{aligned}
\delta_j(a_j)f_j^{\varepsilon\prime}&>0 &&\text{on } I_j\cap(a_j-\rho,a_j+\rho),\\
\delta_{j+1}(a_j)f_{j+1}^{\varepsilon\prime}&>0 &&\text{on } I_{j+1}\cap(a_j-\rho,a_j+\rho). \tag*{(59)}
\end{aligned} δ j ( a j ) f j ε ′ δ j + 1 ( a j ) f j + 1 ε ′ > 0 > 0 on I j ∩ ( a j − ρ , a j + ρ ) , on I j + 1 ∩ ( a j − ρ , a j + ρ ) . (59) Moreover, since I j ∩ I j + 1 ≠ ∅ I_j\cap I_{j+1}\ne\varnothing I j ∩ I j + 1 = ∅ a j a_j a j a j a_j a j I j I_j I j I j + 1 I_{j+1} I j + 1 a j a_j a j σ j + 1 = − σ j \sigma_{j+1}=-\sigma_j σ j + 1 = − σ j
δ j ( a j ) δ j + 1 ( a j ) = − 1. \delta_j(a_j)\delta_{j+1}(a_j)=-1. δ j ( a j ) δ j + 1 ( a j ) = − 1. Therefore, calling z j ε = f j ε ( a j ) = f j + 1 ε ( a j ) z_j^\varepsilon=f_j^\varepsilon(a_j)=f_{j+1}^\varepsilon(a_j) z j ε = f j ε ( a j ) = f j + 1 ε ( a j )
( f j ε − z j ε ) ( f j + 1 ε − z j ε ) < 0 in ( a j − ρ , a j + ρ ) ∩ I j ∩ I j + 1 , (f_j^\varepsilon-z_j^\varepsilon)(f_{j+1}^\varepsilon-z_j^\varepsilon)<0\quad\text{in }(a_j-\rho,a_j+\rho)\cap I_j\cap I_{j+1}, ( f j ε − z j ε ) ( f j + 1 ε − z j ε ) < 0 in ( a j − ρ , a j + ρ ) ∩ I j ∩ I j + 1 , so that f j ε ( r ) ≠ f j + 1 ε ( r ) f_j^\varepsilon(r)\ne f_{j+1}^\varepsilon(r) f j ε ( r ) = f j + 1 ε ( r ) r ∈ ( a j − ρ , a j + ρ ) ∩ I j ∩ I j + 1 r\in(a_j-\rho,a_j+\rho)\cap I_j\cap I_{j+1} r ∈ ( a j − ρ , a j + ρ ) ∩ I j ∩ I j + 1 j = 1 , … , N − 1 j=1,\ldots,N-1 j = 1 , … , N − 1 U j = ( a j − ρ , a j + ρ ) U_j=(a_j-\rho,a_j+\rho) U j = ( a j − ρ , a j + ρ ) j ≠ h j\ne h j = h K j , h ⊂ I j ‾ ∩ I h ‾ K_{j,h}\subset\overline{I_j}\cap\overline{I_h} K j , h ⊂ I j ∩ I h
K j , h = { I j ‾ ∩ I h ‾ ∖ U j , h = j + 1 , I j ‾ ∩ I h ‾ ∖ U h , h = j − 1 , I j ‾ ∩ I h ‾ , ∣ j − h ∣ > 1. K_{j,h}=
\begin{cases}
\overline{I_j}\cap\overline{I_h}\setminus U_j, & h=j+1,\\
\overline{I_j}\cap\overline{I_h}\setminus U_h, & h=j-1,\\
\overline{I_j}\cap\overline{I_h}, & |j-h|>1.
\end{cases} K j , h = ⎩ ⎨ ⎧ I j ∩ I h ∖ U j , I j ∩ I h ∖ U h , I j ∩ I h , h = j + 1 , h = j − 1 , ∣ j − h ∣ > 1. The limit profiles f j f_j f j f h f_h f h K j , h K_{j,h} K j , h γ \gamma γ η > 0 \eta>0 η > 0
∣ f j ( r ) − f h ( r ) ∣ ≥ η \lvert f_j(r)-f_h(r)\rvert\geq\eta ∣ f j ( r ) − f h ( r )∣ ≥ η for every j ≠ h j\neq h j = h r ∈ K j , h r\in K_{j,h} r ∈ K j , h f j ε f_j^\varepsilon f j ε f j f_j f j I j ‾ \overline{I_j} I j N N N ε > 0 \varepsilon>0 ε > 0
sup I j ‾ ∣ f j ε − f j ∣ < η 3 for every j = 1 , … , N . \sup_{\overline{I_j}}\lvert f_j^\varepsilon-f_j\rvert<\frac{\eta}{3}
\qquad\text{for every }j=1,\ldots,N. I j sup ∣ f j ε − f j ∣ < 3 η for every j = 1 , … , N . Therefore, for every j ≠ h j\neq h j = h r ∈ K j , h r\in K_{j,h} r ∈ K j , h ∣ A + B − C ∣ ≥ ∣ A ∣ − ∣ B ∣ − ∣ C ∣ \lvert A+B-C\rvert\geq\lvert A\rvert-\lvert B\rvert-\lvert C\rvert ∣ A + B − C ∣ ≥ ∣ A ∣ − ∣ B ∣ − ∣ C ∣ A = f j ( r ) − f h ( r ) A=f_j(r)-f_h(r) A = f j ( r ) − f h ( r ) B = f j ε ( r ) − f j ( r ) B=f_j^\varepsilon(r)-f_j(r) B = f j ε ( r ) − f j ( r ) C = f h ε ( r ) − f h ( r ) C=f_h^\varepsilon(r)-f_h(r) C = f h ε ( r ) − f h ( r )
∣ f j ε ( r ) − f h ε ( r ) ∣ ≥ ∣ f j ( r ) − f h ( r ) ∣ − ∣ f j ε ( r ) − f j ( r ) ∣ − ∣ f h ε ( r ) − f h ( r ) ∣ ≥ η − η 3 − η 3 > 0. \begin{aligned}
\lvert f_j^\varepsilon(r)-f_h^\varepsilon(r)\rvert\geq\lvert f_j(r)-f_h(r)\rvert-\lvert f_j^\varepsilon(r)-f_j(r)\rvert-\lvert f_h^\varepsilon(r)-f_h(r)\rvert\\
&\geq\eta-\frac{\eta}{3}-\frac{\eta}{3}>0.
\end{aligned} ∣ f j ε ( r ) − f h ε ( r )∣ ≥ ∣ f j ( r ) − f h ( r )∣ − ∣ f j ε ( r ) − f j ( r )∣ − ∣ f h ε ( r ) − f h ( r )∣ ≥ η − 3 η − 3 η > 0. Thus no intersections occur outside the prescribed junctions.
Step III: Gluing and regularity of the approximating boundary. Set
Σ ε = ⋃ j = 1 N Σ j ε . \Sigma_\varepsilon=\bigcup_{j=1}^{N}\Sigma_j^\varepsilon. Σ ε = j = 1 ⋃ N Σ j ε . By construction, the closures of two consecutive pieces meet along the corresponding junction circle:
Σ j ε ‾ ∩ Σ j + 1 ε ‾ = C j ε , j = 1 , … , N − 1 , \overline{\Sigma_j^\varepsilon}\cap\overline{\Sigma_{j+1}^\varepsilon}=C_j^\varepsilon,\qquad j=1,\ldots,N-1, Σ j ε ∩ Σ j + 1 ε = C j ε , j = 1 , … , N − 1 , where
C j ε = { ( x , y , z j ε ) ∈ R 3 : x 2 + y 2 = a j 2 } , z j ε = f j ε ( a j ) = f j + 1 ε ( a j ) . C_j^\varepsilon=\{(x,y,z_j^\varepsilon)\in\mathbb{R}^3:x^2+y^2=a_j^2\},\qquad z_j^\varepsilon=f_j^\varepsilon(a_j)=f_{j+1}^\varepsilon(a_j). C j ε = {( x , y , z j ε ) ∈ R 3 : x 2 + y 2 = a j 2 } , z j ε = f j ε ( a j ) = f j + 1 ε ( a j ) . The closures of the first and last pieces contain the two poles on the rotation axis. By Step II, no other intersections occur. Hence Σ ε \Sigma_\varepsilon Σ ε C 2 C^2 C 2 Ω ε \Omega_\varepsilon Ω ε R 3 ∖ Σ ε \mathbb{R}^3\setminus\Sigma_\varepsilon R 3 ∖ Σ ε
∂ Ω ε = Σ ε . \partial\Omega_\varepsilon=\Sigma_\varepsilon. ∂ Ω ε = Σ ε . In the interior of each interval I j I_j I j f j ε ∈ C 2 ( I j ) f_j^\varepsilon\in C^2(I_j) f j ε ∈ C 2 ( I j ) C 2 C^2 C 2 I j I_j I j r = 0 r=0 r = 0 a j a_j a j j = 1 , … , N − 1 j=1,\ldots,N-1 j = 1 , … , N − 1
For j = 1 , N j=1,N j = 1 , N I j I_j I j r = 0 r=0 r = 0 Σ j ε \Sigma_j^\varepsilon Σ j ε C 2 C^2 C 2
Let q = ( q 1 , q 2 ) ∈ R 2 q=(q_1,q_2)\in\mathbb{R}^2 q = ( q 1 , q 2 ) ∈ R 2 r ( q ) ∈ I j r(q)\in I_j r ( q ) ∈ I j i , ℓ ∈ { 1 , 2 } i,\ell\in\{1,2\} i , ℓ ∈ { 1 , 2 } r ( q ) > 0 r(q)>0 r ( q ) > 0
∂ i ∂ ℓ ( f j ε ( r ( q ) ) ) = q i q ℓ r ( q ) 2 ( f j ε ′ ′ ( r ( q ) ) − f j ε ′ ( r ( q ) ) r ( q ) ) + δ i ℓ f j ε ′ ( r ( q ) ) r ( q ) . \partial_i\partial_\ell\bigl(f_j^\varepsilon(r(q))\bigr)=\frac{q_iq_\ell}{r(q)^2}\left(f_j^{\varepsilon\prime\prime}(r(q))-\frac{f_j^{\varepsilon\prime}(r(q))}{r(q)}\right)+\delta_{i\ell}\frac{f_j^{\varepsilon\prime}(r(q))}{r(q)}. ∂ i ∂ ℓ ( f j ε ( r ( q )) ) = r ( q ) 2 q i q ℓ ( f j ε ′′ ( r ( q )) − r ( q ) f j ε ′ ( r ( q )) ) + δ i ℓ r ( q ) f j ε ′ ( r ( q )) . Using the expansion of f j ε f_j^\varepsilon f j ε 0 0 0 r ( q ) → 0 r(q)\to0 r ( q ) → 0 r − 1 ( I j ) ∋ q → 0 r^{-1}(I_j)\ni q\to0 r − 1 ( I j ) ∋ q → 0
lim q → 0 ∂ i ∂ ℓ ( f j ε ( r ( q ) ) ) = − σ j φ ( 0 ) ε 2 δ i ℓ . \lim_{q\to0}\partial_i\partial_\ell\bigl(f_j^\varepsilon(r(q))\bigr)=-\frac{\sigma_j\varphi(0)\varepsilon}{2}\delta_{i\ell}. q → 0 lim ∂ i ∂ ℓ ( f j ε ( r ( q )) ) = − 2 σ j φ ( 0 ) ε δ i ℓ . Thus the second derivatives extend continuously at the axis, and Σ j ε \Sigma_j^\varepsilon Σ j ε C 2 C^2 C 2
It remains to prove regularity at the junction circles C j ε C_j^\varepsilon C j ε j = 1 , … , N − 1 j=1,\ldots,N-1 j = 1 , … , N − 1 f j ε ′ f_j^{\varepsilon\prime} f j ε ′ f j + 1 ε ′ f_{j+1}^{\varepsilon\prime} f j + 1 ε ′ a j a_j a j Ω ε \Omega_\varepsilon Ω ε C j ε C_j^\varepsilon C j ε z z z (59) , there exists ρ > 0 \rho>0 ρ > 0 f j ε f_j^\varepsilon f j ε f j + 1 ε f_{j+1}^\varepsilon f j + 1 ε U j = ( a j − ρ , a j + ρ ) U_j=(a_j-\rho,a_j+\rho) U j = ( a j − ρ , a j + ρ ) (31) implies that
f j ε ( I j ∩ U j ) ∩ f j + 1 ε ( I j + 1 ∩ U j ) = { z j ε } , f_j^\varepsilon(I_j\cap U_j)\cap f_{j+1}^\varepsilon(I_{j+1}\cap U_j)=\{z_j^\varepsilon\}, f j ε ( I j ∩ U j ) ∩ f j + 1 ε ( I j + 1 ∩ U j ) = { z j ε } , where z j ε = f j ε ( a j ) z_j^\varepsilon=f_j^\varepsilon(a_j) z j ε = f j ε ( a j ) H j ε = f j ε ( I j ∩ U j ) ∪ f j + 1 ε ( I j + 1 ∩ U j ) H_j^\varepsilon=f_j^\varepsilon(I_j\cap U_j)\cup f_{j+1}^\varepsilon(I_{j+1}\cap U_j) H j ε = f j ε ( I j ∩ U j ) ∪ f j + 1 ε ( I j + 1 ∩ U j ) h j ε : H j ε → R + h_j^\varepsilon:H_j^\varepsilon\to\mathbb{R}^+ h j ε : H j ε → R +
h j ε ( z ) = { ( f j ε ) − 1 ( z ) if z ∈ f j ε ( I j ∩ U j ) , ( f j + 1 ε ) − 1 ( z ) if z ∈ f j + 1 ε ( I j + 1 ∩ U j ) h_j^\varepsilon(z)=
\begin{cases}
(f_j^\varepsilon)^{-1}(z) & \text{if }z\in f_j^\varepsilon(I_j\cap U_j),\\
(f_{j+1}^\varepsilon)^{-1}(z) & \text{if }z\in f_{j+1}^\varepsilon(I_{j+1}\cap U_j)
\end{cases} h j ε ( z ) = { ( f j ε ) − 1 ( z ) ( f j + 1 ε ) − 1 ( z ) if z ∈ f j ε ( I j ∩ U j ) , if z ∈ f j + 1 ε ( I j + 1 ∩ U j ) is well defined and C 2 C^2 C 2 H j ε H_j^\varepsilon H j ε h j ε h_j^\varepsilon h j ε C 2 C^2 C 2 z j ε z_j^\varepsilon z j ε (29) implies that ( h j ε ) ′ ( z j ε ) = 0 (h_j^\varepsilon)'(z_j^\varepsilon)=0 ( h j ε ) ′ ( z j ε ) = 0 C 1 C^1 C 1 z j ε z_j^\varepsilon z j ε h j ε h_j^\varepsilon h j ε z ≠ z j ε z\ne z_j^\varepsilon z = z j ε g ε g_\varepsilon g ε (48) as
( f j ε ) ′ ′ ( r ) = − 1 r 2 4 + ε 2 { σ j φ ( r ) ( ( f j ε ) ′ ( r ) 2 + r 2 4 + ε 2 ) 3 2 + ( f j ε ) ′ ( r ) 3 r ( 1 + ε 2 ( f j ε ) ′ ( r ) 2 ) } . (f_j^\varepsilon)''(r)=-\frac{1}{\frac{r^2}{4}+\varepsilon^2}\left\{\sigma_j\varphi(r)\left((f_j^\varepsilon)'(r)^2+\frac{r^2}{4}+\varepsilon^2\right)^{\frac{3}{2}}+\frac{(f_j^\varepsilon)'(r)^3}{r}\left(1+\frac{\varepsilon^2}{(f_j^\varepsilon)'(r)^2}\right)\right\}. ( f j ε ) ′′ ( r ) = − 4 r 2 + ε 2 1 { σ j φ ( r ) ( ( f j ε ) ′ ( r ) 2 + 4 r 2 + ε 2 ) 2 3 + r ( f j ε ) ′ ( r ) 3 ( 1 + ( f j ε ) ′ ( r ) 2 ε 2 ) } . so that, letting r z = h j ε ( z ) r_z=h_j^\varepsilon(z) r z = h j ε ( z ) z ∈ f j ( I j ∩ U j ) z\in f_j(I_j\cap U_j) z ∈ f j ( I j ∩ U j )
( h j ε ) ′ ′ ( z ) = − ( f j ε ) ′ ′ ( r z ) ( f j ε ) ′ ( r z ) 3 = 1 r z 2 4 + ε 2 { σ j φ ( r z ) sgn ( ( f j ε ) ′ ( r z ) ) ( 1 + r z 2 4 + ε 2 ( f j ε ) ′ ( r z ) 2 ) 3 2 + 1 r z ( 1 + ε 2 ( f j ε ) ′ ( r z ) 2 ) } . \begin{aligned}
(h_j^\varepsilon)''(z)=-\frac{(f_j^\varepsilon)''(r_z)}{(f_j^\varepsilon)'(r_z)^3} \\
&=\frac{1}{\frac{r_z^2}{4}+\varepsilon^2}\left\{\sigma_j\varphi(r_z)\operatorname{sgn}((f_j^\varepsilon)'(r_z))\left(1+\frac{\frac{r_z^2}{4}+\varepsilon^2}{(f_j^\varepsilon)'(r_z)^2}\right)^{\frac{3}{2}}+\frac{1}{r_z}\left(1+\frac{\varepsilon^2}{(f_j^\varepsilon)'(r_z)^2}\right)\right\}.
\end{aligned} ( h j ε ) ′′ ( z ) = − ( f j ε ) ′ ( r z ) 3 ( f j ε ) ′′ ( r z ) = 4 r z 2 + ε 2 1 ⎩ ⎨ ⎧ σ j φ ( r z ) sgn (( f j ε ) ′ ( r z )) ( 1 + ( f j ε ) ′ ( r z ) 2 4 r z 2 + ε 2 ) 2 3 + r z 1 ( 1 + ( f j ε ) ′ ( r z ) 2 ε 2 ) ⎭ ⎬ ⎫ . Since h j ε ( z ) → a j h_j^\varepsilon(z)\to a_j h j ε ( z ) → a j ∣ ( f j ε ) ′ ( h j ε ( z ) ) ∣ → + ∞ \lvert(f_j^\varepsilon)'(h_j^\varepsilon(z))\rvert\to+\infty ∣( f j ε ) ′ ( h j ε ( z ))∣ → + ∞ z → z j ε z\to z_j^\varepsilon z → z j ε
lim z → z j ε z ≠ f j ε ( I j ∩ U j ) ( h j ε ) ′ ′ ( z ) = 1 a j 2 4 + ε 2 ( σ j δ j ( a j ) φ ( a j ) + 1 a j ) , \lim_{\substack{z\to z_j^\varepsilon\\z\ne f_j^\varepsilon(I_j\cap U_j)}}(h_j^\varepsilon)''(z)=\frac{1}{\frac{a_j^2}{4}+\varepsilon^2}\left(\sigma_j\delta_j(a_j)\varphi(a_j)+\frac{1}{a_j}\right), z → z j ε z = f j ε ( I j ∩ U j ) lim ( h j ε ) ′′ ( z ) = 4 a j 2 + ε 2 1 ( σ j δ j ( a j ) φ ( a j ) + a j 1 ) , lim z → z j ε z ≠ f j + 1 ε ( I j + 1 ∩ U j ) ( h j ε ) ′ ′ ( z ) = 1 a j 2 4 + ε 2 ( σ j + 1 δ j + 1 ( a j ) φ ( a j ) + 1 a j ) . \lim_{\substack{z\to z_j^\varepsilon\\z\ne f_{j+1}^\varepsilon(I_{j+1}\cap U_j)}}(h_j^\varepsilon)''(z)=\frac{1}{\frac{a_j^2}{4}+\varepsilon^2}\left(\sigma_{j+1}\delta_{j+1}(a_j)\varphi(a_j)+\frac{1}{a_j}\right). z → z j ε z = f j + 1 ε ( I j + 1 ∩ U j ) lim ( h j ε ) ′′ ( z ) = 4 a j 2 + ε 2 1 ( σ j + 1 δ j + 1 ( a j ) φ ( a j ) + a j 1 ) . By (31) we therefore deduce that
lim z → z j ε + ( h j ε ) ′ ′ ( z ) = lim z → z j ε − ( h j ε ) ′ ′ ( z ) , \lim_{z\to z_j^{\varepsilon+}}(h_j^\varepsilon)''(z)=\lim_{z\to z_j^{\varepsilon-}}(h_j^\varepsilon)''(z), z → z j ε + lim ( h j ε ) ′′ ( z ) = z → z j ε − lim ( h j ε ) ′′ ( z ) , hence implying C 2 C^2 C 2 h j ε h_j^\varepsilon h j ε ∂ Ω ε \partial\Omega_\varepsilon ∂ Ω ε C 2 C^2 C 2 C j C_j C j F ( x , y , z ) = x 2 + y 2 − ( h j ε ( z ) ) 2 F(x,y,z)=x^2+y^2-(h_j^\varepsilon(z))^2 F ( x , y , z ) = x 2 + y 2 − ( h j ε ( z ) ) 2 C 2 C^2 C 2 C j C_j C j ( 2 x , 2 y , − 2 h j ε ( z ) ( h j ε ) ′ ( z ) ) ≠ 0 (2x,2y,-2h_j^\varepsilon(z)(h_j^\varepsilon)'(z))\ne0 ( 2 x , 2 y , − 2 h j ε ( z ) ( h j ε ) ′ ( z )) = 0
This proves that Σ ε = ∂ Ω ε \Sigma_\varepsilon=\partial\Omega_\varepsilon Σ ε = ∂ Ω ε C 2 C^2 C 2 f j ε f_j^\varepsilon f j ε f j f_j f j I j ‾ \overline{I_j} I j j = 1 , … , N j=1,\ldots,N j = 1 , … , N
d H ( ∂ Ω ε , ∂ Ω ) ⟶ 0 as ε → 0 + . d_H(\partial\Omega_\varepsilon,\partial\Omega)\longrightarrow0\qquad\text{as }\varepsilon\to0^+. d H ( ∂ Ω ε , ∂ Ω ) ⟶ 0 as ε → 0 + . This concludes the proof. □
Proof of Theorem 1.2
We are now in a position to prove Theorem 1.2. We include here the proof of a very well-known polynomial approximation result. We are going to employ it below in order to approximate γ 1 \gamma_1 γ 1 k ∈ N k \in\mathbb{N} k ∈ N f ( k ) f^{(k)} f ( k ) k k k C k C^k C k f f f
Lemma 3.15 (Approximation with fixed boundary data). Let a < b a < b a < b h ∈ C 2 ( [ a , b ] ) h \in C^2([a,b]) h ∈ C 2 ([ a , b ]) { P n } n ≥ 1 \{P_n\}_{n\geq1} { P n } n ≥ 1
P n ( k ) ( a ) = h ( k ) ( a ) , P n ( k ) ( b ) = h ( k ) ( b ) for k = 0 , 1 , 2 , (60) P_n^{(k)}(a)=h^{(k)}(a), \qquad P_n^{(k)}(b)=h^{(k)}(b) \qquad\text{for } k=0,1,2, \tag*{(60)} P n ( k ) ( a ) = h ( k ) ( a ) , P n ( k ) ( b ) = h ( k ) ( b ) for k = 0 , 1 , 2 , (60) and
lim n → + ∞ P n = h in C 2 ( [ a , b ] ) . (61) \lim_{n\to+\infty} P_n=h \quad\text{in } C^2([a,b]). \tag*{(61)} n → + ∞ lim P n = h in C 2 ([ a , b ]) . (61) Proof. By a generalization of Stone-Weierstrass Theorem (see, for instance, [30 Theorem 1.6.2] ), if h ∈ C 2 ( [ a , b ] ) h \in C^2([a,b]) h ∈ C 2 ([ a , b ]) Q n Q_n Q n
Q n → h in C 2 ( [ a , b ] ) . Q_n \to h \quad\text{in } C^2([a,b]). Q n → h in C 2 ([ a , b ]) . However, these polynomials Q n Q_n Q n h h h k = 0 , 1 , 2 k=0,1,2 k = 0 , 1 , 2
δ k , n a = h ( k ) ( a ) − Q n ( k ) ( a ) , δ k , n b = h ( k ) ( b ) − Q n ( k ) ( b ) . \delta^a_{k,n}=h^{(k)}(a)-Q_n^{(k)}(a), \qquad\delta^b_{k,n}=h^{(k)}(b)-Q_n^{(k)}(b). δ k , n a = h ( k ) ( a ) − Q n ( k ) ( a ) , δ k , n b = h ( k ) ( b ) − Q n ( k ) ( b ) . Since Q n → h Q_n \to h Q n → h C 2 C^2 C 2 δ k , n a → 0 \delta^a_{k,n}\to0 δ k , n a → 0 δ k , n b → 0 \delta^b_{k,n}\to0 δ k , n b → 0 n → ∞ n\to\infty n → ∞
We now construct a correction polynomial R n ( x ) R_n(x) R n ( x ) Q n Q_n Q n [35 Theorem 2.1.5.2] and in the sequel, we introduce polynomials in [ 0 , 1 ] [0,1] [ 0 , 1 ] η 0 , η 1 , η 2 \eta_0,\eta_1,\eta_2 η 0 , η 1 , η 2
η j ( k ) = δ j k (Kronecker delta) , for j , k ∈ { 0 , 1 , 2 } (62) \eta_j^{(k)}=\delta_{jk}\quad\text{(Kronecker delta)}, \qquad\text{for }j,k\in\{0,1,2\} \tag*{(62)} η j ( k ) = δ j k (Kronecker delta) , for j , k ∈ { 0 , 1 , 2 } (62) at t = 0 t=0 t = 0 t = 1 t=1 t = 1
η 0 ( t ) = 1 − 10 t 3 + 15 t 4 − 6 t 5 , \eta_0(t)=1-10t^3+15t^4-6t^5, η 0 ( t ) = 1 − 10 t 3 + 15 t 4 − 6 t 5 , η 1 ( t ) = t − 6 t 3 + 8 t 4 − 3 t 5 , \eta_1(t)=t-6t^3+8t^4-3t^5, η 1 ( t ) = t − 6 t 3 + 8 t 4 − 3 t 5 , η 2 ( t ) = 1 2 t 2 − 3 2 t 3 + 3 2 t 4 − 1 2 t 5 \eta_2(t)=\frac{1}{2}t^2-\frac{3}{2}t^3+\frac{3}{2}t^4-\frac{1}{2}t^5 η 2 ( t ) = 2 1 t 2 − 2 3 t 3 + 2 3 t 4 − 2 1 t 5 satisfy the requirements. Consider also the polynomials
η ~ j ( t ) = ( − 1 ) j η j ( 1 − t ) \widetilde{\eta}_j(t)=(-1)^j\eta_j(1-t) η j ( t ) = ( − 1 ) j η j ( 1 − t ) that by symmetry satisfy the conditions (3.39) at t = 1 t=1 t = 1 t = 0 t=0 t = 0
We can thus define, for L = b − a L=b-a L = b − a
R n ( x ) = ∑ j = 0 2 ( δ j , n a L j η j ( x − a L ) + δ j , n b L j η ~ j ( x − a L ) ) . R_n(x)=\sum_{j=0}^{2}\left(\delta^a_{j,n}L^j\eta_j\left(\frac{x-a}{L}\right)+\delta^b_{j,n}L^j\widetilde{\eta}_j\left(\frac{x-a}{L}\right)\right). R n ( x ) = j = 0 ∑ 2 ( δ j , n a L j η j ( L x − a ) + δ j , n b L j η j ( L x − a ) ) . Observe that R n ( k ) ( a ) = δ k , n a R_n^{(k)}(a)=\delta^a_{k,n} R n ( k ) ( a ) = δ k , n a R n ( k ) ( b ) = δ k , n b R_n^{(k)}(b)=\delta^b_{k,n} R n ( k ) ( b ) = δ k , n b P n = Q n + R n P_n=Q_n+R_n P n = Q n + R n δ k , n a , b → 0 \delta^{a,b}_{k,n}\to0 δ k , n a , b → 0 R n → 0 R_n\to0 R n → 0 C 2 ( [ a , b ] ) C^2([a,b]) C 2 ([ a , b ]) n → + ∞ n\to+\infty n → + ∞
We finally conclude the proof of Theorem 1.2, that from this point becomes fairly standard.
Proof of Theorem 1.2. Let γ ∈ C 2 ( [ 0 , 1 ] , R + 2 ) \gamma\in C^{2}([0,1],\mathbb{R}_{+}^{2}) γ ∈ C 2 ([ 0 , 1 ] , R + 2 ) ∂ Ω \partial\Omega ∂ Ω 0 < a < b < 1 0<a<b<1 0 < a < b < 1 γ 1 ′ ≠ 0 \gamma_{1}'\ne0 γ 1 ′ = 0 [ 0 , a ] ∪ [ b , 1 ] [0,a]\cup[b,1] [ 0 , a ] ∪ [ b , 1 ]
Let P n P_n P n γ 1 \gamma_1 γ 1
γ n ( t ) = { γ ( t ) , t ∈ [ 0 , a ] ∪ [ b , 1 ] , ( P n ( t ) , γ 2 ( t ) ) , t ∈ [ a , b ] . \gamma_n(t)=
\begin{cases}
\gamma(t), & t\in[0,a]\cup[b,1],\\
(P_n(t),\gamma_2(t)), & t\in[a,b].
\end{cases} γ n ( t ) = { γ ( t ) , ( P n ( t ) , γ 2 ( t )) , t ∈ [ 0 , a ] ∪ [ b , 1 ] , t ∈ [ a , b ] . For n n n γ n \gamma_n γ n Ω n \Omega^n Ω n
d H ( ∂ Ω n , ∂ Ω ) < 1 n , c 1 − 1 n ≤ H n ≤ c 2 + 1 n . d_H(\partial\Omega^n,\partial\Omega)<\frac{1}{n},\qquad c_1-\frac{1}{n}\le H^n\le c_2+\frac{1}{n}. d H ( ∂ Ω n , ∂ Ω ) < n 1 , c 1 − n 1 ≤ H n ≤ c 2 + n 1 . where H n H^n H n ∂ Ω n \partial\Omega^n ∂ Ω n n n n Ω ε n \Omega_\varepsilon^n Ω ε n
d H ( ∂ Ω ε n , ∂ Ω n ) ⟶ 0 as ε → 0 + , d_H(\partial\Omega_\varepsilon^n,\partial\Omega^n)\longrightarrow0\qquad\text{as }\varepsilon\to0^+, d H ( ∂ Ω ε n , ∂ Ω n ) ⟶ 0 as ε → 0 + , and whose g ε g_\varepsilon g ε H n H^n H n ε n → 0 \varepsilon_n\to0 ε n → 0
d H ( ∂ Ω ε n , ∂ Ω n ) < 1 n for 0 < ε ≤ ε n , d_H(\partial\Omega_\varepsilon^n,\partial\Omega^n)<\frac{1}{n}\qquad\text{for }0<\varepsilon\le\varepsilon_n, d H ( ∂ Ω ε n , ∂ Ω n ) < n 1 for 0 < ε ≤ ε n , and define Ω ε = Ω ε n \Omega_\varepsilon=\Omega_\varepsilon^n Ω ε = Ω ε n ε ∈ ( ε n + 1 , ε n ] \varepsilon\in(\varepsilon_{n+1},\varepsilon_n] ε ∈ ( ε n + 1 , ε n ]
d H ( ∂ Ω ε , ∂ Ω ) < 2 n , c 1 − 1 n ≤ H ε ≤ c 2 + 1 n d_H(\partial\Omega_\varepsilon,\partial\Omega)<\frac{2}{n},\qquad c_1-\frac{1}{n}\le H_\varepsilon\le c_2+\frac{1}{n} d H ( ∂ Ω ε , ∂ Ω ) < n 2 , c 1 − n 1 ≤ H ε ≤ c 2 + n 1 whenever ε ∈ ( ε n + 1 , ε n ] \varepsilon\in(\varepsilon_{n+1},\varepsilon_n] ε ∈ ( ε n + 1 , ε n ]
Remark 3.16. If Ω = P R \Omega=P_R Ω = P R R > 0 R>0 R > 0 (7) , then the family { Ω ε } ε > 0 \{\Omega_\varepsilon\}_{\varepsilon>0} { Ω ε } ε > 0 [36 , 22 ] . For the surface ∂ P R \partial P_R ∂ P R ψ \psi ψ
ψ ( r ) = ∫ 0 r s φ ( s ) d s = ∫ 0 r s 2 R d s = r 2 R . \psi(r)=\int_0^r s\varphi(s)\,ds=\int_0^r s\frac{2}{R}\,ds=\frac{r^2}{R}. ψ ( r ) = ∫ 0 r s φ ( s ) d s = ∫ 0 r s R 2 d s = R r 2 . Applying our construction with the prescribed curvature φ = 2 / R \varphi=2/R φ = 2/ R
f ε ( r ) = ∫ r R s 2 s 2 + 4 ε 2 R 2 − s 2 d s , r ∈ [ 0 , R ] . f_\varepsilon(r)=\int_r^R\frac{s}{2}\sqrt{\frac{s^2+4\varepsilon^2}{R^2-s^2}}\,ds,\qquad r\in[0,R]. f ε ( r ) = ∫ r R 2 s R 2 − s 2 s 2 + 4 ε 2 d s , r ∈ [ 0 , R ] . corresponding to the generating profile of the Tomter spheres of radius R R R H ε \mathbb{H}_\varepsilon H ε [36 ] ).
The main existence results
The proof of our main existence theorem boils down to obtaining gradient bounds for u ε δ u_\varepsilon^\delta u ε δ ( M C F δ ε ) (\mathrm{MCF}_\delta^\varepsilon) ( MCF δ ε ) δ \delta δ ε \varepsilon ε (4) .
Right invariant maximum principle
The first step we carry out consists in a maximum principle for ∣ ∇ ~ ε u ε δ ∣ 2 |\widetilde{\nabla}_\varepsilon u_\varepsilon^\delta|^2 ∣ ∇ ε u ε δ ∣ 2 ∣ ∇ ~ ε u ε δ ∣ 2 |\widetilde{\nabla}_\varepsilon u_\varepsilon^\delta|^2 ∣ ∇ ε u ε δ ∣ 2 ∇ ~ \widetilde{\nabla} ∇ ∇ \nabla ∇ (16) . We do not expect a similar phenomenon to hold for ∣ ∇ ε u ε δ ∣ 2 |\nabla_\varepsilon u_\varepsilon^\delta|^2 ∣ ∇ ε u ε δ ∣ 2 [23 Proposition 3.4] , that we extend here to include Riemannian Heisenberg metrics.
Proposition 4.1 (Right Bochner-type identity). Let ε > 0 \varepsilon>0 ε > 0 f ∈ C 3 ( R 3 ) f\in C^3(\mathbb{R}^3) f ∈ C 3 ( R 3 )
Δ ε ( ∣ ∇ ~ ε f ∣ 2 ) = 2 [ ∣ ∇ ~ ε ( X f ) ∣ 2 + ∣ ∇ ~ ε ( Y f ) ∣ 2 + ε 2 ∣ ∇ ~ ε ( Z f ) ∣ 2 ] + 2 ⟨ ∇ ~ ε f , ∇ ~ ε ( Δ ε f ) ⟩ . (63) \Delta_\varepsilon(|\widetilde{\nabla}_\varepsilon f|^2)=2\left[|\widetilde{\nabla}_\varepsilon(Xf)|^2+|\widetilde{\nabla}_\varepsilon(Yf)|^2+\varepsilon^2|\widetilde{\nabla}_\varepsilon(Zf)|^2\right]+2\langle\widetilde{\nabla}_\varepsilon f,\widetilde{\nabla}_\varepsilon(\Delta_\varepsilon f)\rangle. \tag*{(63)} Δ ε ( ∣ ∇ ε f ∣ 2 ) = 2 [ ∣ ∇ ε ( X f ) ∣ 2 + ∣ ∇ ε ( Y f ) ∣ 2 + ε 2 ∣ ∇ ε ( Z f ) ∣ 2 ] + 2 ⟨ ∇ ε f , ∇ ε ( Δ ε f )⟩ . (63) Proof. By a direct computation
Δ ε ( ∣ ∇ ~ ε f ∣ 2 ) = X X ( ∣ ∇ ~ ε f ∣ 2 ) + Y Y ( ∣ ∇ ~ ε f ∣ 2 ) + ε 2 Z Z ( ∣ ∇ ~ ε f ∣ 2 ) = X X ( ∇ ~ ε f ⋅ ∇ ~ ε f ) + Y Y ( ∇ ~ ε f ⋅ ∇ ~ ε f ) + ε 2 Z Z ( ∇ ~ ε f ⋅ ∇ ~ ε f ) = X ( 2 ∇ ~ ε f ⋅ X ( ∇ ~ ε f ) ) + Y ( 2 ∇ ~ ε f ⋅ Y ( ∇ ~ ε f ) ) + ε 2 Z ( 2 ∇ ~ ε f ⋅ Z ( ∇ ~ ε f ) ) = 2 [ ∣ X ( ∇ ~ ε f ) ∣ 2 + ∇ ~ ε f ⋅ X X ( ∇ ~ ε f ) + ∣ Y ( ∇ ~ ε f ) ∣ 2 + ∇ ~ ε f ⋅ Y Y ( ∇ ~ ε f ) + ε 2 ∣ Z ( ∇ ~ ε f ) ∣ 2 + ε 2 ∇ ~ ε f ⋅ Z Z ( ∇ ~ ε f ) ] . \begin{aligned}
\Delta_{\varepsilon}(|\widetilde{\nabla}_{\varepsilon}f|^2)=XX(|\widetilde{\nabla}_{\varepsilon}f|^2)+YY(|\widetilde{\nabla}_{\varepsilon}f|^2)+\varepsilon^2ZZ(|\widetilde{\nabla}_{\varepsilon}f|^2) \\
&=XX(\widetilde{\nabla}_{\varepsilon}f\cdot\widetilde{\nabla}_{\varepsilon}f)+YY(\widetilde{\nabla}_{\varepsilon}f\cdot\widetilde{\nabla}_{\varepsilon}f)+\varepsilon^2ZZ(\widetilde{\nabla}_{\varepsilon}f\cdot\widetilde{\nabla}_{\varepsilon}f) \\
&=X(2\widetilde{\nabla}_{\varepsilon}f\cdot X(\widetilde{\nabla}_{\varepsilon}f))+Y(2\widetilde{\nabla}_{\varepsilon}f\cdot Y(\widetilde{\nabla}_{\varepsilon}f))+\varepsilon^2Z(2\widetilde{\nabla}_{\varepsilon}f\cdot Z(\widetilde{\nabla}_{\varepsilon}f)) \\
&=2\left[|X(\widetilde{\nabla}_{\varepsilon}f)|^2+\widetilde{\nabla}_{\varepsilon}f\cdot XX(\widetilde{\nabla}_{\varepsilon}f)+|Y(\widetilde{\nabla}_{\varepsilon}f)|^2+\widetilde{\nabla}_{\varepsilon}f\cdot YY(\widetilde{\nabla}_{\varepsilon}f)+\varepsilon^2|Z(\widetilde{\nabla}_{\varepsilon}f)|^2+\varepsilon^2\widetilde{\nabla}_{\varepsilon}f\cdot ZZ(\widetilde{\nabla}_{\varepsilon}f)\right].
\end{aligned} Δ ε ( ∣ ∇ ε f ∣ 2 ) = X X ( ∣ ∇ ε f ∣ 2 ) + Y Y ( ∣ ∇ ε f ∣ 2 ) + ε 2 Z Z ( ∣ ∇ ε f ∣ 2 ) = X X ( ∇ ε f ⋅ ∇ ε f ) + Y Y ( ∇ ε f ⋅ ∇ ε f ) + ε 2 Z Z ( ∇ ε f ⋅ ∇ ε f ) = X ( 2 ∇ ε f ⋅ X ( ∇ ε f )) + Y ( 2 ∇ ε f ⋅ Y ( ∇ ε f )) + ε 2 Z ( 2 ∇ ε f ⋅ Z ( ∇ ε f )) = 2 [ ∣ X ( ∇ ε f ) ∣ 2 + ∇ ε f ⋅ X X ( ∇ ε f ) + ∣ Y ( ∇ ε f ) ∣ 2 + ∇ ε f ⋅ Y Y ( ∇ ε f ) + ε 2 ∣ Z ( ∇ ε f ) ∣ 2 + ε 2 ∇ ε f ⋅ Z Z ( ∇ ε f ) ] . As observed in (2.5), the left-invariant vector fields { X , Y , Z } \{X,Y,Z\} { X , Y , Z } ∇ ~ ε \widetilde{\nabla}_{\varepsilon} ∇ ε
X ( ∇ ~ ε f ) = ∇ ~ ε ( X f ) , Y ( ∇ ~ ε f ) = ∇ ~ ε ( Y f ) , Z ( ∇ ~ ε f ) = ∇ ~ ε ( Z f ) . X(\widetilde{\nabla}_{\varepsilon}f)=\widetilde{\nabla}_{\varepsilon}(Xf),\qquad Y(\widetilde{\nabla}_{\varepsilon}f)=\widetilde{\nabla}_{\varepsilon}(Yf),\qquad Z(\widetilde{\nabla}_{\varepsilon}f)=\widetilde{\nabla}_{\varepsilon}(Zf). X ( ∇ ε f ) = ∇ ε ( X f ) , Y ( ∇ ε f ) = ∇ ε ( Y f ) , Z ( ∇ ε f ) = ∇ ε ( Z f ) . It follows that Δ ε \Delta_{\varepsilon} Δ ε ∇ ~ ε \widetilde{\nabla}_{\varepsilon} ∇ ε
Δ ε ( ∣ ∇ ~ ε f ∣ 2 ) = 2 [ ∣ ∇ ~ ε ( X f ) ∣ 2 + ∣ ∇ ~ ε ( Y f ) ∣ 2 + ε 2 ∣ ∇ ~ ε ( Z f ) ∣ 2 ] + 2 ⟨ ∇ ~ ε f , ∇ ~ ε ( Δ ε f ) ⟩ . \Delta_{\varepsilon}(|\widetilde{\nabla}_{\varepsilon}f|^2)=2\left[|\widetilde{\nabla}_{\varepsilon}(Xf)|^2+|\widetilde{\nabla}_{\varepsilon}(Yf)|^2+\varepsilon^2|\widetilde{\nabla}_{\varepsilon}(Zf)|^2\right]+2\langle\widetilde{\nabla}_{\varepsilon}f,\widetilde{\nabla}_{\varepsilon}(\Delta_{\varepsilon}f)\rangle. Δ ε ( ∣ ∇ ε f ∣ 2 ) = 2 [ ∣ ∇ ε ( X f ) ∣ 2 + ∣ ∇ ε ( Y f ) ∣ 2 + ε 2 ∣ ∇ ε ( Z f ) ∣ 2 ] + 2 ⟨ ∇ ε f , ∇ ε ( Δ ε f )⟩ . Proposition 4.2. Let D ⊂ R 3 D\subset\mathbb{R}^3 D ⊂ R 3 u ε δ ∈ C ∞ ( D ‾ ) u_{\varepsilon}^{\delta}\in C^{\infty}(\overline{D}) u ε δ ∈ C ∞ ( D ) ( M C F ε δ ) (\mathrm{MCF}_{\varepsilon}^{\delta}) ( MCF ε δ ) D D D
∥ ∇ ~ ε u ε δ ∥ L ∞ ( D ) ≤ ∥ ∇ ~ ε u ε δ ∥ L ∞ ( ∂ D ) . (64) \|\widetilde{\nabla}_{\varepsilon}u_{\varepsilon}^{\delta}\|_{L^{\infty}(D)}\leq\|\widetilde{\nabla}_{\varepsilon}u_{\varepsilon}^{\delta}\|_{L^{\infty}(\partial D)}. \tag*{(64)} ∥ ∇ ε u ε δ ∥ L ∞ ( D ) ≤ ∥ ∇ ε u ε δ ∥ L ∞ ( ∂ D ) . (64) Proof. Applying (4.1) to u ε δ u_{\varepsilon}^{\delta} u ε δ ( M C F ε δ ) (\mathrm{MCF}_{\varepsilon}^{\delta}) ( MCF ε δ )
Δ ε φ = 2 A − 2 B ∣ ∇ ε u ε δ ∣ 2 + δ 2 (65) \Delta_{\varepsilon}\varphi=2\mathcal{A}-\frac{2\mathcal{B}}{|\nabla_{\varepsilon}u_{\varepsilon}^{\delta}|^2+\delta^2} \tag*{(65)} Δ ε φ = 2 A − ∣ ∇ ε u ε δ ∣ 2 + δ 2 2 B (65) − 2 ∇ ε ∇ ε u ε δ ( ∇ ε u ε δ , ∇ ε u ε δ ) ( ∣ ∇ ε u ε δ ∣ 2 + δ 2 ) 2 ⟨ ∇ ε u ε δ , ∇ ε φ ⟩ + ∇ ε ∇ ε φ ( ∇ ε u ε δ , ∇ ε u ε δ ) ∣ ∇ ε u ε δ ∣ 2 + δ 2 + 2 ∇ ε ∇ ε u ε δ ( ∇ ε u ε δ , ∇ ε φ ) ∣ ∇ ε u ε δ ∣ 2 + δ 2 , -2\frac{\nabla_{\varepsilon}\nabla_{\varepsilon}u_{\varepsilon}^{\delta}(\nabla_{\varepsilon}u_{\varepsilon}^{\delta},\nabla_{\varepsilon}u_{\varepsilon}^{\delta})}{(|\nabla_{\varepsilon}u_{\varepsilon}^{\delta}|^2+\delta^2)^2}\langle\nabla_{\varepsilon}u_{\varepsilon}^{\delta},\nabla_{\varepsilon}\varphi\rangle+\frac{\nabla_{\varepsilon}\nabla_{\varepsilon}\varphi(\nabla_{\varepsilon}u_{\varepsilon}^{\delta},\nabla_{\varepsilon}u_{\varepsilon}^{\delta})}{|\nabla_{\varepsilon}u_{\varepsilon}^{\delta}|^2+\delta^2}+\frac{2\nabla_{\varepsilon}\nabla_{\varepsilon}u_{\varepsilon}^{\delta}(\nabla_{\varepsilon}u_{\varepsilon}^{\delta},\nabla_{\varepsilon}\varphi)}{|\nabla_{\varepsilon}u_{\varepsilon}^{\delta}|^2+\delta^2}, − 2 ( ∣ ∇ ε u ε δ ∣ 2 + δ 2 ) 2 ∇ ε ∇ ε u ε δ ( ∇ ε u ε δ , ∇ ε u ε δ ) ⟨ ∇ ε u ε δ , ∇ ε φ ⟩ + ∣ ∇ ε u ε δ ∣ 2 + δ 2 ∇ ε ∇ ε φ ( ∇ ε u ε δ , ∇ ε u ε δ ) + ∣ ∇ ε u ε δ ∣ 2 + δ 2 2 ∇ ε ∇ ε u ε δ ( ∇ ε u ε δ , ∇ ε φ ) , where we let φ = ∣ ∇ ~ ε u ε δ ∣ 2 \varphi=|\widetilde{\nabla}_{\varepsilon}u_{\varepsilon}^{\delta}|^2 φ = ∣ ∇ ε u ε δ ∣ 2
A = ∣ ∇ ~ ε ( X u ε δ ) ∣ 2 + ∣ ∇ ~ ε ( Y u ε δ ) ∣ 2 + ε 2 ∣ ∇ ~ ε ( Z u ε δ ) ∣ 2 \mathcal{A}=|\widetilde{\nabla}_{\varepsilon}(Xu_{\varepsilon}^{\delta})|^2+|\widetilde{\nabla}_{\varepsilon}(Yu_{\varepsilon}^{\delta})|^2+\varepsilon^2|\widetilde{\nabla}_{\varepsilon}(Zu_{\varepsilon}^{\delta})|^2 A = ∣ ∇ ε ( X u ε δ ) ∣ 2 + ∣ ∇ ε ( Y u ε δ ) ∣ 2 + ε 2 ∣ ∇ ε ( Z u ε δ ) ∣ 2 and
B = ⟨ ∇ ε u ε δ , ∇ ~ ε ( X u ε δ ) ⟩ 2 + ⟨ ∇ ε u ε δ , ∇ ~ ε ( Y u ε δ ) ⟩ 2 + ε 2 ⟨ ∇ ε u ε δ , ∇ ~ ε ( Z u ε δ ) ⟩ 2 . \mathcal{B}=\langle\nabla_{\varepsilon}u_{\varepsilon}^{\delta},\widetilde{\nabla}_{\varepsilon}(Xu_{\varepsilon}^{\delta})\rangle^2+\langle\nabla_{\varepsilon}u_{\varepsilon}^{\delta},\widetilde{\nabla}_{\varepsilon}(Yu_{\varepsilon}^{\delta})\rangle^2+\varepsilon^2\langle\nabla_{\varepsilon}u_{\varepsilon}^{\delta},\widetilde{\nabla}_{\varepsilon}(Zu_{\varepsilon}^{\delta})\rangle^2. B = ⟨ ∇ ε u ε δ , ∇ ε ( X u ε δ ) ⟩ 2 + ⟨ ∇ ε u ε δ , ∇ ε ( Y u ε δ ) ⟩ 2 + ε 2 ⟨ ∇ ε u ε δ , ∇ ε ( Z u ε δ ) ⟩ 2 . We now verify that the positive term 2 A 2\mathcal{A} 2 A B \mathcal{B} B ⟨ v , w ⟩ 2 ≤ ∣ v ∣ 2 ∣ w ∣ 2 \langle v,w\rangle^2\leq|v|^2|w|^2 ⟨ v , w ⟩ 2 ≤ ∣ v ∣ 2 ∣ w ∣ 2 B \mathcal{B} B
⟨ ∇ ε u ε δ , ∇ ~ ε ( V u ε δ ) ⟩ 2 ≤ ∣ ∇ ε u ε δ ∣ 2 ∣ ∇ ~ ε ( V u ε δ ) ∣ 2 , \langle\nabla_{\varepsilon}u_{\varepsilon}^{\delta},\widetilde{\nabla}_{\varepsilon}(Vu_{\varepsilon}^{\delta})\rangle^2\leq|\nabla_{\varepsilon}u_{\varepsilon}^{\delta}|^2|\widetilde{\nabla}_{\varepsilon}(Vu_{\varepsilon}^{\delta})|^2, ⟨ ∇ ε u ε δ , ∇ ε ( V u ε δ ) ⟩ 2 ≤ ∣ ∇ ε u ε δ ∣ 2 ∣ ∇ ε ( V u ε δ ) ∣ 2 , for any vector field V ∈ { X , Y , ε Z } V\in\{X,Y,\varepsilon Z\} V ∈ { X , Y , εZ }
B ≤ ∣ ∇ ε u ε δ ∣ 2 A . \mathcal{B}\leq|\nabla_{\varepsilon}u_{\varepsilon}^{\delta}|^2\mathcal{A}. B ≤ ∣ ∇ ε u ε δ ∣ 2 A . Consequently we obtain the estimate:
2 B ∣ ∇ ε u ε δ ∣ 2 + δ 2 ≤ 2 ∣ ∇ ε u ε δ ∣ 2 ∣ ∇ ε u ε δ ∣ 2 + δ 2 A ≤ 2 A , (66) \frac{2\mathcal{B}}{|\nabla_{\varepsilon}u_{\varepsilon}^{\delta}|^2+\delta^2}\leq\frac{2|\nabla_{\varepsilon}u_{\varepsilon}^{\delta}|^2}{|\nabla_{\varepsilon}u_{\varepsilon}^{\delta}|^2+\delta^2}\mathcal{A}\leq2\mathcal{A}, \tag*{(66)} ∣ ∇ ε u ε δ ∣ 2 + δ 2 2 B ≤ ∣ ∇ ε u ε δ ∣ 2 + δ 2 2∣ ∇ ε u ε δ ∣ 2 A ≤ 2 A , (66) that, coupled with (65) and (66) , results in
L φ = Δ ε φ − ∇ ε ∇ ε φ ( ∇ ε u ε δ , ∇ ε u ε δ ) ∣ ∇ ε u ε δ ∣ 2 + δ 2 + 2 ∇ ε ∇ ε u ε δ ( ∇ ε u ε δ , ∇ ε u ε δ ) ( ∣ ∇ ε u ε δ ∣ 2 + δ 2 ) 2 ⟨ ∇ ε u ε δ , ∇ ε φ ⟩ − 2 ∇ ε ∇ ε u ε δ ( ∇ ε u ε δ , ∇ ε φ ) ∣ ∇ ε u ε δ ∣ 2 + δ 2 ≥ 0. \mathcal{L}\varphi= \Delta_{\varepsilon}\varphi- \frac{\nabla_{\varepsilon}\nabla_{\varepsilon}\varphi(\nabla_{\varepsilon}u_{\varepsilon}^{\delta},\nabla_{\varepsilon}u_{\varepsilon}^{\delta})}{|\nabla_{\varepsilon}u_{\varepsilon}^{\delta}|^{2}+\delta^{2}} + 2\frac{\nabla_{\varepsilon}\nabla_{\varepsilon}u_{\varepsilon}^{\delta}(\nabla_{\varepsilon}u_{\varepsilon}^{\delta},\nabla_{\varepsilon}u_{\varepsilon}^{\delta})}{(|\nabla_{\varepsilon}u_{\varepsilon}^{\delta}|^{2}+\delta^{2})^{2}}\langle\nabla_{\varepsilon}u_{\varepsilon}^{\delta},\nabla_{\varepsilon}\varphi\rangle- \frac{2\nabla_{\varepsilon}\nabla_{\varepsilon}u_{\varepsilon}^{\delta}(\nabla_{\varepsilon}u_{\varepsilon}^{\delta},\nabla_{\varepsilon}\varphi)}{|\nabla_{\varepsilon}u_{\varepsilon}^{\delta}|^{2}+\delta^{2}} \geq0. L φ = Δ ε φ − ∣ ∇ ε u ε δ ∣ 2 + δ 2 ∇ ε ∇ ε φ ( ∇ ε u ε δ , ∇ ε u ε δ ) + 2 ( ∣ ∇ ε u ε δ ∣ 2 + δ 2 ) 2 ∇ ε ∇ ε u ε δ ( ∇ ε u ε δ , ∇ ε u ε δ ) ⟨ ∇ ε u ε δ , ∇ ε φ ⟩ − ∣ ∇ ε u ε δ ∣ 2 + δ 2 2 ∇ ε ∇ ε u ε δ ( ∇ ε u ε δ , ∇ ε φ ) ≥ 0. The operator L \mathcal{L} L (64) by the maximum principle. □
With the above result in mind, the following subsections will be devoted to finding uniform upper bounds for ∇ ~ u ε δ \widetilde{\nabla}u_{\varepsilon}^{\delta} ∇ u ε δ
Letting δ → 0 \delta\to0 δ → 0
In the following statement, we gather various properties of solutions u ε δ u_{\varepsilon}^{\delta} u ε δ ( M C F δ ε ) (\mathrm{MCF}_{\delta}^{\varepsilon}) ( MCF δ ε ) u ε u_{\varepsilon} u ε u ε u_{\varepsilon} u ε ( M C F ε ) (\mathrm{MCF}_{\varepsilon}) ( MCF ε ) g ε g_{\varepsilon} g ε η \eta η T p R 3 T_{p}\mathbb{R}^{3} T p R 3 ∣ η ∣ ≤ 1 |\eta| \leq1 ∣ η ∣ ≤ 1
Theorem 4.3. Let D ⊂ R 3 D \subset\mathbb{R}^{3} D ⊂ R 3 ε > 0 \varepsilon> 0 ε > 0 H ε ( p ) > 0 H_{\varepsilon}(p) > 0 H ε ( p ) > 0 p ∈ ∂ D p \in\partial D p ∈ ∂ D
(i) for any δ > 0 \delta> 0 δ > 0 u ε δ ∈ C ∞ ( D ‾ ) u_{\varepsilon}^{\delta} \in C^{\infty}(\overline{D}) u ε δ ∈ C ∞ ( D ) ( M C F δ ε ) (\mathrm{MCF}_{\delta}^{\varepsilon}) ( MCF δ ε )
(ii) There exists a function u ε ∈ C 0 ( D ‾ ) u_{\varepsilon} \in C^{0}(\overline{D}) u ε ∈ C 0 ( D ) u ε δ → u ε u_{\varepsilon}^{\delta} \to u_{\varepsilon} u ε δ → u ε C 0 ( K ) C^{0}(K) C 0 ( K ) δ → 0 \delta\to0 δ → 0 K ⊂ D K \subset D K ⊂ D
(iii) u ε u_{\varepsilon} u ε ( M C F ε ) (\mathrm{MCF}_{\varepsilon}) ( MCF ε )
(iv) ∣ ∇ ε δ u ε δ ∣ → ∣ ∇ ε u ε ∣ |\nabla_{\varepsilon}^{\delta}u_{\varepsilon}^{\delta}| \to|\nabla_{\varepsilon}u_{\varepsilon}| ∣ ∇ ε δ u ε δ ∣ → ∣ ∇ ε u ε ∣ K ⊂ Reg ( u ε ) K \subset\operatorname{Reg}(u_{\varepsilon}) K ⊂ Reg ( u ε )
Reg ( u ε ) = { p ∈ D ∣ u ε is smooth in p and ∣ ∇ ε u ε ∣ ( p ) ≠ 0 } . \operatorname{Reg}(u_{\varepsilon})=\{p\in D\mid u_{\varepsilon}\text{ is smooth in }p\text{ and }|\nabla_{\varepsilon}u_{\varepsilon}|(p)\ne0\}. Reg ( u ε ) = { p ∈ D ∣ u ε is smooth in p and ∣ ∇ ε u ε ∣ ( p ) = 0 } . (v) There exists a tubular neighborhood U r = { p ∈ D ∣ d ε ( p , ∂ D ) < r } U_{r}=\{p\in D\mid d_{\varepsilon}(p,\partial D)<r\} U r = { p ∈ D ∣ d ε ( p , ∂ D ) < r } u ε ∈ C ∞ ( U r ‾ ) u_{\varepsilon}\in C^{\infty}(\overline{U_{r}}) u ε ∈ C ∞ ( U r ) U r ⊂ Reg ( u ε ) U_{r}\subset\operatorname{Reg}(u_{\varepsilon}) U r ⊂ Reg ( u ε )
Proof. The Euclidean version of items (i), (ii), (iii) and (v) are all worked out in the fundamental paper by Evans-Spruck [18 Section 7.3] . The claimed Riemannian version of (i), (ii) and (iii) follow with minor modifications of [18 Theorem 7.4] once suitable C 1 ( D ‾ ) C^{1}(\overline{D}) C 1 ( D ) δ \delta δ u ε δ u_{\varepsilon}^{\delta} u ε δ [34 Lemma 3.1, Lemma 3.2 and Lemma 3.3] , that in fact deal with more general flows. Property (v) follows from the consistency with the classical mean curvature flow, namely, there exists t ˉ > 0 \bar{t}>0 t ˉ > 0 { u ε = t ˉ } = Σ t \{u_{\varepsilon}=\bar{t}\}=\Sigma_{t} { u ε = t ˉ } = Σ t 0 ≤ t < t ˉ 0\leq t<\bar{t} 0 ≤ t < t ˉ { Σ t } t ∈ [ 0 , t ˉ ) \{\Sigma_{t}\}_{t\in[0,\bar{t})} { Σ t } t ∈ [ 0 , t ˉ ) ∂ D \partial D ∂ D [34 Lemma 4.2] for a precise statement. We focus now on the local smooth convergence on an arbitrary compact set K ⊂ Reg ( u ε ) K\subset\operatorname{Reg}(u_{\varepsilon}) K ⊂ Reg ( u ε ) [38 Theorem 3] . Consider the product manifold D × R D\times\mathbb{R} D × R g ε + d s 2 g_{\varepsilon}+ds^{2} g ε + d s 2 d s 2 ds^{2} d s 2 R \mathbb{R} R U δ , U : D × R → R U_{\delta},U:D\times\mathbb{R}\to\mathbb{R} U δ , U : D × R → R
U δ ( p , s ) = u ε δ ( p ) − δ s , U ( p , s ) = u ε ( p ) . U_{\delta}(p,s)=u_{\varepsilon}^{\delta}(p)-\delta s,\qquad U(p,s)=u_{\varepsilon}(p). U δ ( p , s ) = u ε δ ( p ) − δ s , U ( p , s ) = u ε ( p ) . As observed first in [18 Section 7.1] , the level sets Γ t δ = { U δ = t } \Gamma_{t}^{\delta}=\{U_{\delta}=t\} Γ t δ = { U δ = t } U δ U_{\delta} U δ ( D × R , g ε + d s 2 ) (D\times\mathbb{R},g_{\varepsilon}+ds^{2}) ( D × R , g ε + d s 2 ) U δ U_{\delta} U δ
− Δ U δ + ∇ ∇ U δ ( ∇ U δ , ∇ U δ ) ∣ ∇ U δ ∣ 2 = 1 , (67) -\Delta U_{\delta}+\frac{\nabla\nabla U_{\delta}(\nabla U_{\delta},\nabla U_{\delta})}{|\nabla U_{\delta}|^{2}}=1, \tag*{(67)} − Δ U δ + ∣∇ U δ ∣ 2 ∇∇ U δ ( ∇ U δ , ∇ U δ ) = 1 , (67) where the quantities are computed in terms of the Levi Civita connection of g ε + d s 2 g_{\varepsilon}+ds^{2} g ε + d s 2 [34 Section 5] (see [28 Proposition 2.5] for a precise statement in the setting of flat R n \mathbb{R}^{n} R n { Γ t δ } t ≥ 0 \{\Gamma_{t}^{\delta}\}_{t\geq0} { Γ t δ } t ≥ 0 { Γ t } t ≥ 0 \{\Gamma_{t}\}_{t\geq0} { Γ t } t ≥ 0 Γ t = { u ε = t } × R \Gamma_{t}=\{u_{\varepsilon}=t\}\times\mathbb{R} Γ t = { u ε = t } × R p ∈ Reg ( u ε ) p\in\operatorname{Reg}(u_{\varepsilon}) p ∈ Reg ( u ε ) ( p , s , t ) (p,s,t) ( p , s , t ) Γ t × R × R + \Gamma_{t}\times\mathbb{R}\times\mathbb{R}^{+} Γ t × R × R + D × R × R + D\times\mathbb{R}\times\mathbb{R}^{+} D × R × R + s ∈ R s\in\mathbb{R} s ∈ R [37 Theorem 7.3] , we deduce that there exists a neighborhood V ( p , t ) V_{(p,t)} V ( p , t ) ( p , t ) (p,t) ( p , t ) s ˉ \bar{s} s ˉ Γ τ δ → Γ τ \Gamma_{\tau}^{\delta}\to\Gamma_{\tau} Γ τ δ → Γ τ V ( p , t ) V_{(p,t)} V ( p , t ) Γ τ δ \Gamma_{\tau}^{\delta} Γ τ δ D × R D\times\mathbb{R} D × R (67) , by
H ε δ = 1 ∣ ∇ ε u ε δ ∣ 2 + δ 2 . H_{\varepsilon}^{\delta}=\frac{1}{\sqrt{|\nabla_{\varepsilon}u_{\varepsilon}^{\delta}|^{2}+\delta^{2}}}. H ε δ = ∣ ∇ ε u ε δ ∣ 2 + δ 2 1 . Similarly, the mean curvature of the level set Γ τ \Gamma_{\tau} Γ τ ( MCF ε ) (\operatorname{MCF}_{\varepsilon}) ( MCF ε )
H ε = 1 ∣ ∇ ε u ε ∣ . H_{\varepsilon}=\frac{1}{|\nabla_{\varepsilon}u_{\varepsilon}|}. H ε = ∣ ∇ ε u ε ∣ 1 . We deduce, that, given p ∈ Reg ( u ε ) p\in\operatorname{Reg}(u_{\varepsilon}) p ∈ Reg ( u ε ) V p V_{p} V p p p p ∣ ∇ ε u ε δ ∣ → ∣ ∇ ε u ε ∣ |\nabla_{\varepsilon}u_{\varepsilon}^{\delta}|\to|\nabla_{\varepsilon}u_{\varepsilon}| ∣ ∇ ε u ε δ ∣ → ∣ ∇ ε u ε ∣ V p V_{p} V p Reg ( u ε ) \operatorname{Reg}(u_{\varepsilon}) Reg ( u ε )
Remark 4.4. It has been proved by Colding-Minicozzi [14 Corollary 2.2] that the arrival time function u u u D D D C 1 , 1 C^{1,1} C 1 , 1
Reg ( u ) = { p ∈ D ∣ ∇ u ( p ) ≠ 0 } . \operatorname{Reg}(u)=\{p\in D\mid\nabla u(p)\ne0\}. Reg ( u ) = { p ∈ D ∣ ∇ u ( p ) = 0 } . We do not need this sharper description in our work.
Stable gradient estimates
We derive here suitable gradient bounds for u ε δ u_{\varepsilon}^{\delta} u ε δ ε \varepsilon ε δ \delta δ
We are now ready to establish the uniform gradient estimate for the limit solution u ε u_{\varepsilon} u ε
Proposition 4.5 (Global gradient estimate-axisymmetric case). Let D ⊂ R 3 D\subset\mathbb{R}^{3} D ⊂ R 3 D D D g ε g_{\varepsilon} g ε H ε ( ∂ D ) ≥ c ε > 0 H_{\varepsilon}(\partial D)\geq c_{\varepsilon}>0 H ε ( ∂ D ) ≥ c ε > 0
Let u ε u_{\varepsilon} u ε ( MCF ε ) (\operatorname{MCF}_{\varepsilon}) ( MCF ε ) Theorem 4.3 . Then,
∣ u ε ( p ) − u ε ( q ) ∣ ≤ ( sup ∂ D ∣ ∇ ε u ε ∣ ) d ( p , q ) for all p , q ∈ D ‾ . (68) |u_{\varepsilon}(p)-u_{\varepsilon}(q)|\leq\left(\sup_{\partial D}|\nabla_{\varepsilon}u_{\varepsilon}|\right)d(p,q)\quad\text{for all }p,q\in\overline{D}. \tag*{(68)} ∣ u ε ( p ) − u ε ( q ) ∣ ≤ ( ∂ D sup ∣ ∇ ε u ε ∣ ) d ( p , q ) for all p , q ∈ D . (68) Furthermore, if c ε ≥ c 0 > 0 c_{\varepsilon}\geq c_{0}>0 c ε ≥ c 0 > 0 c 0 c_{0} c 0 ε \varepsilon ε C > 0 C>0 C > 0 ε \varepsilon ε
∣ u ε ( p ) − u ε ( q ) ∣ ≤ 1 c 0 d ( p , q ) for all p , q ∈ D ‾ . (69) |u_{\varepsilon}(p)-u_{\varepsilon}(q)|\leq\frac{1}{c_{0}}d(p,q)\quad\text{for all }p,q\in\overline{D}. \tag*{(69)} ∣ u ε ( p ) − u ε ( q ) ∣ ≤ c 0 1 d ( p , q ) for all p , q ∈ D . (69) Proof. By Theorem 4.3 , (v), equation ( MCF ε ) (\operatorname{MCF}_{\varepsilon}) ( MCF ε ) ∂ D \partial D ∂ D ∂ D \partial D ∂ D
sup ∂ D ∣ ∇ ε u ε ∣ = 1 min ∂ D H ε < 1 c ε . \sup_{\partial D}|\nabla_{\varepsilon}u_{\varepsilon}|=\frac{1}{\min_{\partial D}H_{\varepsilon}}<\frac{1}{c_{\varepsilon}}. ∂ D sup ∣ ∇ ε u ε ∣ = min ∂ D H ε 1 < c ε 1 . By the continuity of ∇ ε u ε \nabla_{\varepsilon}u_{\varepsilon} ∇ ε u ε η ~ > 0 \widetilde{\eta}>0 η > 0
Γ η = { p ∈ D ∣ u ε ( p ) = η } \Gamma_{\eta}=\{p\in D\mid u_{\varepsilon}(p)=\eta\} Γ η = { p ∈ D ∣ u ε ( p ) = η } is disjoint from the critical set for each η < η ~ \eta< \tilde{\eta} η < η ~ ∣ ∇ ε u ε ∣ ≠ 0 |\nabla_{\varepsilon}u_{\varepsilon}| \ne0 ∣ ∇ ε u ε ∣ = 0 Γ η \Gamma_{\eta} Γ η
We consider now the approximators u ε δ u_{\varepsilon}^{\delta} u ε δ η < η ~ \eta< \tilde{\eta} η < η ~ { u ε ≥ η } \{u_{\varepsilon} \ge\eta\} { u ε ≥ η }
sup { u ε ≥ η } ∣ ∇ ε u ε δ ∣ ≤ sup Γ η ∣ ∇ ε u ε δ ∣ . \sup_{\{u_{\varepsilon} \ge\eta\}} |\nabla_{\varepsilon}u_{\varepsilon}^{\delta}| \le\sup_{\Gamma_{\eta}} |\nabla_{\varepsilon}u_{\varepsilon}^{\delta}|. { u ε ≥ η } sup ∣ ∇ ε u ε δ ∣ ≤ Γ η sup ∣ ∇ ε u ε δ ∣. The above inequality implies, letting δ → 0 \delta\to0 δ → 0 Theorem 4.3 , that, for any two points p p p q q q { u ε > η } \{u_{\varepsilon} > \eta\} { u ε > η }
∣ u ε ( p ) − u ε ( q ) ∣ ≤ sup Γ η ∣ ∇ ε u ε ∣ d ε ( p , q ) ≤ sup Γ η ∣ ∇ ε u ε ∣ d ( p , q ) . |u_{\varepsilon}(p)-u_{\varepsilon}(q)| \le\sup_{\Gamma_{\eta}}|\nabla_{\varepsilon}u_{\varepsilon}|d_{\varepsilon}(p,q) \le\sup_{\Gamma_{\eta}}|\nabla_{\varepsilon}u_{\varepsilon}|d(p,q). ∣ u ε ( p ) − u ε ( q ) ∣ ≤ Γ η sup ∣ ∇ ε u ε ∣ d ε ( p , q ) ≤ Γ η sup ∣ ∇ ε u ε ∣ d ( p , q ) . We also used Proposition 2.6 in the second inequality. Letting η → 0 \eta\to0 η → 0 Theorem 4.3 yields (4.6). Using that on ∂ D \partial D ∂ D ∣ ∇ u ε ∣ = H ε − 1 |\nabla u_{\varepsilon}|=H_{\varepsilon}^{-1} ∣∇ u ε ∣ = H ε − 1
Proposition 4.6. Let D ⊂ R 3 D \subset\mathbb{R}^{3} D ⊂ R 3 D D D H ( ∂ D ) ≥ c > 0 H(\partial D) \ge c > 0 H ( ∂ D ) ≥ c > 0 u ε u_{\varepsilon} u ε ε _{\varepsilon} ε D D D C > 0 C > 0 C > 0 ε \varepsilon ε δ \delta δ
∥ ∇ ~ ε u ε δ ∥ L ∞ ( D ) ≤ C . (70) \|\widetilde{\nabla}_{\varepsilon}u_{\varepsilon}^{\delta}\|_{L^{\infty}(D)} \le C. \tag*{(70)} ∥ ∇ ε u ε δ ∥ L ∞ ( D ) ≤ C . (70) Once estimate (4.8) is established, it implies that the family { u ε } \{u_{\varepsilon}\} { u ε } d ~ ε \tilde{d}_{\varepsilon} d ~ ε
Now we can prove the uniform right-invariant Lipschitz estimates of the viscosity solutions { u ε } \{u_{\varepsilon}\} { u ε } ε _{\varepsilon} ε
Proof of Proposition 4.6. By Theorem 4.2, the maximum of the gradient is attained on the boundary. Thus, to conclude the proof, we only need to establish a uniform bound for ∥ ∇ ~ ε u ε δ ∥ L ∞ ( ∂ D ) \|\widetilde{\nabla}_{\varepsilon}u_{\varepsilon}^{\delta}\|_{L^{\infty}(\partial D)} ∥ ∇ ε u ε δ ∥ L ∞ ( ∂ D ) ε \varepsilon ε δ \delta δ
We consider the regularized operator
M δ v = Δ ε v − ∇ ε ∇ ε v ( ∇ ε v , ∇ ε v ) ∣ ∇ ε v ∣ 2 + δ 2 . M_{\delta}v=\Delta_{\varepsilon}v-\frac{\nabla_{\varepsilon}\nabla_{\varepsilon}v(\nabla_{\varepsilon}v,\nabla_{\varepsilon}v)}{|\nabla_{\varepsilon}v|^{2}+\delta^{2}}. M δ v = Δ ε v − ∣ ∇ ε v ∣ 2 + δ 2 ∇ ε ∇ ε v ( ∇ ε v , ∇ ε v ) . By definition, M δ u ε δ = − 1 M_{\delta}u_{\varepsilon}^{\delta}=-1 M δ u ε δ = − 1 d E d_{E} d E ∂ D \partial D ∂ D
d E ( p ) = min q ∈ ∂ D ∣ p − q ∣ , d_{E}(p)=\min_{q\in\partial D}|p-q|, d E ( p ) = q ∈ ∂ D min ∣ p − q ∣ , and let
w = λ g ( d E ) . w=\lambda g(d_{E}). w = λ g ( d E ) . We are going to choose λ > 0 \lambda>0 λ > 0 s 0 > 0 s_{0}>0 s 0 > 0 g = g ( s ) ∈ C 2 ( [ 0 , s 0 ) ) g=g(s)\in C^{2}([0,s_{0})) g = g ( s ) ∈ C 2 ([ 0 , s 0 )) w w w u ε δ u_{\varepsilon}^{\delta} u ε δ ∂ D \partial D ∂ D
A direct computation gives:
∇ ε w = λ g ′ ∇ ε d \nabla_{\varepsilon}w=\lambda g'\nabla_{\varepsilon}d ∇ ε w = λ g ′ ∇ ε d
∇ ε ∇ ε w = λ [ g ′ ∇ ε ∇ ε d + g ′ ′ ∇ ε d ⊗ ∇ ε d ] \nabla_{\varepsilon}\nabla_{\varepsilon}w=\lambda\left[g'\nabla_{\varepsilon}\nabla_{\varepsilon}d+g''\nabla_{\varepsilon}d\otimes\nabla_{\varepsilon}d\right] ∇ ε ∇ ε w = λ [ g ′ ∇ ε ∇ ε d + g ′′ ∇ ε d ⊗ ∇ ε d ]
Δ ε w = λ [ g ′ Δ ε d + g ′ ′ ∣ ∇ ε d ∣ 2 ] \Delta_{\varepsilon}w=\lambda[g'\Delta_{\varepsilon}d+g''|\nabla_{\varepsilon}d|^2] Δ ε w = λ [ g ′ Δ ε d + g ′′ ∣ ∇ ε d ∣ 2 ]
∇ ε ∇ ε w ( ∇ ε w , ∇ ε w ) = λ 3 [ g ′ 3 ∇ ε ∇ ε d ( ∇ ε d , ∇ ε d ) + g ′ ′ g ′ 2 ∣ ∇ ε d ∣ 4 ] \nabla_{\varepsilon}\nabla_{\varepsilon}w(\nabla_{\varepsilon}w,\nabla_{\varepsilon}w)=\lambda^3[g'^3\nabla_{\varepsilon}\nabla_{\varepsilon}d(\nabla_{\varepsilon}d,\nabla_{\varepsilon}d)+g''g'^2|\nabla_{\varepsilon}d|^4] ∇ ε ∇ ε w ( ∇ ε w , ∇ ε w ) = λ 3 [ g ′3 ∇ ε ∇ ε d ( ∇ ε d , ∇ ε d ) + g ′′ g ′2 ∣ ∇ ε d ∣ 4 ]
that yield
M δ w = λ [ g ′ Δ ε d + g ′ ′ ∣ ∇ ε d ∣ 2 ] − λ 3 g ′ 3 ∇ ε ∇ ε d ( ∇ ε d , ∇ ε d ) + g ′ ′ g ′ 2 ∣ ∇ ε d ∣ 4 λ 2 g ′ 2 ∣ ∇ ε d ∣ 2 + δ 2 = λ g ′ [ Δ ε d − λ 2 g ′ 2 ∇ ε ∇ ε d ( ∇ ε d , ∇ ε d ) λ 2 g ′ 2 ∣ ∇ ε d ∣ 2 + δ 2 ] + λ δ 2 g ′ ′ ∣ ∇ ε d ∣ 2 λ 2 g ′ 2 ∣ ∇ ε d ∣ 2 + δ 2 . (71) \begin{aligned}
M\delta w=\lambda[g'\Delta_{\varepsilon}d+g''|\nabla_{\varepsilon}d|^2]-\lambda^3\frac{g'^3\nabla_{\varepsilon}\nabla_{\varepsilon}d(\nabla_{\varepsilon}d,\nabla_{\varepsilon}d)+g''g'^2|\nabla_{\varepsilon}d|^4}{\lambda^2g'^2|\nabla_{\varepsilon}d|^2+\delta^2} \\
&=\lambda g'\left[\Delta_{\varepsilon}d-\lambda^2g'^2\frac{\nabla_{\varepsilon}\nabla_{\varepsilon}d(\nabla_{\varepsilon}d,\nabla_{\varepsilon}d)}{\lambda^2g'^2|\nabla_{\varepsilon}d|^2+\delta^2}\right]+\frac{\lambda\delta^2g''|\nabla_{\varepsilon}d|^2}{\lambda^2g'^2|\nabla_{\varepsilon}d|^2+\delta^2}. \tag*{(71)}
\end{aligned} M δ w = λ [ g ′ Δ ε d + g ′′ ∣ ∇ ε d ∣ 2 ] − λ 3 λ 2 g ′2 ∣ ∇ ε d ∣ 2 + δ 2 g ′3 ∇ ε ∇ ε d ( ∇ ε d , ∇ ε d ) + g ′′ g ′2 ∣ ∇ ε d ∣ 4 = λ g ′ [ Δ ε d − λ 2 g ′2 λ 2 g ′2 ∣ ∇ ε d ∣ 2 + δ 2 ∇ ε ∇ ε d ( ∇ ε d , ∇ ε d ) ] + λ 2 g ′2 ∣ ∇ ε d ∣ 2 + δ 2 λ δ 2 g ′′ ∣ ∇ ε d ∣ 2 . (71) Let us analyze first the term inside the brackets, defining it as F δ F_{\delta} F δ
F δ ( d E ) = Δ ε d E − λ 2 g ′ 2 λ 2 g ′ 2 ∣ ∇ ε d E ∣ ε 2 + δ 2 ∇ ε ∇ ε d E ( ∇ ε d E , ∇ ε d E ) . F_{\delta}(d_E)=\Delta_{\varepsilon}d_E-\frac{\lambda^2g'^2}{\lambda^2g'^2|\nabla_{\varepsilon}d_E|_{\varepsilon}^2+\delta^2}\nabla_{\varepsilon}\nabla_{\varepsilon}d_E(\nabla_{\varepsilon}d_E,\nabla_{\varepsilon}d_E). F δ ( d E ) = Δ ε d E − λ 2 g ′2 ∣ ∇ ε d E ∣ ε 2 + δ 2 λ 2 g ′2 ∇ ε ∇ ε d E ( ∇ ε d E , ∇ ε d E ) . Observe that F 0 ( d E ) ( p ) = − ∣ ∇ ε d E ∣ ( p ) H ε ( p ) F_0(d_E)(p)=-|\nabla_{\varepsilon}d_E|(p)H_{\varepsilon}(p) F 0 ( d E ) ( p ) = − ∣ ∇ ε d E ∣ ( p ) H ε ( p ) H ε ( p ) H_{\varepsilon}(p) H ε ( p ) g ε g_{\varepsilon} g ε { d E = d E ( p ) } \{d_E=d_E(p)\} { d E = d E ( p )} p p p U s = { p ∈ D ‾ ∣ d E ( p ) < s } U_s=\{p\in\overline{D}\mid d_E(p)<s\} U s = { p ∈ D ∣ d E ( p ) < s } d E d_E d E C 2 C^2 C 2 U s ‾ \overline{U_s} U s [24 Lemma 14.16] ). From ∂ D \partial D ∂ D s s s ∣ H ε ∣ ≤ C |H_{\varepsilon}|\le C ∣ H ε ∣ ≤ C C − 1 ≤ ∣ ∇ ε d E ∣ ε ≤ C C^{-1}\le|\nabla_{\varepsilon}d_E|_{\varepsilon}\le C C − 1 ≤ ∣ ∇ ε d E ∣ ε ≤ C C > 1 C>1 C > 1 ε ≤ 1 \varepsilon\le1 ε ≤ 1
Choosing g g g w w w
g ′ ( s ) > a > 0 for 0 ≤ s < s 0 , (72) g'(s)>a>0\quad\text{for}\quad0\le s<s_0, \tag*{(72)} g ′ ( s ) > a > 0 for 0 ≤ s < s 0 , (72) we observe that
∣ F δ ( d E ) − F 0 ( d E ) ∣ = ∣ ∇ ε ∇ ε d E ( ∇ ε d E , ∇ ε d E ) ∣ ∣ ∇ ε d E ∣ ε 2 δ 2 λ 2 g ′ ( d E ) 2 ∣ ∇ ε d E ∣ 2 + δ 2 ≤ C δ 2 |F_{\delta}(d_E)-F_0(d_E)|=\frac{|\nabla_{\varepsilon}\nabla_{\varepsilon}d_E(\nabla_{\varepsilon}d_E,\nabla_{\varepsilon}d_E)|}{|\nabla_{\varepsilon}d_E|_{\varepsilon}^2}\frac{\delta^2}{\lambda^2g'(d_E)^2|\nabla_{\varepsilon}d_E|^2+\delta^2}\le C\delta^2 ∣ F δ ( d E ) − F 0 ( d E ) ∣ = ∣ ∇ ε d E ∣ ε 2 ∣ ∇ ε ∇ ε d E ( ∇ ε d E , ∇ ε d E ) ∣ λ 2 g ′ ( d E ) 2 ∣ ∇ ε d E ∣ 2 + δ 2 δ 2 ≤ C δ 2 on U s 0 U_{s_0} U s 0 C > 0 C>0 C > 0 ε \varepsilon ε λ ≥ 1 \lambda\ge1 λ ≥ 1
This bound on C C C ε ≤ 1 \varepsilon\le1 ε ≤ 1 g ε g_{\varepsilon} g ε { X , Y , ε Z } \{X,Y,\varepsilon Z\} { X , Y , εZ } ∇ ε ∇ ε d E \nabla_{\varepsilon}\nabla_{\varepsilon}d_E ∇ ε ∇ ε d E 1 / ε 1/\varepsilon 1/ ε
∇ ε ∇ ε d E ( X , ε Z ) = ε X ( Z d E ) + 1 2 ε Y d E , ∇ ε ∇ ε d E ( Y , ε Z ) = ε Y ( Z d E ) − 1 2 ε X d E . \nabla_{\varepsilon}\nabla_{\varepsilon}d_E(X,\varepsilon Z)=\varepsilon X(Zd_E)+\frac{1}{2\varepsilon}Yd_E,\qquad
\nabla_{\varepsilon}\nabla_{\varepsilon}d_E(Y,\varepsilon Z)=\varepsilon Y(Zd_E)-\frac{1}{2\varepsilon}Xd_E. ∇ ε ∇ ε d E ( X , εZ ) = εX ( Z d E ) + 2 ε 1 Y d E , ∇ ε ∇ ε d E ( Y , εZ ) = ε Y ( Z d E ) − 2 ε 1 X d E . Contracted twice with ∇ ε d E \nabla_{\varepsilon}d_E ∇ ε d E
1 ε [ ( X d E ) ( Z d E ) ( Y d E ) − ( Y d E ) ( Z d E ) ( X d E ) ] = 0 , \frac{1}{\varepsilon}\left[(Xd_E)(Zd_E)(Yd_E)-(Yd_E)(Zd_E)(Xd_E)\right]=0, ε 1 [ ( X d E ) ( Z d E ) ( Y d E ) − ( Y d E ) ( Z d E ) ( X d E ) ] = 0 , which cancels exactly. The remaining terms involve only Euclidean second derivatives of d E d_E d E U s 0 ‾ \overline{U_{s_0}} U s 0 ε ≤ 1 \varepsilon\le1 ε ≤ 1 ∇ ε ∇ ε d E ( ∇ ε d E , ∇ ε d E ) \nabla_{\varepsilon}\nabla_{\varepsilon}d_E(\nabla_{\varepsilon}d_E,\nabla_{\varepsilon}d_E) ∇ ε ∇ ε d E ( ∇ ε d E , ∇ ε d E ) C C C ε ≤ 1 \varepsilon\le1 ε ≤ 1
Using again the request (72) , we deduce that the first summand in the rightmost hand side of (71) can be bounded by
λ g ′ ( d E ) [ Δ ε d E − λ 2 ( g ′ ( d E ) ) 2 ∇ ε ∇ ε d E ( ∇ ε d E , ∇ ε d E ) λ 2 ( g ′ ( d E ) ) 2 ∣ ∇ ε d E ∣ 2 + δ 2 ] ≤ λ g ′ ( d E ) ( − H ε ∣ ∇ ε d E ∣ + C δ 2 ) . (73) \lambda g'(d_E)\left[\Delta_{\varepsilon}d_E-\lambda^2(g'(d_E))^2\frac{\nabla_{\varepsilon}\nabla_{\varepsilon}d_E(\nabla_{\varepsilon}d_E,\nabla_{\varepsilon}d_E)}{\lambda^2(g'(d_E))^2|\nabla_{\varepsilon}d_E|^2+\delta^2}\right]\le\lambda g'(d_E)(-H_{\varepsilon}|\nabla_{\varepsilon}d_E|+C\delta^2). \tag*{(73)} λ g ′ ( d E ) [ Δ ε d E − λ 2 ( g ′ ( d E ) ) 2 λ 2 ( g ′ ( d E ) ) 2 ∣ ∇ ε d E ∣ 2 + δ 2 ∇ ε ∇ ε d E ( ∇ ε d E , ∇ ε d E ) ] ≤ λ g ′ ( d E ) ( − H ε ∣ ∇ ε d E ∣ + C δ 2 ) . (73) Observe that, by the strict mean convexity of ∂ D \partial D ∂ D s 0 s_0 s 0 H ε ( p ) > 2 c > 0 H_{\varepsilon}(p)>2c>0 H ε ( p ) > 2 c > 0 p ∈ U s 0 ‾ p\in\overline{U_{s_0}} p ∈ U s 0 δ 0 > 0 \delta_0>0 δ 0 > 0 − c < 0 -c<0 − c < 0 0 < δ ≤ δ 0 0<\delta\le\delta_0 0 < δ ≤ δ 0 p ∈ U s 0 p\in U_{s_0} p ∈ U s 0 δ \delta δ (73) yields
λ g ′ ( d E ) [ Δ ε d E − λ 2 ( g ′ ( d E ) ) 2 ∇ ε ∇ ε d E ( ∇ ε d E , ∇ ε d E ) λ 2 ( g ′ ( d E ) ) 2 ∣ ∇ ε d E ∣ 2 + δ 2 ] ≤ − C λ (74) \lambda g'(d_E)\left[\Delta_{\varepsilon}d_E-\lambda^2(g'(d_E))^2\frac{\nabla_{\varepsilon}\nabla_{\varepsilon}d_E(\nabla_{\varepsilon}d_E,\nabla_{\varepsilon}d_E)}{\lambda^2(g'(d_E))^2|\nabla_{\varepsilon}d_E|^2+\delta^2}\right]\le-C\lambda \tag*{(74)} λ g ′ ( d E ) [ Δ ε d E − λ 2 ( g ′ ( d E ) ) 2 λ 2 ( g ′ ( d E ) ) 2 ∣ ∇ ε d E ∣ 2 + δ 2 ∇ ε ∇ ε d E ( ∇ ε d E , ∇ ε d E ) ] ≤ − C λ (74) for any λ ≥ 1 \lambda\ge1 λ ≥ 1 C > 0 C>0 C > 0 ε \varepsilon ε
The second term in (71) can be bounded assuming g ′ ′ > 0 g''>0 g ′′ > 0
λ δ 2 g ′ ′ ∣ ∇ ε d ∣ 2 λ 2 g ′ 2 ∣ ∇ ε d ∣ 2 + δ 2 ≤ λ δ 2 g ′ ′ ∣ ∇ ε d ∣ 2 λ 2 g ′ 2 ∣ ∇ ε d ∣ 2 = δ 2 g ′ ′ λ g ′ 2 . (75) \frac{\lambda\delta^2 g''|\nabla\varepsilon d|^2}{\lambda^2 g'^2|\nabla\varepsilon d|^2+\delta^2}\leq\frac{\lambda\delta^2 g''|\nabla\varepsilon d|^2}{\lambda^2 g'^2|\nabla\varepsilon d|^2}=\frac{\delta^2g''}{\lambda g'^2}. \tag*{(75)} λ 2 g ′2 ∣∇ ε d ∣ 2 + δ 2 λ δ 2 g ′′ ∣∇ ε d ∣ 2 ≤ λ 2 g ′2 ∣∇ ε d ∣ 2 λ δ 2 g ′′ ∣∇ ε d ∣ 2 = λ g ′2 δ 2 g ′′ . (75) So, choosing g g g
g ′ ′ = ( g ′ ) 2 , (76) g''=(g')^2, \tag*{(76)} g ′′ = ( g ′ ) 2 , (76) we obtain, by plugging (74) and (75) into (71)
M δ w < C ( 1 λ − λ ) M_\delta w<C\left(\frac{1}{\lambda}-\lambda\right) M δ w < C ( λ 1 − λ ) for any 0 < δ ≤ δ 0 < 1 0<\delta\leq\delta_0<1 0 < δ ≤ δ 0 < 1 λ ≥ 1 \lambda\geq1 λ ≥ 1 C C C δ \delta δ ε \varepsilon ε λ \lambda λ
M δ w < − 1 = M δ u ε δ in { 0 < d < s 0 } . (77) M_\delta w<-1=M_\delta u^\delta_\varepsilon\quad\text{in }\{0<d<s_0\}. \tag*{(77)} M δ w < − 1 = M δ u ε δ in { 0 < d < s 0 } . (77) Finally, we request
g = 0 on ∂ D , lim s → s 0 g ( s ) → + ∞ . (78) g=0\quad\text{on }\partial D,\qquad\lim_{s\to s_0}g(s)\to+\infty. \tag*{(78)} g = 0 on ∂ D , s → s 0 lim g ( s ) → + ∞. (78) Thus, by (77) and (78) we can apply the comparison principle for M δ M_\delta M δ [19 Proposition 4.3] ) in U ‾ s \overline{U}_s U s s s s s 0 s_0 s 0
u ε δ ≤ w in U ‾ s . u^\delta_\varepsilon\leq w\quad\text{in }\overline{U}_s. u ε δ ≤ w in U s . An explicit function g g g (72) , (76) , and (78) is g : [ 0 , s 0 ) → R g:[0,s_0)\to\mathbb{R} g : [ 0 , s 0 ) → R
g ( s ) = log ( s 0 s 0 − s ) . g(s)=\log\left(\frac{s_0}{s_0-s}\right). g ( s ) = log ( s 0 − s s 0 ) . To evaluate the normal derivative at the boundary, consider an inward Euclidean normal curve γ ( t ) \gamma(t) γ ( t ) p ∈ ∂ D p\in\partial D p ∈ ∂ D d E ( γ ( t ) ) = t d_E(\gamma(t))=t d E ( γ ( t )) = t u ε δ ( p ) = w ( p ) = 0 u^\delta_\varepsilon(p)=w(p)=0 u ε δ ( p ) = w ( p ) = 0
u ε δ ( γ ( t ) ) − u ε δ ( γ ( 0 ) ) t ≤ w ( γ ( t ) ) − w ( γ ( 0 ) ) t = λ g ( t ) − g ( 0 ) t . \frac{u^\delta_\varepsilon(\gamma(t))-u^\delta_\varepsilon(\gamma(0))}{t}\leq\frac{w(\gamma(t))-w(\gamma(0))}{t}=\lambda\frac{g(t)-g(0)}{t}. t u ε δ ( γ ( t )) − u ε δ ( γ ( 0 )) ≤ t w ( γ ( t )) − w ( γ ( 0 )) = λ t g ( t ) − g ( 0 ) . Taking the limit as t → 0 + t\to0^+ t → 0 + ∂ ν E u ε δ ( p ) ≤ λ g ′ ( 0 ) = λ / s 0 \partial_{\nu_E}u^\delta_\varepsilon(p)\leq\lambda g'(0)=\lambda/s_0 ∂ ν E u ε δ ( p ) ≤ λ g ′ ( 0 ) = λ / s 0 u ε δ > 0 u^\delta_\varepsilon>0 u ε δ > 0 D D D u ε δ ( γ ( 0 ) ) = 0 u^\delta_\varepsilon(\gamma(0))=0 u ε δ ( γ ( 0 )) = 0 ≥ 0 \geq0 ≥ 0 t → 0 + t\to0^+ t → 0 +
∣ ∂ ν E u ε δ ( p ) ∣ ≤ λ s 0 . \left|\partial_{\nu_E}u^\delta_\varepsilon(p)\right|\leq\frac{\lambda}{s_0}. ∂ ν E u ε δ ( p ) ≤ s 0 λ . Because u ε δ u^\delta_\varepsilon u ε δ ∂ D \partial D ∂ D ∇ ~ u ε δ ( p ) = ( ∂ ν E u ε δ ( p ) ) ∇ ~ d E ( p ) \widetilde{\nabla}u^\delta_\varepsilon(p)=(\partial_{\nu_E}u^\delta_\varepsilon(p))\widetilde{\nabla}d_E(p) ∇ u ε δ ( p ) = ( ∂ ν E u ε δ ( p )) ∇ d E ( p ) ∂ D \partial D ∂ D X ~ , Y ~ , Z ~ \widetilde{X},\widetilde{Y},\widetilde{Z} X , Y , Z ∣ ∇ ~ d E ∣ g ~ ε ≤ C 0 |\widetilde{\nabla}d_E|_{\widetilde{g}_\varepsilon}\leq C_0 ∣ ∇ d E ∣ g ε ≤ C 0 ∂ D \partial D ∂ D C 0 > 0 C_0>0 C 0 > 0 ε ≤ 1 \varepsilon\leq1 ε ≤ 1
∥ ∇ ~ u ε δ ∥ L ∞ ( ∂ D ) ≤ λ s 0 C 0 = : C , \|\widetilde{\nabla}u^\delta_\varepsilon\|_{L^\infty(\partial D)}\leq\frac{\lambda}{s_0}C_0=:C, ∥ ∇ u ε δ ∥ L ∞ ( ∂ D ) ≤ s 0 λ C 0 =: C , completing the proof. □
Proposition 4.6 , (ii) in Theorem 4.3 and Proposition 2.6 immediately imply the following.
Corollary 4.7 (Right-invariant Lipschitz bound for the viscosity solution). Let u ε u_\varepsilon u ε ε _\varepsilon ε D D D u ε u_\varepsilon u ε
∣ u ε ( p ) − u ε ( q ) ∣ ≤ C d ~ ( p , q ) ∀ p , q ∈ D ‾ , |u_\varepsilon(p)-u_\varepsilon(q)|\leq C\widetilde{d}(p,q)\quad\forall p,q\in\overline{D}, ∣ u ε ( p ) − u ε ( q ) ∣ ≤ C d ( p , q ) ∀ p , q ∈ D , where C C C Proposition 4.6 , independent of ε \varepsilon ε
Letting ε → 0 \varepsilon\to0 ε → 0
Having established the uniform gradient estimates in both the axisymmetric and the non-characteristic scenarios, we are now in the position to prove the global convergence of the Riemannian solutions to suitable sub-Riemannian limit solutions.
The following result holds in general for converging solutions to the Riemannian Heisenberg mean-convex mean curvature flow, and substantially follows [18 ] and [9 ] .
Proposition 4.8. Let D ⊂ R 3 D \subset\mathbb{R}^{3} D ⊂ R 3 C 2 C^{2} C 2 ε > 0 \varepsilon> 0 ε > 0 u ε u_\varepsilon u ε
− Δ ε u ε + ∇ ε ∇ ε u ε ( ∇ ε u ε , ∇ ε u ε ) ∣ ∇ ε u ε ∣ 2 = 1 -\Delta_\varepsilon u_\varepsilon+\frac{\nabla_\varepsilon\nabla_\varepsilon u_\varepsilon(\nabla_\varepsilon u_\varepsilon,\nabla_\varepsilon u_\varepsilon)}{|\nabla_\varepsilon u_\varepsilon|^{2}}=1 − Δ ε u ε + ∣ ∇ ε u ε ∣ 2 ∇ ε ∇ ε u ε ( ∇ ε u ε , ∇ ε u ε ) = 1 in D D D u ε → u u_\varepsilon\to u u ε → u ε → 0 + \varepsilon\to0^{+} ε → 0 + u ∈ C 0 ( D ) u \in C^{0}(D) u ∈ C 0 ( D ) u u u (4) in the viscosity sense given by Definition 2.8.
Proof. We assume for notational simplicity that the whole sequence u ε → u u_\varepsilon\to u u ε → u u u u Φ ∈ C 2 ( D ) \Phi\in C^{2}(D) Φ ∈ C 2 ( D ) u − Φ u-\Phi u − Φ p ∈ D p \in D p ∈ D u ε → u u_\varepsilon\to u u ε → u { p ε } ⊂ D \{p_\varepsilon\} \subset D { p ε } ⊂ D u ε − Φ u_\varepsilon-\Phi u ε − Φ p ε p_\varepsilon p ε p ε → p p_\varepsilon\to p p ε → p ε → 0 + \varepsilon\to0^{+} ε → 0 + [15 Lemma 6.1] ). Assume first that ∇ Φ ( p ) ≠ 0 \nabla\Phi(p) \ne0 ∇Φ ( p ) = 0 ε \varepsilon ε ∇ ε Φ ( p ε ) ≠ 0 \nabla_\varepsilon\Phi(p_\varepsilon) \ne0 ∇ ε Φ ( p ε ) = 0 u ε u_\varepsilon u ε
1 ≥ − Δ ε Φ + ∇ ε ∇ ε Φ ( ∇ ε Φ , ∇ ε Φ ) ∣ ∇ ε Φ ∣ 2 at p ε , (79) 1 \ge-\Delta_\varepsilon\Phi+\frac{\nabla_\varepsilon\nabla_\varepsilon\Phi(\nabla_\varepsilon\Phi,\nabla_\varepsilon\Phi)}{|\nabla_\varepsilon\Phi|^{2}}\quad\text{at }p_\varepsilon, \tag*{(79)} 1 ≥ − Δ ε Φ + ∣ ∇ ε Φ ∣ 2 ∇ ε ∇ ε Φ ( ∇ ε Φ , ∇ ε Φ ) at p ε , (79) that passed to the limit as ε → 0 + \varepsilon\to0^{+} ε → 0 +
1 ≥ − Δ Φ ( p ) + ∇ ∇ Φ ( ∇ Φ , ∇ Φ ) ∣ ∇ Φ ∣ 2 ( p ) . 1 \ge-\Delta\Phi(p)+\frac{\nabla\nabla\Phi(\nabla\Phi,\nabla\Phi)}{|\nabla\Phi|^{2}}(p). 1 ≥ − ΔΦ ( p ) + ∣∇Φ ∣ 2 ∇∇Φ ( ∇Φ , ∇Φ ) ( p ) . On the other hand, assume ∇ Φ ( p ) = 0 \nabla\Phi(p)=0 ∇Φ ( p ) = 0 η \eta η (21) is satisfied. Up taking a further subsequence, we can suppose that either ∇ ε Φ ( p ε ) = 0 \nabla_\varepsilon\Phi(p_\varepsilon)=0 ∇ ε Φ ( p ε ) = 0 ε > 0 \varepsilon>0 ε > 0 ∇ ε Φ ( p ε ) ≠ 0 \nabla_\varepsilon\Phi(p_\varepsilon)\ne0 ∇ ε Φ ( p ε ) = 0 ε > 0 \varepsilon>0 ε > 0
∇ ε ∇ ε Φ = ( X 2 Φ X Y Φ − 1 2 Z Φ ε X Z Φ + 1 2 ε Y Φ Y X Φ + 1 2 Z Φ Y 2 Φ ε Y Z Φ − 1 2 ε X Φ ε Z X Φ + 1 2 ε Y Φ ε Z Y Φ − 1 2 ε X Φ ε 2 Z 2 Φ ) \nabla_\varepsilon\nabla_\varepsilon\Phi=
\begin{pmatrix}
X^{2}\Phi& XY\Phi-\frac{1}{2}Z\Phi& \varepsilon XZ\Phi+\frac{1}{2\varepsilon}Y\Phi\\
YX\Phi+\frac{1}{2}Z\Phi& Y^{2}\Phi& \varepsilon YZ\Phi-\frac{1}{2\varepsilon}X\Phi\\
\varepsilon ZX\Phi+\frac{1}{2\varepsilon}Y\Phi& \varepsilon ZY\Phi-\frac{1}{2\varepsilon}X\Phi& \varepsilon^{2}Z^{2}\Phi
\end{pmatrix} ∇ ε ∇ ε Φ = X 2 Φ Y X Φ + 2 1 Z Φ εZ X Φ + 2 ε 1 Y Φ X Y Φ − 2 1 Z Φ Y 2 Φ εZ Y Φ − 2 ε 1 X Φ εX Z Φ + 2 ε 1 Y Φ ε Y Z Φ − 2 ε 1 X Φ ε 2 Z 2 Φ with respect to the orthonormal frame X , Y , ε Z X,Y,\varepsilon Z X , Y , εZ
Case 1. ∇ ε Φ ( p ε ) = 0 \nabla_\varepsilon\Phi(p_\varepsilon)=0 ∇ ε Φ ( p ε ) = 0 ε > 0 \varepsilon>0 ε > 0 u ε u_\varepsilon u ε η ε \eta_\varepsilon η ε ∣ η ε ∣ ≤ 1 |\eta_\varepsilon|\le1 ∣ η ε ∣ ≤ 1
1 ≥ − Δ ε Φ + ∇ ε ∇ ε Φ ( η ε , η ε ) at p ε , 1 \ge-\Delta_\varepsilon\Phi+\nabla_\varepsilon\nabla_\varepsilon\Phi(\eta_\varepsilon,\eta_\varepsilon)\quad\text{at }p_\varepsilon, 1 ≥ − Δ ε Φ + ∇ ε ∇ ε Φ ( η ε , η ε ) at p ε , Observe that ∣ v ∣ 1 ≤ ∣ v ∣ ε |v|_{1}\le|v|_{\varepsilon} ∣ v ∣ 1 ≤ ∣ v ∣ ε v ∈ T p R 3 v\in T_{p}\mathbb{R}^{3} v ∈ T p R 3 p ∈ R 3 p\in\mathbb{R}^{3} p ∈ R 3 η ε → η ~ \eta_\varepsilon\to\widetilde{\eta} η ε → η η ~ \widetilde{\eta} η ∣ η ~ ∣ 1 ≤ 1 |\widetilde{\eta}|_{1}\le1 ∣ η ∣ 1 ≤ 1 η \eta η g 1 g_{1} g 1 η ~ \widetilde{\eta} η D \mathcal{D} D η ∈ D \eta\in\mathcal{D} η ∈ D ∣ η ∣ ≤ ∣ η ∣ 1 ≤ 1 |\eta|\le|\eta|_{1}\le1 ∣ η ∣ ≤ ∣ η ∣ 1 ≤ 1 X Φ ( p ε ) = Y Φ ( p ε ) = Z Φ ( p ε ) = 0 X\Phi(p_\varepsilon)=Y\Phi(p_\varepsilon)=Z\Phi(p_\varepsilon)=0 X Φ ( p ε ) = Y Φ ( p ε ) = Z Φ ( p ε ) = 0
lim ε → 0 ∇ ε ∇ ε Φ ( p ε ) ( η ε , η ε ) = ∇ ∇ Φ ( p ) ( η , η ) , (80) \lim_{\varepsilon\to0}\nabla_\varepsilon\nabla_\varepsilon\Phi(p_\varepsilon)(\eta_\varepsilon,\eta_\varepsilon)=\nabla\nabla\Phi(p)(\eta,\eta), \tag*{(80)} ε → 0 lim ∇ ε ∇ ε Φ ( p ε ) ( η ε , η ε ) = ∇∇Φ ( p ) ( η , η ) , (80) and this implies that (21) is satisfied for such η \eta η Case 2. ∇ ε Φ ( p ε ) ≠ 0 \nabla_{\varepsilon}\Phi(p_{\varepsilon}) \ne0 ∇ ε Φ ( p ε ) = 0 ε > 0 \varepsilon> 0 ε > 0 η ε : = ∇ ε Φ / ∣ ∇ ε Φ ∣ \eta_{\varepsilon} := \nabla_{\varepsilon}\Phi/|\nabla_{\varepsilon}\Phi| η ε := ∇ ε Φ/∣ ∇ ε Φ∣ p ε p_{\varepsilon} p ε η \eta η g 1 g_{1} g 1 η ~ \widetilde{\eta} η η ε \eta_{\varepsilon} η ε T ε = ∇ ε ∇ ε Φ T^{\varepsilon} = \nabla_{\varepsilon}\nabla_{\varepsilon}\Phi T ε = ∇ ε ∇ ε Φ p ε p_{\varepsilon} p ε T = ∇ ∇ Φ T = \nabla\nabla\Phi T = ∇∇Φ T ε ( η ε , η ε ) T^{\varepsilon}(\eta_{\varepsilon},\eta_{\varepsilon}) T ε ( η ε , η ε ) T ( η , η ) T(\eta,\eta) T ( η , η ) i , j = 1 , 2 i,j = 1,2 i , j = 1 , 2
T ε ( η ε , η ε ) = ( X 2 Φ X Y Φ ε X Z Φ Y X Φ Y 2 Φ ε Y Z Φ ε Z X Φ ε Z Y Φ ε 2 Z 2 Φ ) ( η ε , η ε ) . T^{\varepsilon}(\eta_{\varepsilon},\eta_{\varepsilon}) =
\begin{pmatrix}
X^{2}\Phi& XY\Phi& \varepsilon XZ\Phi\\
YX\Phi& Y^{2}\Phi& \varepsilon YZ\Phi\\
\varepsilon ZX\Phi& \varepsilon ZY\Phi& \varepsilon^{2}Z^{2}\Phi
\end{pmatrix}
(\eta_{\varepsilon},\eta_{\varepsilon}). T ε ( η ε , η ε ) = X 2 Φ Y X Φ εZ X Φ X Y Φ Y 2 Φ εZ Y Φ εX Z Φ ε Y Z Φ ε 2 Z 2 Φ ( η ε , η ε ) . This implies that (4.18) again holds for this choice of η ε \eta_{\varepsilon} η ε
This concludes the proof in the case where p p p p p p u − Φ u-\Phi u − Φ Φ ~ = Φ + ψ \widetilde{\Phi} = \Phi+\psi Φ = Φ + ψ ψ ( q ) \psi(q) ψ ( q ) ψ ( q ) = ∣ p − 1 q ∣ 4 \psi(q)=|p^{-1}q|^{4} ψ ( q ) = ∣ p − 1 q ∣ 4 p p p u − Φ ~ u-\widetilde{\Phi} u − Φ ψ \psi ψ p p p
∇ ψ ( p ) = 0 and ∇ ∇ ψ ( p ) = 0 , (81) \nabla\psi(p)=0 \quad\text{and} \quad\nabla\nabla\psi(p)=0, \tag*{(81)} ∇ ψ ( p ) = 0 and ∇∇ ψ ( p ) = 0 , (81) Moreover, it also holds that
Δ ψ ( p ) = 0. (82) \Delta\psi(p)=0. \tag*{(82)} Δ ψ ( p ) = 0. (82) Using that (2.9) or (2.10) is satisfied for Φ ~ \widetilde{\Phi} Φ Φ \Phi Φ u u u u u u
We are finally in a position to prove our main Theorem 1.1.
Proof of Theorem 1.1. Let Ω ε \Omega_{\varepsilon} Ω ε Ω ε \Omega_{\varepsilon} Ω ε ( A ) (A) ( A ) Ω ε = Ω \Omega_{\varepsilon}=\Omega Ω ε = Ω ( B ) (B) ( B ) Ω ε \Omega_{\varepsilon} Ω ε { Ω ε i } i \{\Omega_{\varepsilon}^{i}\}_{i} { Ω ε i } i Ω ε i → Ω ε \Omega_{\varepsilon}^{i}\to\Omega_{\varepsilon} Ω ε i → Ω ε C 2 C^{2} C 2 i → + ∞ i\to+\infty i → + ∞ Ω ε i \Omega_{\varepsilon}^{i} Ω ε i ( A ) (A) ( A ) ∂ Ω ε \partial\Omega_{\varepsilon} ∂ Ω ε H ε i ≥ c > 0 H_{\varepsilon}^{i}\ge c>0 H ε i ≥ c > 0 c c c ε , i \varepsilon,i ε , i ( B ) (B) ( B ) K ⊂ Ω K\subset\Omega K ⊂ Ω ε > 0 \varepsilon>0 ε > 0 i i i K ⊂ Ω ε i K\subset\Omega_{\varepsilon}^{i} K ⊂ Ω ε i ∂ Ω ε → ∂ Ω \partial\Omega_{\varepsilon}\to\partial\Omega ∂ Ω ε → ∂ Ω u ε i u_{\varepsilon}^{i} u ε i Ω ε i \Omega_{\varepsilon}^{i} Ω ε i i , ε i,\varepsilon i , ε i → + ∞ i\to+\infty i → + ∞ u ε u_{\varepsilon} u ε ε \varepsilon ε K K K K K K u ε u_{\varepsilon} u ε u u u C l o c 0 ( Ω ) C^{0}_{\mathrm{loc}}(\Omega) C loc 0 ( Ω ) u u u p ∈ Ω p\in\Omega p ∈ Ω q ε ∈ ∂ Ω ε q_{\varepsilon}\in\partial\Omega_{\varepsilon} q ε ∈ ∂ Ω ε ∣ u ε ( p ) ∣ ≤ C d ∗ ( p , q ε ) |u_{\varepsilon}(p)|\le Cd^{*}(p,q_{\varepsilon}) ∣ u ε ( p ) ∣ ≤ C d ∗ ( p , q ε ) d ∗ d^{*} d ∗ d d d d ~ \widetilde{d} d ( A ) (A) ( A ) ( B ) (B) ( B ) u ε → u u_{\varepsilon}\to u u ε → u Ω \Omega Ω ∂ Ω ε → ∂ Ω \partial\Omega_{\varepsilon}\to\partial\Omega ∂ Ω ε → ∂ Ω ∣ u ( p ) ∣ ≤ C d ∗ ( p , q ) |u(p)|\le Cd^{*}(p,q) ∣ u ( p ) ∣ ≤ C d ∗ ( p , q ) p ∈ Ω p\in\Omega p ∈ Ω q ∈ ∂ Ω q\in\partial\Omega q ∈ ∂ Ω u u u ∂ Ω \partial\Omega ∂ Ω
Mean cuvature blow-up at characteristic points
In this section we will prove results in turn leading to Theorem 1.4. In fact, we are showing that any time a horizontal mean curvature flow forms a foliation that is, roughly speaking, C 2 C^{2} C 2 u u u H H H C 2 C^{2} C 2 u u u
This blow-up is the key obstruction to self-similarity: a self-similar solution reproduces rescaled copies of ∂ Ω \partial\Omega ∂ Ω
Generalized C 2 C^{2} C 2
The following generalized notion of C 2 C^{2} C 2
Definition 5.1 (Generalized C 2 C^{2} C 2 F = E 2 ∖ E 1 ‾ ⊂ R 3 F=E_{2}\setminus\overline{E_{1}}\subset\mathbb{R}^{3} F = E 2 ∖ E 1 ⊂ R 3 E 1 E_{1} E 1 E 2 E_{2} E 2 C 2 C^{2} C 2 E 1 ‾ ⊂ E 2 \overline{E_{1}}\subset E_{2} E 1 ⊂ E 2 { Σ λ } λ ∈ Λ \{\Sigma_{\lambda}\}_{\lambda\in\Lambda} { Σ λ } λ ∈ Λ F ‾ \overline{F} F generalized C 2 C^{2} C 2 of F ‾ \overline{F} F
R : F ‾ ⟶ [ 0 , ∞ ) R:\overline{F}\longrightarrow[0,\infty) R : F ⟶ [ 0 , ∞ ) such that:
(i) Λ = [ λ min , λ max ] \Lambda=[\lambda_{\min},\lambda_{\max}] Λ = [ λ m i n , λ m a x ] 0 ≤ λ min < λ max 0\leq\lambda_{\min}<\lambda_{\max} 0 ≤ λ m i n < λ m a x ∂ E 1 = Σ λ min , ∂ E 2 = Σ λ max \partial E_{1}=\Sigma_{\lambda_{\min}},\partial E_{2}=\Sigma_{\lambda_{\max}} ∂ E 1 = Σ λ m i n , ∂ E 2 = Σ λ m a x
(ii) for every p ∈ F ‾ p\in\overline{F} p ∈ F λ ∈ Λ \lambda\in\Lambda λ ∈ Λ p ∈ Σ λ = { p ∈ F ‾ : R ( p ) = λ } p\in\Sigma_{\lambda}=\{p\in\overline{F}:R(p)=\lambda\} p ∈ Σ λ = { p ∈ F : R ( p ) = λ }
(iii) there exists an open, dense set A ⊆ F A\subseteq F A ⊆ F R ∈ C 2 ( A ) R\in C^{2}(A) R ∈ C 2 ( A ) ∇ e u c l R ≠ 0 \nabla_{\mathrm{eucl}}R\ne0 ∇ eucl R = 0 A A A
A ∩ Σ λ ≠ ∅ for every λ ∈ Λ with Σ λ ⊂ F ; A\cap\Sigma_{\lambda}\ne\varnothing\qquad\text{for every }\lambda\in\Lambda\text{ with }\Sigma_{\lambda}\subset F; A ∩ Σ λ = ∅ for every λ ∈ Λ with Σ λ ⊂ F ; (iv) setting F λ = { p ∈ F ‾ : R ( p ) < λ } F_{\lambda}=\{p\in\overline{F}:R(p)<\lambda\} F λ = { p ∈ F : R ( p ) < λ }
Σ λ = ∂ ( E 1 ∪ F λ ) \Sigma_{\lambda}=\partial(E_{1}\cup F_{\lambda}) Σ λ = ∂ ( E 1 ∪ F λ ) Observe that a classical C 2 C^{2} C 2 A = F A=F A = F
We find convenient to isolate the following geometrically meaningful quantities.
Definition 5.2 (Dilation generator and transversal set). Let D ( x , y , z ) = ( x , y , 2 z ) D(x,y,z)=(x,y,2z) D ( x , y , z ) = ( x , y , 2 z ) δ λ \delta_{\lambda} δ λ D ( p ) = d d λ ∣ λ = 1 δ λ ( p ) D(p)=\left.\frac{d}{d\lambda}\right|_{\lambda=1}\delta_{\lambda}(p) D ( p ) = d λ d λ = 1 δ λ ( p ) Ω ⊂ R 3 \Omega\subset\mathbb{R}^{3} Ω ⊂ R 3 C 1 C^{1} C 1 ν e u c l \nu_{\mathrm{eucl}} ν eucl ∂ Ω \partial\Omega ∂ Ω
G = { p ∈ ∂ Ω ∣ ⟨ D ( p ) , ν e u c l ( p ) ⟩ ≠ 0 } . G=\{p\in\partial\Omega\mid\langle D(p),\nu_{\mathrm{eucl}}(p)\rangle\ne0\}. G = { p ∈ ∂ Ω ∣ ⟨ D ( p ) , ν eucl ( p )⟩ = 0 } . The motivation behind the introduction of the above weakened version lies in the following observation.
Lemma 5.3. Let Ω ⊂ R 3 \Omega\subset\mathbb{R}^{3} Ω ⊂ R 3 C 2 C^{2} C 2 ∂ Ω \partial\Omega ∂ Ω λ ∈ ( 0 , 1 ] \lambda\in(0,1] λ ∈ ( 0 , 1 ]
Σ λ = δ λ ( ∂ Ω ) , F λ = δ λ ( Ω ) , \Sigma_{\lambda}=\delta_{\lambda}(\partial\Omega),\qquad F_{\lambda}=\delta_{\lambda}(\Omega), Σ λ = δ λ ( ∂ Ω ) , F λ = δ λ ( Ω ) , and assume
F λ 1 ‾ ⊂ F λ 2 for every 0 < λ 1 < λ 2 ≤ 1. \overline{F_{\lambda_{1}}}\subset F_{\lambda_{2}}\qquad\text{for every }0<\lambda_{1}<\lambda_{2}\leq1. F λ 1 ⊂ F λ 2 for every 0 < λ 1 < λ 2 ≤ 1. Then, for any 0 < λ ‾ < 1 0<\overline{\lambda}<1 0 < λ < 1 { Σ λ } λ ∈ [ λ ‾ , 1 ] \{\Sigma_{\lambda}\}_{\lambda\in[\overline{\lambda},1]} { Σ λ } λ ∈ [ λ , 1 ] C 2 C^{2} C 2 Ω ‾ ∖ F λ ‾ \overline{\Omega}\setminus F_{\overline{\lambda}} Ω ∖ F λ
A = ⋃ λ ∈ ( λ ‾ , 1 ) δ λ ( G ) . (83) A=\bigcup_{\lambda\in(\overline{\lambda},1)}\delta_{\lambda}(G). \tag*{(83)} A = λ ∈ ( λ , 1 ) ⋃ δ λ ( G ) . (83) Proof. The assumed nesting, condition (iv) is satisfied.
Step 1 Here we show properties (i) and (ii) of Definition 5.1, Observe first that
⋃ λ ∈ [ λ ‾ , 1 ] Σ λ = Ω ‾ ∖ F λ ‾ ‾ . \bigcup_{\lambda\in[\overline{\lambda},1]}\Sigma_{\lambda}=\overline{\Omega}\setminus\overline{F_{\overline{\lambda}}}. λ ∈ [ λ , 1 ] ⋃ Σ λ = Ω ∖ F λ . By the strict nesting F ‾ λ 1 ⊂ F λ 2 \overline{F}_{\lambda_1} \subset F_{\lambda_2} F λ 1 ⊂ F λ 2 λ 1 < λ 2 \lambda_1 < \lambda_2 λ 1 < λ 2 Σ λ = ∂ F λ \Sigma_\lambda= \partial F_\lambda Σ λ = ∂ F λ p ∈ Ω ‾ ∖ F λ ‾ p \in\overline{\Omega} \setminus F_{\overline{\lambda}} p ∈ Ω ∖ F λ Σ λ \Sigma_\lambda Σ λ
R : Ω ‾ ∖ F λ ‾ → [ 0 , 1 ] , R ( p ) = λ ⟺ p ∈ Σ λ . R : \overline{\Omega} \setminus F_{\overline{\lambda}} \to[0,1], \qquad R(p) = \lambda\Longleftrightarrow p \in\Sigma_\lambda. R : Ω ∖ F λ → [ 0 , 1 ] , R ( p ) = λ ⟺ p ∈ Σ λ . For 0 ≤ α < β ≤ 1 0 \leq\alpha< \beta\leq1 0 ≤ α < β ≤ 1
R − 1 ( ( α , β ) ) = F β ∖ F α ‾ , R^{-1}\bigl((\alpha,\beta)\bigr) = F_\beta\setminus\overline{F_\alpha}, R − 1 ( ( α , β ) ) = F β ∖ F α , which is an open set (the difference between the open set F β = δ β ( Ω ) F_\beta= \delta_\beta(\Omega) F β = δ β ( Ω ) F α ‾ \overline{F_\alpha} F α R R R Σ 1 = ∂ Ω = { R = 1 } \Sigma_1 = \partial\Omega= \{R = 1\} Σ 1 = ∂ Ω = { R = 1 } F λ = { R < λ } F_\lambda= \{R < \lambda\} F λ = { R < λ } λ = λ ‾ \lambda= \overline{\lambda} λ = λ Σ ‾ λ ‾ = ∂ F λ ‾ = { R = λ ‾ } \overline{\Sigma}_{\overline{\lambda}} = \partial F_{\overline{\lambda}} = \{R = \overline{\lambda}\} Σ λ = ∂ F λ = { R = λ }
Setting F = Ω ∖ F λ ‾ ‾ F = \Omega\setminus\overline{F_{\overline{\lambda}}} F = Ω ∖ F λ F λ ‾ ‾ = F λ ‾ ∪ ∂ F λ ‾ = { R < λ ‾ } ∪ { R = λ ‾ } = { R ≤ λ ‾ } \overline{F_{\overline{\lambda}}} = F_{\overline{\lambda}} \cup\partial F_{\overline{\lambda}} = \{R < \overline{\lambda}\} \cup\{R = \overline{\lambda}\} = \{R \leq\overline{\lambda}\} F λ = F λ ∪ ∂ F λ = { R < λ } ∪ { R = λ } = { R ≤ λ }
F = { R < 1 } ∖ { R ≤ λ ‾ } = { λ ‾ < R < 1 } , F = \{R < 1\} \setminus\{R \leq\overline{\lambda}\} = \{\overline{\lambda} < R < 1\}, F = { R < 1 } ∖ { R ≤ λ } = { λ < R < 1 } , F ‾ = { λ ‾ ≤ R ≤ 1 } = Ω ‾ ∖ F λ ‾ , \overline{F} = \{\overline{\lambda} \leq R \leq1\} = \overline{\Omega} \setminus F_{\overline{\lambda}}, F = { λ ≤ R ≤ 1 } = Ω ∖ F λ , and consequently
∂ F = F ‾ ∖ F = { R = λ ‾ } ∪ { R = 1 } = Σ λ ‾ ∪ Σ 1 . \partial F = \overline{F} \setminus F = \{R = \overline{\lambda}\} \cup\{R = 1\} = \Sigma_{\overline{\lambda}} \cup\Sigma_1. ∂ F = F ∖ F = { R = λ } ∪ { R = 1 } = Σ λ ∪ Σ 1 . This is exactly property (i) of Definition 5.1 (with λ min = λ ‾ \lambda_{\min} = \overline{\lambda} λ m i n = λ λ max = 1 \lambda_{\max} = 1 λ m a x = 1
Step 2 (the transversal set is dense on ∂ Ω \partial\Omega ∂ Ω Let ν e u c l \nu_{\mathrm{eucl}} ν eucl C 2 C^2 C 2 ∂ Ω \partial\Omega ∂ Ω
G = { p ∈ ∂ Ω ∣ ⟨ D ( p ) , ν e u c l ( p ) ⟩ ≠ 0 } G = \{p \in\partial\Omega\mid\langle D(p), \nu_{\mathrm{eucl}}(p)\rangle\ne0\} G = { p ∈ ∂ Ω ∣ ⟨ D ( p ) , ν eucl ( p )⟩ = 0 } is dense in ∂ Ω = Σ 1 \partial\Omega= \Sigma_1 ∂ Ω = Σ 1 G ≠ ∅ G \ne\varnothing G = ∅ G G G Σ 1 \Sigma_1 Σ 1 B ⊂ Σ 1 B \subset\Sigma_1 B ⊂ Σ 1 ⟨ D ( p ) , ν e u c l ( p ) ⟩ = 0 \langle D(p), \nu_{\mathrm{eucl}}(p)\rangle= 0 ⟨ D ( p ) , ν eucl ( p )⟩ = 0 p ∈ B p \in B p ∈ B λ \lambda λ 1 1 1 δ λ ( p ) \delta_\lambda(p) δ λ ( p ) ∂ Ω \partial\Omega ∂ Ω Σ λ ⋐ Ω \Sigma_\lambda\Subset\Omega Σ λ ⋐ Ω
Step 3. Here, we show that the union of the dilated copies of the set G G G A A A g : R 3 → R g : \mathbb{R}^3 \to\mathbb{R} g : R 3 → R C 2 C^2 C 2 Ω \Omega Ω ∂ Ω = { p : g ( p ) = 0 } \partial\Omega= \{p : g(p) = 0\} ∂ Ω = { p : g ( p ) = 0 } g < 0 g < 0 g < 0 Ω \Omega Ω ∇ e u c l g ≠ 0 \nabla_{\mathrm{eucl}}g \ne0 ∇ eucl g = 0 ∂ Ω \partial\Omega ∂ Ω F : ( H ∖ { 0 } ) × ( 0 , 1 ] → R F : (\mathbb{H} \setminus\{0\}) \times(0,1] \to\mathbb{R} F : ( H ∖ { 0 }) × ( 0 , 1 ] → R
F ( p , λ ) = g ( δ 1 / λ ( p ) ) . F(p,\lambda) = g(\delta_{1/\lambda}(p)). F ( p , λ ) = g ( δ 1/ λ ( p )) . By definition, Σ λ = δ λ ( ∂ Ω ) = { p ∈ H ∖ { 0 } ∣ F ( p , λ ) = 0 } \Sigma_\lambda= \delta_\lambda(\partial\Omega) = \{p \in\mathbb{H} \setminus\{0\} \mid F(p,\lambda) = 0\} Σ λ = δ λ ( ∂ Ω ) = { p ∈ H ∖ { 0 } ∣ F ( p , λ ) = 0 } R ( p ) R(p) R ( p ) p ∈ Σ R ( p ) p \in\Sigma_{R(p)} p ∈ Σ R ( p )
F ( p , R ( p ) ) = 0 for all p ∈ Ω ‾ ∖ F λ ‾ . (84) F(p,R(p)) = 0 \qquad\text{for all } p \in\overline{\Omega} \setminus F_{\overline{\lambda}}. \tag*{(84)} F ( p , R ( p )) = 0 for all p ∈ Ω ∖ F λ . (84) To prove that R R R C 2 C^2 C 2 ∂ λ F \partial_\lambda F ∂ λ F p ∈ Σ λ p \in\Sigma_\lambda p ∈ Σ λ q = δ 1 / λ ( p ) ∈ ∂ Ω q = \delta_{1/\lambda}(p) \in\partial\Omega q = δ 1/ λ ( p ) ∈ ∂ Ω q = ( x / λ , y / λ , z / λ 2 ) = : ( x q , y q , z q ) q = (x/\lambda,y/\lambda,z/\lambda^2) =: (x_q,y_q,z_q) q = ( x / λ , y / λ , z / λ 2 ) =: ( x q , y q , z q )
∂ λ F ( p , λ ) = ∂ x g ( q ) ( − x λ 2 ) + ∂ y g ( q ) ( − y λ 2 ) + ∂ z g ( q ) ( − 2 z λ 3 ) = − 1 λ ( x q ∂ x g ( q ) + y q ∂ y g ( q ) + 2 z q ∂ z g ( q ) ) = − 1 λ ⟨ D ( q ) , ∇ e u c l g ( q ) ⟩ . \begin{aligned}
\partial_\lambda F(p,\lambda) = \partial_x g(q)\left(-\frac{x}{\lambda^2}\right) + \partial_y g(q)\left(-\frac{y}{\lambda^2}\right) + \partial_z g(q)\left(-\frac{2z}{\lambda^3}\right) \\
&= -\frac{1}{\lambda}\left(x_q\partial_x g(q) + y_q\partial_y g(q) + 2z_q\partial_z g(q)\right) \\
&= -\frac{1}{\lambda}\langle D(q),\nabla_{\mathrm{eucl}}g(q)\rangle.
\end{aligned} ∂ λ F ( p , λ ) = ∂ x g ( q ) ( − λ 2 x ) + ∂ y g ( q ) ( − λ 2 y ) + ∂ z g ( q ) ( − λ 3 2 z ) = − λ 1 ( x q ∂ x g ( q ) + y q ∂ y g ( q ) + 2 z q ∂ z g ( q ) ) = − λ 1 ⟨ D ( q ) , ∇ eucl g ( q )⟩ . Now, assume that q ∈ G q \in G q ∈ G G G G D ( q ) D(q) D ( q ) ∂ Ω \partial\Omega ∂ Ω ⟨ D ( q ) , ∇ e u c l g ( q ) ⟩ ≠ 0 \langle D(q),\nabla_{\mathrm{eucl}}g(q)\rangle\ne0 ⟨ D ( q ) , ∇ eucl g ( q )⟩ = 0 ∂ λ F ( p , λ ) ≠ 0 \partial_\lambda F(p,\lambda) \ne0 ∂ λ F ( p , λ ) = 0
By the implicit function theorem applied to (5.2), the function R ( p ) R(p) R ( p ) C 2 C^2 C 2 p p p F ( p , R ( p ) ) = 0 F(p,R(p)) = 0 F ( p , R ( p )) = 0 p p p
∇ e u c l R ( p ) = − ∇ e u c l F ( p , λ ) ∂ λ F ( p , λ ) . \nabla_{\mathrm{eucl}}R(p) = -\frac{\nabla_{\mathrm{eucl}}F(p,\lambda)}{\partial_\lambda F(p,\lambda)}. ∇ eucl R ( p ) = − ∂ λ F ( p , λ ) ∇ eucl F ( p , λ ) . By the chain rule, ∇ e u c l F ( p , λ ) = ( ∂ x g ( q ) λ , ∂ y g ( q ) λ , ∂ z g ( q ) λ 2 ) \nabla_{\mathrm{eucl}}\mathcal{F}(p,\lambda)=\left(\frac{\partial_xg(q)}{\lambda},\frac{\partial_yg(q)}{\lambda},\frac{\partial_zg(q)}{\lambda^2}\right) ∇ eucl F ( p , λ ) = ( λ ∂ x g ( q ) , λ ∂ y g ( q ) , λ 2 ∂ z g ( q ) ) ∇ e u c l g ( q ) ≠ 0 \nabla_{\mathrm{eucl}}g(q)\ne0 ∇ eucl g ( q ) = 0 g g g ∇ e u c l F ( p , λ ) ≠ 0 \nabla_{\mathrm{eucl}}\mathcal{F}(p,\lambda)\ne0 ∇ eucl F ( p , λ ) = 0 ∇ e u c l R ( p ) ≠ 0 \nabla_{\mathrm{eucl}}R(p)\ne0 ∇ eucl R ( p ) = 0
Let now A A A
By the results of Step 3, R ∈ C 2 ( A ) R\in C^2(A) R ∈ C 2 ( A ) ∇ e u c l R ≠ 0 \nabla_{\mathrm{eucl}}R\ne0 ∇ eucl R = 0 A A A G G G ∂ Ω \partial\Omega ∂ Ω ( λ ˉ , 1 ) (\bar{\lambda},1) ( λ ˉ , 1 ) G × ( λ ˉ , 1 ) G\times(\bar{\lambda},1) G × ( λ ˉ , 1 ) ∂ Ω × ( 0 , 1 ] \partial\Omega\times(0,1] ∂ Ω × ( 0 , 1 ] ( q , λ ) ↦ δ λ ( q ) (q,\lambda)\mapsto\delta_\lambda(q) ( q , λ ) ↦ δ λ ( q ) A A A A ⊆ Ω ∖ F λ ‾ = : F A\subseteq\Omega\setminus\overline{F_\lambda}=:F A ⊆ Ω ∖ F λ =: F G × ( λ ˉ , 1 ) G\times(\bar{\lambda},1) G × ( λ ˉ , 1 ) ∂ Ω × [ λ ˉ , 1 ] \partial\Omega\times[\bar{\lambda},1] ∂ Ω × [ λ ˉ , 1 ] A A A Ω ‾ ∖ F λ ‾ \overline{\Omega}\setminus\overline{F_\lambda} Ω ∖ F λ A ∩ Σ λ = δ λ ( G ) ≠ ∅ A\cap\Sigma_\lambda=\delta_\lambda(G)\ne\varnothing A ∩ Σ λ = δ λ ( G ) = ∅ λ ∈ ( λ ˉ , 1 ) \lambda\in(\bar{\lambda},1) λ ∈ ( λ ˉ , 1 ) F F F R R R
Lemma 5.4. Let Ω ⊂ R 3 \Omega\subset\mathbb{R}^3 Ω ⊂ R 3 u ∈ C 0 ( Ω ‾ ) u\in C^0(\overline{\Omega}) u ∈ C 0 ( Ω ) u u u { Γ t = { u = t } } t ∈ [ t 1 , t 2 ] \{\Gamma_t=\{u=t\}\}_{t\in[t_1,t_2]} { Γ t = { u = t } } t ∈ [ t 1 , t 2 ] C 2 C^2 C 2 Ω [ t 1 , t 2 ] = { p ∈ Ω ‾ ∣ u ( p ) ∈ [ t 1 , t 2 ] } \Omega_{[t_1,t_2]}=\{p\in\overline{\Omega}\mid u(p)\in[t_1,t_2]\} Ω [ t 1 , t 2 ] = { p ∈ Ω ∣ u ( p ) ∈ [ t 1 , t 2 ]} λ ∈ [ λ 1 , λ 2 ] \lambda\in[\lambda_1,\lambda_2] λ ∈ [ λ 1 , λ 2 ]
Then, there exists a continuous function R ∈ C 0 ( Ω [ t 1 , t 2 ] ; [ λ 1 , λ 2 ] ) R\in C^0(\Omega_{[t_1,t_2]};[\lambda_1,\lambda_2]) R ∈ C 0 ( Ω [ t 1 , t 2 ] ; [ λ 1 , λ 2 ]) f : [ λ 1 , λ 2 ] → [ t 1 , t 2 ] f:[\lambda_1,\lambda_2]\to[t_1,t_2] f : [ λ 1 , λ 2 ] → [ t 1 , t 2 ]
u = f ∘ R in Ω [ t 1 , t 2 ] . u=f\circ R\quad\text{in }\Omega_{[t_1,t_2]}. u = f ∘ R in Ω [ t 1 , t 2 ] . Proof. By Definition 5.1, the assumption that { Γ t } t ∈ [ t 1 , t 2 ] \{\Gamma_t\}_{t\in[t_1,t_2]} { Γ t } t ∈ [ t 1 , t 2 ] C 2 C^2 C 2 Ω [ t 1 , t 2 ] \Omega_{[t_1,t_2]} Ω [ t 1 , t 2 ] R : Ω [ t 1 , t 2 ] → [ λ 1 , λ 2 ] R:\Omega_{[t_1,t_2]}\to[\lambda_1,\lambda_2] R : Ω [ t 1 , t 2 ] → [ λ 1 , λ 2 ] Σ λ = { R = λ } \Sigma_\lambda=\{R=\lambda\} Σ λ = { R = λ } Ω [ t 1 , t 2 ] \Omega_{[t_1,t_2]} Ω [ t 1 , t 2 ] { Γ t } t ∈ [ t 1 , t 2 ] \{\Gamma_t\}_{t\in[t_1,t_2]} { Γ t } t ∈ [ t 1 , t 2 ] λ ∈ [ λ 1 , λ 2 ] \lambda\in[\lambda_1,\lambda_2] λ ∈ [ λ 1 , λ 2 ] t ∈ [ t 1 , t 2 ] t\in[t_1,t_2] t ∈ [ t 1 , t 2 ] Σ λ = Γ t \Sigma_\lambda=\Gamma_t Σ λ = Γ t
Since the sets Γ t \Gamma_t Γ t u u u Ω [ t 1 , t 2 ] = u − 1 ( [ t 1 , t 2 ] ) \Omega_{[t_1,t_2]}=u^{-1}([t_1,t_2]) Ω [ t 1 , t 2 ] = u − 1 ([ t 1 , t 2 ]) t t t λ \lambda λ Σ λ = Γ t = Γ t ′ \Sigma_\lambda=\Gamma_t=\Gamma_{t'} Σ λ = Γ t = Γ t ′ t ≠ t ′ t\ne t' t = t ′ Γ t ∩ Γ t ′ ≠ ∅ \Gamma_t\cap\Gamma_{t'}\ne\varnothing Γ t ∩ Γ t ′ = ∅
f : [ λ 1 , λ 2 ] → [ t 1 , t 2 ] , f ( λ ) = t ⟺ Σ λ = Γ t . f:[\lambda_1,\lambda_2]\to[t_1,t_2],\qquad f(\lambda)=t\Longleftrightarrow\Sigma_\lambda=\Gamma_t. f : [ λ 1 , λ 2 ] → [ t 1 , t 2 ] , f ( λ ) = t ⟺ Σ λ = Γ t . By construction, for every p ∈ Ω [ t 1 , t 2 ] p\in\Omega_{[t_1,t_2]} p ∈ Ω [ t 1 , t 2 ] λ = R ( p ) \lambda=R(p) λ = R ( p ) p ∈ Σ λ = Γ f ( λ ) p\in\Sigma_\lambda=\Gamma_{f(\lambda)} p ∈ Σ λ = Γ f ( λ ) u ( p ) = f ( λ ) = f ( R ( p ) ) u(p)=f(\lambda)=f(R(p)) u ( p ) = f ( λ ) = f ( R ( p ))
u = f ∘ R in Ω [ t 1 , t 2 ] . u=f\circ R\quad\text{in }\Omega_{[t_1,t_2]}. u = f ∘ R in Ω [ t 1 , t 2 ] . We now prove that f f f f ( λ a ) = f ( λ b ) = t f(\lambda_a)=f(\lambda_b)=t f ( λ a ) = f ( λ b ) = t Σ λ a = Γ t = Σ λ b \Sigma_{\lambda_a}=\Gamma_t=\Sigma_{\lambda_b} Σ λ a = Γ t = Σ λ b Σ λ \Sigma_\lambda Σ λ Ω [ t 1 , t 2 ] \Omega_{[t_1,t_2]} Ω [ t 1 , t 2 ] λ a = λ b \lambda_a=\lambda_b λ a = λ b
Moreover, for every t ∈ [ t 1 , t 2 ] t\in[t_1,t_2] t ∈ [ t 1 , t 2 ] Γ t \Gamma_t Γ t Σ λ \Sigma_\lambda Σ λ t = f ( λ ) t=f(\lambda) t = f ( λ ) λ \lambda λ f f f f f f λ , λ j ∈ [ λ 1 , λ 2 ] \lambda,\lambda_j\in[\lambda_1,\lambda_2] λ , λ j ∈ [ λ 1 , λ 2 ] λ j → λ \lambda_j\to\lambda λ j → λ j → + ∞ j\to+\infty j → + ∞ f ( λ j ) ↛ f ( λ ) f(\lambda_j)\not\to f(\lambda) f ( λ j ) → f ( λ ) η > 0 \eta>0 η > 0 ∣ f ( λ j ) − f ( λ ) ∣ ≥ η \lvert f(\lambda_j)-f(\lambda)\rvert\ge\eta ∣ f ( λ j ) − f ( λ )∣ ≥ η j j j p j ∈ { R = λ j } p_j\in\{R=\lambda_j\} p j ∈ { R = λ j } Ω [ t 1 , t 2 ] \Omega_{[t_1,t_2]} Ω [ t 1 , t 2 ] p j k p_{j_k} p j k p ∈ Ω [ t 1 , t 2 ] p\in\Omega_{[t_1,t_2]} p ∈ Ω [ t 1 , t 2 ] R R R R ( p ) = lim k R ( p j k ) = lim k λ j k = λ R(p)=\lim_kR(p_{j_k})=\lim_k\lambda_{j_k}=\lambda R ( p ) = lim k R ( p j k ) = lim k λ j k = λ u u u
f ( λ j k ) = u ( p j k ) → u ( p ) = f ( λ ) , f(\lambda_{j_k})=u(p_{j_k})\to u(p)=f(\lambda), f ( λ j k ) = u ( p j k ) → u ( p ) = f ( λ ) , contradicting ∣ f ( λ j k ) − f ( λ ) ∣ ≥ η \lvert f(\lambda_{j_k})-f(\lambda)\rvert\ge\eta ∣ f ( λ j k ) − f ( λ )∣ ≥ η k k k f f f
It is a standard fact in real analysis that a continuous injective function on an interval is strictly monotonic. Hence f f f [ λ 1 , λ 2 ] [\lambda_1,\lambda_2] [ λ 1 , λ 2 ]
This completes the proof: R ∈ C 0 ( Ω [ t 1 , t 2 ] ; [ λ 1 , λ 2 ] ) R\in C^0(\Omega_{[t_1,t_2]};[\lambda_1,\lambda_2]) R ∈ C 0 ( Ω [ t 1 , t 2 ] ; [ λ 1 , λ 2 ]) A A A f ∈ C 0 ( [ λ 1 , λ 2 ] ; [ t 1 , t 2 ] ) f\in C^0([\lambda_1,\lambda_2];[t_1,t_2]) f ∈ C 0 ([ λ 1 , λ 2 ] ; [ t 1 , t 2 ]) u = f ∘ R u=f\circ R u = f ∘ R
A viscosity approach to the curvature blow-up
Let u ∈ C 0 ( Ω ‾ ) u \in C^{0}(\overline{\Omega}) u ∈ C 0 ( Ω ) Γ t = { p ∈ Ω ‾ : u ( p ) = t } \Gamma_t = \{p \in\overline{\Omega} : u(p) = t\} Γ t = { p ∈ Ω : u ( p ) = t } T ∗ = max Ω ‾ u T^{*} = \max_{\overline{\Omega}} u T ∗ = max Ω u Γ t \Gamma_t Γ t
Theorem 5.5. Let Ω ⊂ R 3 \Omega\subset\mathbb{R}^{3} Ω ⊂ R 3 C 2 C^{2} C 2 u ∈ C 0 ( Ω ‾ ) u \in C^{0}(\overline{\Omega}) u ∈ C 0 ( Ω ) d ~ \widetilde{d} d { Γ t } t ∈ [ t 1 , t 2 ] \{\Gamma_t\}_{t \in[t_1,t_2]} { Γ t } t ∈ [ t 1 , t 2 ] C 2 C^{2} C 2 [ t 1 , t 2 ] ⊂ ( 0 , T ∗ ) [t_1,t_2] \subset(0,T^{*}) [ t 1 , t 2 ] ⊂ ( 0 , T ∗ ) A A A Definition 5.1 (iii). Then, for any p c h a r ∈ A ∩ ⋃ t ∈ [ t 1 , t 2 ] C ( Γ t ) p_{\mathrm{char}} \in A \cap\bigcup_{t \in[t_1,t_2]} C(\Gamma_t) p char ∈ A ∩ ⋃ t ∈ [ t 1 , t 2 ] C ( Γ t ) { p n } n ⊂ Ω [ t 1 , t 2 ] \{p_n\}_{n} \subset\Omega_{[t_1,t_2]} { p n } n ⊂ Ω [ t 1 , t 2 ] p n → p c h a r p_n \to p_{\mathrm{char}} p n → p char
H n ( p n ) ⟶ ∞ as n ⟶ ∞ , H_n(p_n) \longrightarrow\infty\quad\text{as } n \longrightarrow\infty, H n ( p n ) ⟶ ∞ as n ⟶ ∞ , where H n H_n H n Γ τ n \Gamma_{\tau_n} Γ τ n τ n ∈ [ t 1 , t 2 ] \tau_n \in[t_1,t_2] τ n ∈ [ t 1 , t 2 ] p n ∈ Γ τ n p_n \in\Gamma_{\tau_n} p n ∈ Γ τ n
The proof of Theorem 5.5 proceeds by constructing a sequence of non-characteristic points approaching p c h a r p_{\mathrm{char}} p char u u u C 2 C^{2} C 2 u u u
Let g : U → R g : U \to\mathbb{R} g : U → R U ⊂ R 3 U \subset\mathbb{R}^{3} U ⊂ R 3 s > 0 s > 0 s > 0 sup-convolution g s g^{s} g s inf-convolution g s g_s g s
g s ( p ) = sup q ∈ U { g ( q ) − 1 s ∣ p − q ∣ 2 } and g s ( p ) = inf q ∈ U { g ( q ) + 1 s ∣ p − q ∣ 2 } , (85) g^{s}(p) = \sup_{q \in U} \left\{g(q) - \frac{1}{s}|p-q|^{2}\right\}
\quad\text{and}\quad
g_s(p) = \inf_{q \in U} \left\{g(q) + \frac{1}{s}|p-q|^{2}\right\}, \tag*{(85)} g s ( p ) = q ∈ U sup { g ( q ) − s 1 ∣ p − q ∣ 2 } and g s ( p ) = q ∈ U inf { g ( q ) + s 1 ∣ p − q ∣ 2 } , (85) see e.g. [8 Chapter 3.5] . Since the mapping p ↦ g s ( p ) + 1 s ∣ p ∣ 2 p \mapsto g^{s}(p) + \frac{1}{s}|p|^{2} p ↦ g s ( p ) + s 1 ∣ p ∣ 2 g s g^{s} g s g s g_s g s g s g_s g s g s g^{s} g s
Moreover, a standard fact is that assuming the domain to be compact implies that the minimizer in the inf-convolution converges to the evaluated point uniformly as s → 0 s \to0 s → 0 q s ∈ U ‾ q_s \in\overline{U} q s ∈ U g s ( p ) g_s(p) g s ( p ) g g g
∣ p − q s ∣ ≤ 2 M s , M = sup U ‾ ∣ g ∣ < ∞ , |p-q_s| \le\sqrt{2Ms}, \qquad M = \sup_{\overline{U}} |g| < \infty, ∣ p − q s ∣ ≤ 2 M s , M = U sup ∣ g ∣ < ∞ , uniformly in p p p q = p q=p q = p g s ( p ) ≤ g ( p ) ≤ M g_s(p) \le g(p) \le M g s ( p ) ≤ g ( p ) ≤ M g ( q s ) ≥ − M g(q_s) \ge-M g ( q s ) ≥ − M
g s ( p ) = g ( q s ) + 1 s ∣ p − q s ∣ 2 ≥ − M + 1 s ∣ p − q s ∣ 2 . g_s(p) = g(q_s) + \frac{1}{s}|p-q_s|^{2} \ge-M + \frac{1}{s}|p-q_s|^{2}. g s ( p ) = g ( q s ) + s 1 ∣ p − q s ∣ 2 ≥ − M + s 1 ∣ p − q s ∣ 2 . Combining the two inequalities yields ∣ p − q s ∣ 2 ≤ 2 M s |p-q_s|^{2} \le2Ms ∣ p − q s ∣ 2 ≤ 2 M s s → 0 s \to0 s → 0
The following 1-dimensional C 2 C^{2} C 2 C 2 C^{2} C 2 [7 Corollary 12.14] , specialized to the one-dimensional case.
Lemma 5.6. Let f : [ a , b ] → R f : [a,b] \to\mathbb{R} f : [ a , b ] → R r 0 ∈ ( a , b ) r_0 \in(a,b) r 0 ∈ ( a , b ) φ ∈ C 2 ( [ a , b ] ) \varphi\in C^{2}([a,b]) φ ∈ C 2 ([ a , b ])
φ ( r 0 ) = f ( r 0 ) , φ ′ ( r 0 ) = f ′ ( r 0 ) , φ ′ ′ ( r 0 ) = f ′ ′ ( r 0 ) , \varphi(r_0) = f(r_0), \qquad\varphi'(r_0) = f'(r_0), \qquad\varphi''(r_0) = f''(r_0), φ ( r 0 ) = f ( r 0 ) , φ ′ ( r 0 ) = f ′ ( r 0 ) , φ ′′ ( r 0 ) = f ′′ ( r 0 ) , and the difference f − φ f-\varphi f − φ r 0 r_0 r 0
Lemma 5.7. Let f : [ a , b ] → R f : [a,b] \to\mathbb{R} f : [ a , b ] → R r ∈ ( a , b ) r \in(a,b) r ∈ ( a , b ) f f f r r r L > 0 L > 0 L > 0 ( r n ) n ∈ N ⊂ ( a , b ) (r_n)_{n\in\mathbb{N}} \subset(a,b) ( r n ) n ∈ N ⊂ ( a , b ) r n → r r_n \to r r n → r n → ∞ n \to\infty n → ∞ ( ψ n ) n ∈ N ⊂ C 2 ( [ a , b ] ) (\psi_n)_{n\in\mathbb{N}} \subset C^2([a,b]) ( ψ n ) n ∈ N ⊂ C 2 ([ a , b ]) n n n f − ψ n f-\psi_n f − ψ n r n r_n r n
0 < ∣ ψ n ′ ( r n ) ∣ ≤ L , ∀ n . 0 < |\psi_n'(r_n)| \le L, \qquad\forall n. 0 < ∣ ψ n ′ ( r n ) ∣ ≤ L , ∀ n . In the proof of this Lemma we need a technical result about inf-convolutions of strictly decreasing functions, which is the following.
Lemma 5.8. Let f : [ a , b ] → R f : [a,b] \to\mathbb{R} f : [ a , b ] → R s > 0 s > 0 s > 0 f s f_s f s I ⊂ [ a , b ] I \subset[a,b] I ⊂ [ a , b ]
Proof. Assume by contradiction that there exists an open interval I ⊂ ( a , b ) I \subset(a,b) I ⊂ ( a , b ) C ∈ R C \in\mathbb{R} C ∈ R f s ( l ) = C f_s(l) = C f s ( l ) = C l ∈ I l \in I l ∈ I
For any l ∈ I l \in I l ∈ I m l ∈ [ a , b ] m_l \in[a,b] m l ∈ [ a , b ]
C = f s ( l ) = f ( m l ) + 1 s ( m l − l ) 2 . (86) C = f_s(l) = f(m_l) + \frac{1}{s}(m_l-l)^2. \tag*{(86)} C = f s ( l ) = f ( m l ) + s 1 ( m l − l ) 2 . (86) For any other point l ′ ∈ I l' \in I l ′ ∈ I f s ( l ′ ) f_s(l') f s ( l ′ ) l ′ l' l ′ m l m_l m l
f s ( l ′ ) ≤ f ( m l ) + 1 s ( m l − l ′ ) 2 . f_s(l') \le f(m_l) + \frac{1}{s}(m_l-l')^2. f s ( l ′ ) ≤ f ( m l ) + s 1 ( m l − l ′ ) 2 . Using the assumption that f s ( l ′ ) = C f_s(l') = C f s ( l ′ ) = C
C ≤ f ( m l ) + 1 s ( m l − l ′ ) 2 . C \le f(m_l) + \frac{1}{s}(m_l-l')^2. C ≤ f ( m l ) + s 1 ( m l − l ′ ) 2 . Plugging (5.4) into the inequality above, we obtain:
C ≤ C − 1 s ( m l − l ) 2 + 1 s ( m l − l ′ ) 2 . C \le C - \frac{1}{s}(m_l-l)^2 + \frac{1}{s}(m_l-l')^2. C ≤ C − s 1 ( m l − l ) 2 + s 1 ( m l − l ′ ) 2 . Simplifying C C C s > 0 s > 0 s > 0
( m l − l ) 2 ≤ ( m l − l ′ ) 2 . (m_l-l)^2 \le(m_l-l')^2. ( m l − l ) 2 ≤ ( m l − l ′ ) 2 . Expanding the squares, and factorizing we get:
( l − l ′ ) ( l + l ′ ) ≤ 2 m l ( l − l ′ ) . (l-l')(l+l') \le2m_l(l-l'). ( l − l ′ ) ( l + l ′ ) ≤ 2 m l ( l − l ′ ) . This inequality must hold for any choice of l , l ′ ∈ I l,l' \in I l , l ′ ∈ I l ′ → l − l' \to l^- l ′ → l − m l ≥ l m_l \ge l m l ≥ l m l = l m_l = l m l = l l ∈ I l \in I l ∈ I m l = l m_l = l m l = l
C = f ( m l ) + 1 s ( m l − l ) 2 = f ( l ) + 1 s ( l − l ) 2 = f ( l ) . C = f(m_l) + \frac{1}{s}(m_l-l)^2 = f(l) + \frac{1}{s}(l-l)^2 = f(l). C = f ( m l ) + s 1 ( m l − l ) 2 = f ( l ) + s 1 ( l − l ) 2 = f ( l ) . This implies that f ( l ) = C f(l) = C f ( l ) = C l ∈ I l \in I l ∈ I f f f [ a , b ] [a,b] [ a , b ] f s f_s f s
Now we give the proof of Lemma 5.7.
Proof of Lemma 5.7 . For s > 0 s>0 s > 0 f s : [ a , b ] → R f_s:[a,b]\to\mathbb{R} f s : [ a , b ] → R (85) , which is Lipschitz continuous and twice differentiable almost everywhere.
Let ρ ∈ ( a , b ) \rho\in(a,b) ρ ∈ ( a , b ) f s f_s f s f f f f s ( ρ ) f_s(\rho) f s ( ρ ) m ∈ [ a , b ] m\in[a,b] m ∈ [ a , b ]
f s ( ρ ) = f ( m ) + 1 s ( ρ − m ) 2 . (87) f_s(\rho)=f(m)+\frac{1}{s}(\rho-m)^2. \tag*{(87)} f s ( ρ ) = f ( m ) + s 1 ( ρ − m ) 2 . (87) By Lemma 5.6, there exists φ ∈ C 2 ( [ a , b ] ) \varphi\in C^2([a,b]) φ ∈ C 2 ([ a , b ]) f s − φ f_s-\varphi f s − φ ρ \rho ρ l ∈ ( a , b ) l\in(a,b) l ∈ ( a , b ) c ∈ [ a , b ] c\in[a,b] c ∈ [ a , b ]
f s ( ρ ) − φ ( ρ ) ≤ f s ( l ) − φ ( l ) ≤ f ( c ) + 1 s ( l − c ) 2 − φ ( l ) . f_s(\rho)-\varphi(\rho)\leq f_s(l)-\varphi(l)\leq f(c)+\frac{1}{s}(l-c)^2-\varphi(l). f s ( ρ ) − φ ( ρ ) ≤ f s ( l ) − φ ( l ) ≤ f ( c ) + s 1 ( l − c ) 2 − φ ( l ) . Plugging (87) into the latter and choosing c ∈ [ a , b ] c\in[a,b] c ∈ [ a , b ] l = c + ρ − m ∈ ( a , b ) l=c+\rho-m\in(a,b) l = c + ρ − m ∈ ( a , b )
f ( m ) − φ ( ρ ) ≤ f ( c ) − φ ( c + ρ − m ) . (88) f(m)-\varphi(\rho)\leq f(c)-\varphi(c+\rho-m). \tag*{(88)} f ( m ) − φ ( ρ ) ≤ f ( c ) − φ ( c + ρ − m ) . (88) Since ρ \rho ρ U m U_m U m m m m ρ + c − m ∈ ( a , b ) \rho+c-m\in(a,b) ρ + c − m ∈ ( a , b ) c ∈ U m c\in U_m c ∈ U m ψ ( c ) = φ ( c + ρ − m ) \psi(c)=\varphi(c+\rho-m) ψ ( c ) = φ ( c + ρ − m ) c ∈ U m c\in U_m c ∈ U m ψ ∈ C 2 ( U m ) \psi\in C^2(U_m) ψ ∈ C 2 ( U m ) C 2 ( [ a , b ] ) C^2([a,b]) C 2 ([ a , b ]) ψ ( m ) = φ ( ρ ) \psi(m)=\varphi(\rho) ψ ( m ) = φ ( ρ ) (88) gives
f ( m ) − ψ ( m ) ≤ f ( c ) − ψ ( c ) ∀ c ∈ U m , f(m)-\psi(m)\leq f(c)-\psi(c)\qquad\forall c\in U_m, f ( m ) − ψ ( m ) ≤ f ( c ) − ψ ( c ) ∀ c ∈ U m , so m m m f − ψ f-\psi f − ψ ψ ′ ( m ) = φ ′ ( ρ ) = f s ′ ( ρ ) \psi'(m)=\varphi'(\rho)=f_s'(\rho) ψ ′ ( m ) = φ ′ ( ρ ) = f s ′ ( ρ )
For every fixed s > 0 s>0 s > 0
E s = { ρ ∈ ( a , b ) : f s is twice differentiable at ρ , f s ′ ( ρ ) ≠ 0 } E_s=\{\rho\in(a,b):f_s\text{ is twice differentiable at }\rho,\ f_s'(\rho)\ne0\} E s = { ρ ∈ ( a , b ) : f s is twice differentiable at ρ , f s ′ ( ρ ) = 0 } is dense in ( a , b ) (a,b) ( a , b ) I ⊂ ( a , b ) I\subset(a,b) I ⊂ ( a , b ) f s f_s f s [ a , b ] [a,b] [ a , b ] I I I ξ 0 , ξ ∈ I \xi_0,\xi\in I ξ 0 , ξ ∈ I f s ( ξ ) − f s ( ξ 0 ) = ∫ ξ 0 ξ f s ′ ( t ) d t f_s(\xi)-f_s(\xi_0)=\int_{\xi_0}^{\xi}f_s'(t)\,\mathrm{d}t f s ( ξ ) − f s ( ξ 0 ) = ∫ ξ 0 ξ f s ′ ( t ) d t f s ′ = 0 f_s'=0 f s ′ = 0 I I I f s f_s f s I I I Lemma 5.8 (using strict monotonicity of f f f { t ∈ I : f s ′ ( t ) ≠ 0 } \{t\in I:f_s'(t)\ne0\} { t ∈ I : f s ′ ( t ) = 0 } E s ∩ I ≠ ∅ E_s\cap I\ne\varnothing E s ∩ I = ∅ I I I E s E_s E s ( a , b ) (a,b) ( a , b )
Set σ n = 1 / n \sigma_n=1/n σ n = 1/ n E σ n E_{\sigma_n} E σ n
ρ n ∈ E σ n ∩ ( r − 1 n , r + 1 n ) , \rho_n\in E_{\sigma_n}\cap\left(r-\frac{1}{n},r+\frac{1}{n}\right), ρ n ∈ E σ n ∩ ( r − n 1 , r + n 1 ) , so that ρ n → r \rho_n\to r ρ n → r r n ∈ [ a , b ] r_n\in[a,b] r n ∈ [ a , b ] C 2 C^2 C 2 ψ n \psi_n ψ n f − ψ n f-\psi_n f − ψ n r n r_n r n
ψ n ′ ( r n ) = φ n ′ ( ρ n ) = f σ n ′ ( ρ n ) ≠ 0. \psi_n'(r_n)=\varphi_n'(\rho_n)=f_{\sigma_n}'(\rho_n)\ne0. ψ n ′ ( r n ) = φ n ′ ( ρ n ) = f σ n ′ ( ρ n ) = 0. Recall that such r n r_n r n f σ n ( ρ n ) f_{\sigma_n}(\rho_n) f σ n ( ρ n ) r n → r r_n\to r r n → r f f f L L L V V V r r r ρ n , r n → r \rho_n,r_n\to r ρ n , r n → r n n n ρ n , r n ∈ V \rho_n,r_n\in V ρ n , r n ∈ V n n n l , l ′ ∈ V l,l'\in V l , l ′ ∈ V c = c ∗ + ( l − l ′ ) c=c^*+(l-l') c = c ∗ + ( l − l ′ ) f σ n ( l ) f_{\sigma_n}(l) f σ n ( l ) c ∗ c^* c ∗ f σ n ( l ′ ) f_{\sigma_n}(l') f σ n ( l ′ ) (88) , yielding
f σ n ( l ) − f σ n ( l ′ ) ≤ f ( c ∗ + ( l − l ′ ) ) − f ( c ∗ ) ≤ L ∣ l − l ′ ∣ . f_{\sigma_n}(l)-f_{\sigma_n}(l')\leq f(c^*+(l-l'))-f(c^*)\leq L|l-l'|. f σ n ( l ) − f σ n ( l ′ ) ≤ f ( c ∗ + ( l − l ′ )) − f ( c ∗ ) ≤ L ∣ l − l ′ ∣. Then, f σ n f_{\sigma_n} f σ n L L L r r r n n n ∣ f σ n ′ ( ρ n ) ∣ ≤ L |f_{\sigma_n}'(\rho_n)|\leq L ∣ f σ n ′ ( ρ n ) ∣ ≤ L n n n
∣ ψ n ′ ( r n ) ∣ = ∣ f σ n ′ ( ρ n ) ∣ ≤ L , ∀ n . |\psi_n'(r_n)|=|f_{\sigma_n}'(\rho_n)|\leq L,\qquad\forall n. ∣ ψ n ′ ( r n ) ∣ = ∣ f σ n ′ ( ρ n ) ∣ ≤ L , ∀ n . We now give the proof of Theorem 5.5. The first step is to show that f f f u u u f f f ψ n \psi_n ψ n f f f p c h a r p_{\mathrm{char}} p char ψ n ∘ R \psi_n \circ R ψ n ∘ R p n → p c h a r p_n \to p_{\mathrm{char}} p n → p char p n p_n p n p n p_n p n p c h a r p_{\mathrm{char}} p char
Proof of Theorem 5.5. Let u = f ∘ R u=f\circ R u = f ∘ R f ∈ C 0 ( [ R 1 , R 2 ] ; [ t 1 , t 2 ] ) f\in C^0([R_1,R_2];[t_1,t_2]) f ∈ C 0 ([ R 1 , R 2 ] ; [ t 1 , t 2 ]) R ∈ C 0 ( Ω [ t 1 , t 2 ] ; [ R 1 , R 2 ] ) R\in C^0(\Omega_{[t_1,t_2]};[R_1,R_2]) R ∈ C 0 ( Ω [ t 1 , t 2 ] ; [ R 1 , R 2 ]) A A A r ∈ [ R 1 , R 2 ] r\in[R_1,R_2] r ∈ [ R 1 , R 2 ]
Σ r = Γ f ( r ) . \Sigma_r=\Gamma_{f(r)}. Σ r = Γ f ( r ) . Suppose m ∈ ( R 1 , R 2 ) m\in(R_1,R_2) m ∈ ( R 1 , R 2 ) ψ ∈ C 2 ( [ R 1 , R 2 ] ) \psi\in C^2([R_1,R_2]) ψ ∈ C 2 ([ R 1 , R 2 ]) f − ψ f-\psi f − ψ m m m ψ ′ ( m ) ≠ 0 \psi'(m)\ne0 ψ ′ ( m ) = 0 p ˉ ∈ Σ m ∩ A \bar p\in\Sigma_m\cap A p ˉ ∈ Σ m ∩ A R ( p ˉ ) = m R(\bar p)=m R ( p ˉ ) = m ∇ R ( p ˉ ) ≠ 0 \nabla R(\bar p)\ne0 ∇ R ( p ˉ ) = 0 Ψ = ψ ∘ R ∈ C 2 \Psi=\psi\circ R\in C^2 Ψ = ψ ∘ R ∈ C 2 p ˉ ∈ A \bar p\in A p ˉ ∈ A
Since f − ψ f-\psi f − ψ m m m U m ⊂ ( R 1 , R 2 ) U_m\subset(R_1,R_2) U m ⊂ ( R 1 , R 2 ) m m m
f ( m ) − ψ ( m ) ≤ f ( c ) − ψ ( c ) ∀ c ∈ U m . f(m)-\psi(m)\le f(c)-\psi(c)\qquad\forall c\in U_m. f ( m ) − ψ ( m ) ≤ f ( c ) − ψ ( c ) ∀ c ∈ U m . By continuity of R R R V p ˉ V_{\bar p} V p ˉ p ˉ \bar p p ˉ R ( p ) ∈ U m R(p)\in U_m R ( p ) ∈ U m p ∈ V p ˉ p\in V_{\bar p} p ∈ V p ˉ c = R ( p ) c=R(p) c = R ( p ) u = f ∘ R u=f\circ R u = f ∘ R Ψ = ψ ∘ R \Psi=\psi\circ R Ψ = ψ ∘ R R ( p ˉ ) = m R(\bar p)=m R ( p ˉ ) = m
u ( p ˉ ) − Ψ ( p ˉ ) = f ( m ) − ψ ( m ) ≤ f ( R ( p ) ) − ψ ( R ( p ) ) = u ( p ) − Ψ ( p ) ∀ p ∈ V p ˉ , u(\bar p)-\Psi(\bar p)=f(m)-\psi(m)\le f(R(p))-\psi(R(p))=u(p)-\Psi(p)\qquad\forall p\in V_{\bar p}, u ( p ˉ ) − Ψ ( p ˉ ) = f ( m ) − ψ ( m ) ≤ f ( R ( p )) − ψ ( R ( p )) = u ( p ) − Ψ ( p ) ∀ p ∈ V p ˉ , i.e. u − Ψ u-\Psi u − Ψ p ˉ \bar p p ˉ
By the chain rule, ∇ Ψ = ψ ′ ( m ) ∇ R \nabla\Psi=\psi'(m)\nabla R ∇Ψ = ψ ′ ( m ) ∇ R ψ ′ ( m ) ≠ 0 \psi'(m)\ne0 ψ ′ ( m ) = 0 ∇ R ( p ˉ ) ≠ 0 \nabla R(\bar p)\ne0 ∇ R ( p ˉ ) = 0 ∇ Ψ ( p ˉ ) ≠ 0 \nabla\Psi(\bar p)\ne0 ∇Ψ ( p ˉ ) = 0 u u u Ψ ∈ C 2 \Psi\in C^2 Ψ ∈ C 2 u u u p ˉ \bar p p ˉ ∇ Ψ ( p ˉ ) ≠ 0 \nabla\Psi(\bar p)\ne0 ∇Ψ ( p ˉ ) = 0
1 ≤ − ∣ ∇ Ψ ( p ˉ ) ∣ div ( ∇ Ψ ∣ ∇ Ψ ∣ ) ( p ˉ ) = − ∣ ψ ′ ( m ) ∣ ∣ ∇ R ( p ˉ ) ∣ div ( ψ ′ ( m ) ∣ ψ ′ ( m ) ∣ ∇ R ∣ ∇ R ∣ ) ( p ˉ ) = − ψ ′ ( m ) ∣ ∇ R ( p ˉ ) ∣ div ( ∇ R ∣ ∇ R ∣ ) ( p ˉ ) . (89) \begin{aligned}
1&\le-|\nabla\Psi(\bar p)|\operatorname{div}\left(\frac{\nabla\Psi}{|\nabla\Psi|}\right)(\bar p)\\
&=-|\psi'(m)||\nabla R(\bar p)|\operatorname{div}\left(\frac{\psi'(m)}{|\psi'(m)|}\frac{\nabla R}{|\nabla R|}\right)(\bar p)\\
&=-\psi'(m)|\nabla R(\bar p)|\operatorname{div}\left(\frac{\nabla R}{|\nabla R|}\right)(\bar p). \tag*{(89)}
\end{aligned} 1 ≤ − ∣∇Ψ ( p ˉ ) ∣ div ( ∣∇Ψ∣ ∇Ψ ) ( p ˉ ) = − ∣ ψ ′ ( m ) ∣∣∇ R ( p ˉ ) ∣ div ( ∣ ψ ′ ( m ) ∣ ψ ′ ( m ) ∣∇ R ∣ ∇ R ) ( p ˉ ) = − ψ ′ ( m ) ∣∇ R ( p ˉ ) ∣ div ( ∣∇ R ∣ ∇ R ) ( p ˉ ) . (89) Fix p c h a r ∈ Ω [ t 1 , t 2 ] p_{\mathrm{char}}\in\Omega_{[t_1,t_2]} p char ∈ Ω [ t 1 , t 2 ] R ( p c h a r ) = r 0 ∈ ( R 1 , R 2 ) R(p_{\mathrm{char}})=r_0\in(R_1,R_2) R ( p char ) = r 0 ∈ ( R 1 , R 2 ) p c h a r ∈ C ( Σ r 0 ) p_{\mathrm{char}}\in C(\Sigma_{r_0}) p char ∈ C ( Σ r 0 ) p c h a r ∈ A p_{\mathrm{char}}\in A p char ∈ A
Step 1 (f f f r 0 r_0 r 0 We show that f f f r 0 r_0 r 0 D ~ \widetilde{\mathcal D} D C 2 C^2 C 2 q ∗ q^* q ∗ Σ r 0 ∩ A \Sigma_{r_0}\cap A Σ r 0 ∩ A ∇ ~ R ( q ∗ ) ≠ 0 \widetilde{\nabla}R(q^*)\ne0 ∇ R ( q ∗ ) = 0 Σ r 0 ∩ A \Sigma_{r_0}\cap A Σ r 0 ∩ A A A A Σ r 0 \Sigma_{r_0} Σ r 0 C 2 C^2 C 2 U ⊂ A U\subset A U ⊂ A q ∗ q^* q ∗ C 1 C^1 C 1
W ~ = ∇ ~ R ∣ ∇ ~ R ∣ 2 . \widetilde W=\frac{\widetilde{\nabla}R}{|\widetilde{\nabla}R|^2}. W = ∣ ∇ R ∣ 2 ∇ R . In particular, there exists K > 0 K>0 K > 0 ∣ W ~ ∣ = 1 / ∣ ∇ ~ R ∣ ≤ 1 / K |\widetilde W|=1/|\widetilde{\nabla}R|\le1/K ∣ W ∣ = 1/∣ ∇ R ∣ ≤ 1/ K U U U
For r 1 , r 2 r_1,r_2 r 1 , r 2 r 0 r_0 r 0 W ~ \widetilde W W q 1 ∈ Σ r 1 ∩ U q_1\in\Sigma_{r_1}\cap U q 1 ∈ Σ r 1 ∩ U q 2 ∈ Σ r 2 ∩ U q_2\in\Sigma_{r_2}\cap U q 2 ∈ Σ r 2 ∩ U γ \gamma γ D ~ \widetilde{\mathcal D} D γ \gamma γ A A A ∇ ~ R ⋅ W ~ = 1 \widetilde{\nabla}R\cdot\widetilde W=1 ∇ R ⋅ W = 1
T = ∫ 0 T ∇ ~ R ( γ ( t ) ) ⋅ W ~ ( γ ( t ) ) d t = ∫ 0 T ( R ∘ γ ) ′ ( t ) d t = ∣ r 1 − r 2 ∣ . T=\int_{0}^{T}\widetilde{\nabla}R(\gamma(t))\cdot\widetilde{W}(\gamma(t))\,dt=\int_{0}^{T}(R\circ\gamma)'(t)\,dt=\lvert r_1-r_2\rvert. T = ∫ 0 T ∇ R ( γ ( t )) ⋅ W ( γ ( t )) d t = ∫ 0 T ( R ∘ γ ) ′ ( t ) d t = ∣ r 1 − r 2 ∣ . Hence, we have
∫ 0 T ∣ γ ˙ ( t ) ∣ d t = ∫ 0 T ∣ W ~ ( γ ( t ) ) ∣ d t ≤ ( sup u ∣ W ~ ∣ ) T ≤ ∣ r 1 − r 2 ∣ K . \int_{0}^{T}\lvert\dot{\gamma}(t)\rvert\,dt=\int_{0}^{T}\lvert\widetilde{W}(\gamma(t))\rvert\,dt\leq\left(\sup_{u}\lvert\widetilde{W}\rvert\right)T\leq\frac{\lvert r_1-r_2\rvert}{K}. ∫ 0 T ∣ γ ˙ ( t )∣ d t = ∫ 0 T ∣ W ( γ ( t ))∣ d t ≤ ( u sup ∣ W ∣ ) T ≤ K ∣ r 1 − r 2 ∣ . By (14) for the right invariant structure, this implies: d ~ ( q 1 , q 2 ) ≤ C ∣ r 1 − r 2 ∣ \widetilde{d}(q_1,q_2)\leq C\lvert r_1-r_2\rvert d ( q 1 , q 2 ) ≤ C ∣ r 1 − r 2 ∣ C = 1 / K C=1/K C = 1/ K u u u d ~ \widetilde{d} d L L L U U U
∣ f ( r 1 ) − f ( r 2 ) ∣ = ∣ u ( q 1 ) − u ( q 2 ) ∣ ≤ L d ~ ( q 1 , q 2 ) ≤ L C ∣ r 1 − r 2 ∣ , \lvert f(r_1)-f(r_2)\rvert=\lvert u(q_1)-u(q_2)\rvert\leq L\widetilde{d}(q_1,q_2)\leq LC\lvert r_1-r_2\rvert, ∣ f ( r 1 ) − f ( r 2 )∣ = ∣ u ( q 1 ) − u ( q 2 )∣ ≤ L d ( q 1 , q 2 ) ≤ L C ∣ r 1 − r 2 ∣ , i.e., f f f r 0 r_0 r 0 M = L C M=LC M = L C
Step 2 (touching sequence). By Step 1, f f f r 0 r_0 r 0 M M M Lemma 5.7 , applied to f f f M M M r = r 0 r=r_0 r = r 0 r n → r 0 r_n\to r_0 r n → r 0 ψ n ∈ C 2 ( [ R 1 , R 2 ] ) \psi_n\in C^2([R_1,R_2]) ψ n ∈ C 2 ([ R 1 , R 2 ]) f − ψ n f-\psi_n f − ψ n r n r_n r n
0 < ∣ ψ n ′ ( r n ) ∣ ≤ M ∀ n . 0<\lvert\psi_n'(r_n)\rvert\leq M\qquad\forall n. 0 < ∣ ψ n ′ ( r n )∣ ≤ M ∀ n . Step 3 (non-characteristic points). Having produced, in Step 2, a sequence r n → r 0 r_n\to r_0 r n → r 0 n n n p n p_n p n p c h a r p_{\mathrm{char}} p char Σ r n \Sigma_{r_n} Σ r n p c h a r ∈ A p_{\mathrm{char}}\in A p char ∈ A p c h a r p_{\mathrm{char}} p char Σ r n \Sigma_{r_n} Σ r n Σ r n ∩ A \Sigma_{r_n}\cap A Σ r n ∩ A
Since p c h a r ∈ A p_{\mathrm{char}}\in A p char ∈ A A A A ρ > 0 \rho>0 ρ > 0 B ‾ ( p c h a r , ρ ) ⊂ A \overline{B}(p_{\mathrm{char}},\rho)\subset A B ( p char , ρ ) ⊂ A R ∈ C 2 ( A ) R\in C^2(A) R ∈ C 2 ( A ) ∇ e u c l R ≠ 0 \nabla_{\mathrm{eucl}}R\ne0 ∇ eucl R = 0 A A A
W e u c l = ∇ e u c l R ∣ ∇ e u c l R ∣ R 3 2 W_{\mathrm{eucl}}=\frac{\nabla_{\mathrm{eucl}}R}{\lvert\nabla_{\mathrm{eucl}}R\rvert_{\mathbb{R}^3}^{2}} W eucl = ∣ ∇ eucl R ∣ R 3 2 ∇ eucl R is well defined and of class C 1 C^1 C 1 B ‾ ( p c h a r , ρ ) \overline{B}(p_{\mathrm{char}},\rho) B ( p char , ρ )
M W = sup B ‾ ( p c h a r , ρ ) ∣ W e u c l ∣ R 3 < ∞ . M_W=\sup_{\overline{B}(p_{\mathrm{char}},\rho)}\lvert W_{\mathrm{eucl}}\rvert_{\mathbb{R}^3}<\infty. M W = B ( p char , ρ ) sup ∣ W eucl ∣ R 3 < ∞. Let γ : ( − τ 0 , τ 0 ) → R 3 \gamma:(-\tau_0,\tau_0)\to\mathbb{R}^3 γ : ( − τ 0 , τ 0 ) → R 3 W e u c l W_{\mathrm{eucl}} W eucl γ ( 0 ) = p c h a r \gamma(0)=p_{\mathrm{char}} γ ( 0 ) = p char B ‾ ( p c h a r , ρ ) \overline{B}(p_{\mathrm{char}},\rho) B ( p char , ρ ) ∇ e u c l R ⋅ W e u c l ≡ 1 \nabla_{\mathrm{eucl}}R\cdot W_{\mathrm{eucl}}\equiv1 ∇ eucl R ⋅ W eucl ≡ 1
d d τ R ( γ ( τ ) ) = ∇ e u c l R ( γ ( τ ) ) ⋅ W e u c l ( γ ( τ ) ) = 1 , so R ( γ ( τ ) ) = r 0 + τ , (90) \frac{d}{d\tau}R(\gamma(\tau))=\nabla_{\mathrm{eucl}}R(\gamma(\tau))\cdot W_{\mathrm{eucl}}(\gamma(\tau))=1,\qquad\text{so}\qquad R(\gamma(\tau))=r_0+\tau, \tag*{(90)} d τ d R ( γ ( τ )) = ∇ eucl R ( γ ( τ )) ⋅ W eucl ( γ ( τ )) = 1 , so R ( γ ( τ )) = r 0 + τ , (90) and, as long as γ ( τ ) ∈ B ‾ ( p c h a r , ρ ) \gamma(\tau)\in\overline{B}(p_{\mathrm{char}},\rho) γ ( τ ) ∈ B ( p char , ρ )
∣ γ ( τ ) − p c h a r ∣ R 3 ≤ ∫ 0 ∣ τ ∣ ∣ γ ˙ ( σ ) ∣ R 3 d σ ≤ M W ∣ τ ∣ . (91) \lvert\gamma(\tau)-p_{\mathrm{char}}\rvert_{\mathbb{R}^3}\leq\int_{0}^{\lvert\tau\rvert}\lvert\dot{\gamma}(\sigma)\rvert_{\mathbb{R}^3}\,d\sigma\leq M_W\lvert\tau\rvert. \tag*{(91)} ∣ γ ( τ ) − p char ∣ R 3 ≤ ∫ 0 ∣ τ ∣ ∣ γ ˙ ( σ ) ∣ R 3 d σ ≤ M W ∣ τ ∣ . (91) In particular, for ∣ τ ∣ < ρ / M W \lvert\tau\rvert<\rho/M_W ∣ τ ∣ < ρ / M W B ‾ ( p c h a r , ρ ) ⊂ A \overline{B}(p_{\mathrm{char}},\rho)\subset A B ( p char , ρ ) ⊂ A
Since r n → r 0 r_n\to r_0 r n → r 0 n ˉ \bar{n} n ˉ ∣ r n − r 0 ∣ < ρ / M W \lvert r_n-r_0\rvert<\rho/M_W ∣ r n − r 0 ∣ < ρ / M W n ≥ n ˉ n\geq\bar{n} n ≥ n ˉ n n n
q n = γ ( r n − r 0 ) ∈ B ( p c h a r , ρ ) ⊂ A . q_n=\gamma(r_n-r_0)\in B(p_{\mathrm{char}},\rho)\subset A. q n = γ ( r n − r 0 ) ∈ B ( p char , ρ ) ⊂ A . By (90) , R ( q n ) = r n R(q_n)=r_n R ( q n ) = r n
q n ∈ Σ r n ∩ A ≠ ∅ , q_n\in\Sigma_{r_n}\cap A\ne\varnothing, q n ∈ Σ r n ∩ A = ∅ , and by (91) ,
∣ q n − p c h a r ∣ R 3 ≤ M W ∣ r n − r 0 ∣ ⟶ 0. \lvert q_n-p_{\mathrm{char}}\rvert_{\mathbb{R}^3}\leq M_W\lvert r_n-r_0\rvert\longrightarrow0. ∣ q n − p char ∣ R 3 ≤ M W ∣ r n − r 0 ∣ ⟶ 0. Since Σ r n ∩ A \Sigma_{r_n} \cap A Σ r n ∩ A C 2 C^2 C 2 R ∈ C 2 ( A ) R \in C^2(A) R ∈ C 2 ( A ) ∇ e u c l R ≠ 0 \nabla_{\mathrm{eucl}}R \ne0 ∇ eucl R = 0 A A A ∇ R ≠ 0 \nabla R \ne0 ∇ R = 0 Σ r n ∩ A \Sigma_{r_n} \cap A Σ r n ∩ A
p n ∈ ( Σ r n ∩ A ) ∖ C ( Σ r n ) , ∣ p n − q n ∣ R 3 < 1 n . p_n \in(\Sigma_{r_n} \cap A) \setminus\mathcal{C}(\Sigma_{r_n}), \qquad|p_n-q_n|_{\mathbb{R}^3}<\frac{1}{n}. p n ∈ ( Σ r n ∩ A ) ∖ C ( Σ r n ) , ∣ p n − q n ∣ R 3 < n 1 . By the triangle inequality,
∣ p n − p c h a r ∣ R 3 ≤ ∣ p n − q n ∣ R 3 + ∣ q n − p c h a r ∣ R 3 ⟶ 0 , |p_n-p_{\mathrm{char}}|_{\mathbb{R}^3}\le|p_n-q_n|_{\mathbb{R}^3}+|q_n-p_{\mathrm{char}}|_{\mathbb{R}^3}\longrightarrow0, ∣ p n − p char ∣ R 3 ≤ ∣ p n − q n ∣ R 3 + ∣ q n − p char ∣ R 3 ⟶ 0 , so p n → p c h a r p_n \to p_{\mathrm{char}} p n → p char p n ∈ A p_n \in A p n ∈ A n n n ∇ R ( p n ) ≠ 0 \nabla R(p_n) \ne0 ∇ R ( p n ) = 0 R ∈ C 2 R \in C^2 R ∈ C 2 p n p_n p n p n ∈ A p_n \in A p n ∈ A Ψ n = ψ n ∘ R \Psi_n=\psi_n\circ R Ψ n = ψ n ∘ R C 2 C^2 C 2 p n p_n p n
Since ψ n ′ ( r n ) ≠ 0 \psi_n'(r_n) \ne0 ψ n ′ ( r n ) = 0 ∇ R ( p n ) ≠ 0 \nabla R(p_n) \ne0 ∇ R ( p n ) = 0 (89) holds at p n p_n p n
1 ≤ − ψ n ′ ( r n ) ∣ ∇ R ( p n ) ∣ div ( ∇ R ∣ ∇ R ∣ ) ( p n ) ≤ ∣ ψ n ′ ( r n ) ∣ ∣ ∇ R ( p n ) ∣ ∣ div ( ∇ R ∣ ∇ R ∣ ) ( p n ) ∣ . 1\le-\psi_n'(r_n)|\nabla R(p_n)|\operatorname{div}\left(\frac{\nabla R}{|\nabla R|}\right)(p_n)\le|\psi_n'(r_n)||\nabla R(p_n)|\left|\operatorname{div}\left(\frac{\nabla R}{|\nabla R|}\right)(p_n)\right|. 1 ≤ − ψ n ′ ( r n ) ∣∇ R ( p n ) ∣ div ( ∣∇ R ∣ ∇ R ) ( p n ) ≤ ∣ ψ n ′ ( r n ) ∣∣∇ R ( p n ) ∣ div ( ∣∇ R ∣ ∇ R ) ( p n ) . Since 0 < ∣ ψ n ′ ( r n ) ∣ ≤ M 0<|\psi_n'(r_n)|\le M 0 < ∣ ψ n ′ ( r n ) ∣ ≤ M
∣ div ( ∇ R ∣ ∇ R ∣ ) ( p n ) ∣ ≥ 1 ∣ ψ n ′ ( r n ) ∣ ∣ ∇ R ( p n ) ∣ ≥ 1 M ∣ ∇ R ( p n ) ∣ . (92) \left|\operatorname{div}\left(\frac{\nabla R}{|\nabla R|}\right)(p_n)\right|\ge\frac{1}{|\psi_n'(r_n)||\nabla R(p_n)|}\ge\frac{1}{M|\nabla R(p_n)|}. \tag*{(92)} div ( ∣∇ R ∣ ∇ R ) ( p n ) ≥ ∣ ψ n ′ ( r n ) ∣∣∇ R ( p n ) ∣ 1 ≥ M ∣∇ R ( p n ) ∣ 1 . (92) Step 4 (conclusion). Since p c h a r ∈ A p_{\mathrm{char}}\in A p char ∈ A A A A R ∈ C 2 R\in C^2 R ∈ C 2 p c h a r p_{\mathrm{char}} p char p n → p c h a r p_n\to p_{\mathrm{char}} p n → p char ∇ R \nabla R ∇ R ∇ R ( p n ) → ∇ R ( p c h a r ) \nabla R(p_n)\to\nabla R(p_{\mathrm{char}}) ∇ R ( p n ) → ∇ R ( p char ) R ∈ C 2 R\in C^2 R ∈ C 2 p c h a r p_{\mathrm{char}} p char ∇ e u c l R ≠ 0 \nabla_{\mathrm{eucl}}R\ne0 ∇ eucl R = 0 R R R Σ r 0 \Sigma_{r_0} Σ r 0 p c h a r p_{\mathrm{char}} p char p c h a r ∈ C ( Σ r 0 ) p_{\mathrm{char}}\in\mathcal{C}(\Sigma_{r_0}) p char ∈ C ( Σ r 0 ) ∇ R ( p c h a r ) = 0 \nabla R(p_{\mathrm{char}})=0 ∇ R ( p char ) = 0 ∣ ∇ R ( p n ) ∣ → 0 |\nabla R(p_n)|\to0 ∣∇ R ( p n ) ∣ → 0
∣ div ( ∇ R ∣ ∇ R ∣ ) ( p n ) ∣ ≥ 1 M ∣ ∇ R ( p n ) ∣ ⟶ + ∞ . \left|\operatorname{div}\left(\frac{\nabla R}{|\nabla R|}\right)(p_n)\right|\ge\frac{1}{M|\nabla R(p_n)|}\longrightarrow+\infty. div ( ∣∇ R ∣ ∇ R ) ( p n ) ≥ M ∣∇ R ( p n ) ∣ 1 ⟶ + ∞. Setting τ n : = f ( r n ) ∈ [ t 1 , t 2 ] \tau_n:=f(r_n)\in[t_1,t_2] τ n := f ( r n ) ∈ [ t 1 , t 2 ] p n ∈ Σ r n = Γ τ n p_n\in\Sigma_{r_n}=\Gamma_{\tau_n} p n ∈ Σ r n = Γ τ n
H n ( p n ) = ∣ div ( ∇ R ∣ ∇ R ∣ ) ( p n ) ∣ → ∞ as n → ∞ , H_n(p_n)=\left|\operatorname{div}\left(\frac{\nabla R}{|\nabla R|}\right)(p_n)\right|\to\infty\quad\text{as }n\to\infty, H n ( p n ) = div ( ∣∇ R ∣ ∇ R ) ( p n ) → ∞ as n → ∞ , which is the conclusion of Theorem 5.5. □
Self-similar solutions
We first observe that self similar solutions immediately yield a generalized foliation.
Lemma 5.9. Let u u u λ : [ 0 , T ∗ ) → ( 0 , 1 ] \lambda:[0,T^*)\to(0,1] λ : [ 0 , T ∗ ) → ( 0 , 1 ] Γ t = { p ∈ Ω ‾ ∣ u ( p ) = t } \Gamma_t=\{p\in\overline{\Omega}\mid u(p)=t\} Γ t = { p ∈ Ω ∣ u ( p ) = t } t ˉ ∈ ( 0 , T ∗ ) \bar t\in(0,T^*) t ˉ ∈ ( 0 , T ∗ ) { Γ t } t ∈ [ 0 , t ˉ ] \{\Gamma_t\}_{t\in[0,\bar t]} { Γ t } t ∈ [ 0 , t ˉ ] C 2 C^2 C 2 Ω ‾ ∖ F λ ( t ˉ ) \overline{\Omega}\setminus F_{\lambda(\bar t)} Ω ∖ F λ ( t ˉ )
Proof. By definition of self-similar solution, Γ t = δ λ ( t ) ( ∂ Ω ) = Σ λ ( t ) \Gamma_t=\delta_{\lambda(t)}(\partial\Omega)=\Sigma_{\lambda(t)} Γ t = δ λ ( t ) ( ∂ Ω ) = Σ λ ( t ) t ∈ [ 0 , T ∗ ) t\in[0,T^*) t ∈ [ 0 , T ∗ ) λ \lambda λ { Γ t } t ∈ [ 0 , t ˉ ] \{\Gamma_t\}_{t\in[0,\bar t]} { Γ t } t ∈ [ 0 , t ˉ ] { Σ λ } λ ∈ [ λ ( t ˉ ) , 1 ] \{\Sigma_\lambda\}_{\lambda\in[\lambda(\bar t),1]} { Σ λ } λ ∈ [ λ ( t ˉ ) , 1 ] Σ λ = δ λ ( ∂ Ω ) \Sigma_\lambda=\delta_\lambda(\partial\Omega) Σ λ = δ λ ( ∂ Ω ) F λ 1 ⊂ F λ 2 F_{\lambda_1}\subset F_{\lambda_2} F λ 1 ⊂ F λ 2 λ 1 < λ 2 \lambda_1<\lambda_2 λ 1 < λ 2 u u u Lemma 5.3 , applied with λ ˉ = λ ( t ˉ ) \bar\lambda=\lambda(\bar t) λ ˉ = λ ( t ˉ )
We are then interested in understanding whether the characteristic points of a self-similar generalized foliation are in fact regular. We start with the following simple observation.
Lemma 5.10. Let Ω ⊂ R 3 \Omega\subset\mathbb{R}^3 Ω ⊂ R 3 C 2 C^2 C 2 g : R 3 → R g:\mathbb{R}^3\to\mathbb{R} g : R 3 → R C 2 C^2 C 2 Ω \Omega Ω Lemma 5.3 , i.e. ∂ Ω = { g = 0 } \partial\Omega=\{g=0\} ∂ Ω = { g = 0 } ∇ e u c l g ≠ 0 \nabla_{\mathrm{eucl}}g\ne0 ∇ eucl g = 0 ∂ Ω \partial\Omega ∂ Ω p = ( x , y , z ) ∈ C ( ∂ Ω ) p=(x,y,z)\in\mathcal{C}(\partial\Omega) p = ( x , y , z ) ∈ C ( ∂ Ω ) p p p G G G Definition 5.2 if and only if z ≠ 0 z\ne0 z = 0
Proof. Since p ∈ C ( ∂ Ω ) p \in C(\partial\Omega) p ∈ C ( ∂ Ω ) X g ( p ) = Y g ( p ) = 0 Xg(p)=Yg(p)=0 X g ( p ) = Y g ( p ) = 0
g x ( p ) = y 2 g z ( p ) , g y ( p ) = − x 2 g z ( p ) . g_x(p)=\frac{y}{2}g_z(p), \qquad g_y(p)=-\frac{x}{2}g_z(p). g x ( p ) = 2 y g z ( p ) , g y ( p ) = − 2 x g z ( p ) . so that
∇ e u c l g ( p ) = g z ( p ) ( y 2 , − x 2 , 1 ) . \nabla_{\mathrm{eucl}}g(p)=g_z(p)\left(\frac{y}{2},-\frac{x}{2},1\right). ∇ eucl g ( p ) = g z ( p ) ( 2 y , − 2 x , 1 ) . As ∂ Ω \partial\Omega ∂ Ω C 2 C^2 C 2 ∇ e u c l g ( p ) ≠ 0 \nabla_{\mathrm{eucl}}g(p)\ne0 ∇ eucl g ( p ) = 0 ( y 2 , − x 2 , 1 ) \left(\frac{y}{2},-\frac{x}{2},1\right) ( 2 y , − 2 x , 1 ) 1 ≠ 0 1\ne0 1 = 0 g z ( p ) ≠ 0 g_z(p)\ne0 g z ( p ) = 0
⟨ D ( p ) , ∇ e u c l g ( p ) ⟩ = g z ( p ) ⟨ ( x , y , 2 z ) , ( y 2 , − x 2 , 1 ) ⟩ = g z ( p ) ( x y 2 − x y 2 + 2 z ) = 2 z g z ( p ) . \langle D(p),\nabla_{\mathrm{eucl}}g(p)\rangle=g_z(p)\left\langle(x,y,2z),\left(\frac{y}{2},-\frac{x}{2},1\right)\right\rangle=g_z(p)\left(\frac{xy}{2}-\frac{xy}{2}+2z\right)=2z g_z(p). ⟨ D ( p ) , ∇ eucl g ( p )⟩ = g z ( p ) ⟨ ( x , y , 2 z ) , ( 2 y , − 2 x , 1 ) ⟩ = g z ( p ) ( 2 x y − 2 x y + 2 z ) = 2 z g z ( p ) . Since g z ( p ) ≠ 0 g_z(p)\ne0 g z ( p ) = 0 z = 0 z=0 z = 0 G G G □ \square □
We can finally give the proof of Theorem 1.4.
Proof of Theorem 1.4. Let u ∈ C 0 ( Ω ‾ ) u\in C^0(\overline{\Omega}) u ∈ C 0 ( Ω ) Ω \Omega Ω u u u λ : [ 0 , T ∗ ) → ( 0 , 1 ] \lambda:[0,T^*)\to(0,1] λ : [ 0 , T ∗ ) → ( 0 , 1 ]
Fix t ˉ ∈ ( 0 , T ∗ ) \bar{t}\in(0,T^*) t ˉ ∈ ( 0 , T ∗ ) η > 0 \eta>0 η > 0 t ˉ + η < T ∗ \bar{t}+\eta<T^* t ˉ + η < T ∗
By Lemma 5.9, the level sets { Γ t } t ∈ [ 0 , t ˉ + η ] \{\Gamma_t\}_{t\in[0,\bar{t}+\eta]} { Γ t } t ∈ [ 0 , t ˉ + η ] C 2 C^2 C 2 Ω ‾ ∖ F λ ( t ˉ + η ) \overline{\Omega}\setminus F_{\lambda(\bar{t}+\eta)} Ω ∖ F λ ( t ˉ + η ) A = ⋃ λ ∈ ( λ ( t ˉ + η ) , 1 ) δ λ ( G ) A=\bigcup_{\lambda\in(\lambda(\bar{t}+\eta),1)}\delta_\lambda(G) A = ⋃ λ ∈ ( λ ( t ˉ + η ) , 1 ) δ λ ( G ) λ ˉ = λ ( t ˉ + η ) \bar{\lambda}=\lambda(\bar{t}+\eta) λ ˉ = λ ( t ˉ + η )
Moreover, by Corollary 3.5, the characteristic set C ( ∂ Ω ) C(\partial\Omega) C ( ∂ Ω ) p 1 , p 2 p_1,p_2 p 1 , p 2 z z z p 1 p_1 p 1 p 2 p_2 p 2 0 ∈ ∂ Ω 0\in\partial\Omega 0 ∈ ∂ Ω δ λ ( 0 ) = 0 \delta_\lambda(0)=0 δ λ ( 0 ) = 0 λ > 0 \lambda>0 λ > 0 Γ t = δ λ ( t ) ( ∂ Ω ) \Gamma_t=\delta_{\lambda(t)}(\partial\Omega) Γ t = δ λ ( t ) ( ∂ Ω ) t ∈ [ 0 , T ∗ ) t\in[0,T^*) t ∈ [ 0 , T ∗ ) u u u p 1 p_1 p 1 p 2 p_2 p 2 G G G Γ t \Gamma_t Γ t δ λ ( t ) ( p 1 ) \delta_{\lambda(t)}(p_1) δ λ ( t ) ( p 1 ) δ λ ( t ) ( p 2 ) \delta_{\lambda(t)}(p_2) δ λ ( t ) ( p 2 ) A A A
By Theorem 5.5 applied to p c h a r ∈ A ∩ C ( Γ t ˉ ) p_{\mathrm{char}}\in A\cap C(\Gamma_{\bar{t}}) p char ∈ A ∩ C ( Γ t ˉ ) { p n } n ⊂ Ω [ 0 , t ˉ + η ] \{p_n\}_n\subset\Omega_{[0,\bar{t}+\eta]} { p n } n ⊂ Ω [ 0 , t ˉ + η ]
p n ⟶ p c h a r , H n ( p n ) ⟶ ∞ as n ⟶ ∞ , p_n\longrightarrow p_{\mathrm{char}}, \qquad H_n(p_n)\longrightarrow\infty\quad\text{as }n\longrightarrow\infty, p n ⟶ p char , H n ( p n ) ⟶ ∞ as n ⟶ ∞ , where H n ( p n ) H_n(p_n) H n ( p n ) Γ τ n \Gamma_{\tau_n} Γ τ n p n p_n p n τ n ∈ [ 0 , t ˉ + η ] \tau_n\in[0,\bar{t}+\eta] τ n ∈ [ 0 , t ˉ + η ]
Since Γ τ n = δ λ ( τ n ) ( ∂ Ω ) \Gamma_{\tau_n}=\delta_{\lambda(\tau_n)}(\partial\Omega) Γ τ n = δ λ ( τ n ) ( ∂ Ω ) p n = δ λ n ( q n ) p_n=\delta_{\lambda_n}(q_n) p n = δ λ n ( q n ) q n ∈ ∂ Ω q_n\in\partial\Omega q n ∈ ∂ Ω λ n = λ ( τ n ) \lambda_n=\lambda(\tau_n) λ n = λ ( τ n )
H ( Γ τ n ) ( p n ) = λ n − 1 H ( ∂ Ω ) ( q n ) . H(\Gamma_{\tau_n})(p_n)=\lambda_n^{-1}H(\partial\Omega)(q_n). H ( Γ τ n ) ( p n ) = λ n − 1 H ( ∂ Ω ) ( q n ) . Since the initial datum has bounded curvature, H ( ∂ Ω ) ≤ c H(\partial\Omega)\le c H ( ∂ Ω ) ≤ c ∂ Ω ∖ C ( ∂ Ω ) \partial\Omega\setminus C(\partial\Omega) ∂ Ω ∖ C ( ∂ Ω )
H n ( p n ) = ∣ H ( Γ τ n ) ( p n ) ∣ ≤ c λ n . H_n(p_n)=\lvert H(\Gamma_{\tau_n})(p_n)\rvert\le\frac{c}{\lambda_n}. H n ( p n ) = ∣ H ( Γ τ n ) ( p n )∣ ≤ λ n c . Since τ n ∈ [ 0 , t ˉ + η ] \tau_n\in[0,\bar{t}+\eta] τ n ∈ [ 0 , t ˉ + η ] λ \lambda λ [ 0 , t ˉ + η ] ⊂ [ 0 , T ∗ ) [0,\bar{t}+\eta]\subset[0,T^*) [ 0 , t ˉ + η ] ⊂ [ 0 , T ∗ ) λ n ≥ min [ 0 , t ˉ + η ] λ = : λ min > 0 \lambda_n\ge\min_{[0,\bar{t}+\eta]}\lambda=:\lambda_{\min}>0 λ n ≥ min [ 0 , t ˉ + η ] λ =: λ m i n > 0 n n n
H n ( p n ) ≤ c λ min < ∞ for every n , H_n(p_n)\le\frac{c}{\lambda_{\min}}<\infty\qquad\text{for every }n, H n ( p n ) ≤ λ m i n c < ∞ for every n , uniformly in n n n H n ( p n ) ⟶ ∞ H_n(p_n)\longrightarrow\infty H n ( p n ) ⟶ ∞ □ \square □
References
(G. Bombardieri) Dipartimento di Matematica “Tullio Levi Civita”, Università di Padova, via Trieste 63, 35121 Padova (PD), Italy
Email address: gaia.bombardieri@studenti.unipd.it
(M. Fogagnolo) Dipartimento di Matematica “Tullio Levi Civita”, Università di Padova, via Trieste 63, 35121 Padova (PD), Italy
Email address: mattia.fogagnolo@unipd.it
(V. Franceschi) Dipartimento di Matematica “Tullio Levi Civita”, Università di Padova, via Trieste 63, 35121 Padova (PD), Italy
Email address: valentina.franceschi@unipd.it
