Introduction and main results
In a recent work, the first and the last two authors have addressed a non-local version of the so-called Massari’s problem about sets with prescribed mean curvature. In turn, they have shown regularity of (a tailored version of) almost minimizers for the fractional perimeter (see [13 ] ).
In the present paper, we develop a more comprehensive regularity theory for almost minimal sets to include a wider class of almost minimizers for the s s s n − s n-s n − s
Indeed, non-local almost minimizers theory finds broad utility in a variety of physical and geometric problems such as obstacle and free boundary problems, capillarity problems, and regularity of sets with prescribed non-local mean curvature (see for instance [4 , 12 , 15 ] ).
Another interesting setting in which almost minimizers naturally appear is given by metastable configurations. When a phase-coexistence model or an interface dynamics system is thrown out of its ground state (e.g., by an oscillatory external force or a mass constraint), the resulting configurations do not satisfy a classical stationary equation. Instead, they may become stuck in a near-stationary regime for a long time. Therefore, it is natural to interpret these intermediate states as almost minimal for the functional under consideration, and investigate their stability and regularity.
We stress that both the primary novelty and the main mathematical challenge of the present work lie in allowing for a parameter γ ∈ ( 0 , s ) \gamma\in(0,s) γ ∈ ( 0 , s ) H s H_s H s
However, through a careful analysis and leveraging the lower-order scaling of the error term, we deduce here a bound on H s H_s H s Theorem 3.4 here with [13 Theorem 2.6] ).
We start our discussion by recalling the definition of non-local interaction between disjoint sets introduced in [7 ] . Whenever needed, all subsets of R n \mathbb{R}^n R n
For every couple of disjoint sets A , B ⊆ R n A,B\subseteq\mathbb{R}^n A , B ⊆ R n A A A B B B
L s ( A , B ) : = ∬ A × B d x d y ∣ x − y ∣ n + s . \mathcal{L}_s(A,B):=\iint_{A\times B}\frac{dx\,dy}{\lvert x-y\rvert^{n+s}}. L s ( A , B ) := ∬ A × B ∣ x − y ∣ n + s d x d y . Given an open bounded Lipschitz domain Ω ⊆ R n \Omega\subseteq\mathbb{R}^n Ω ⊆ R n E ⊆ R n E\subseteq\mathbb{R}^n E ⊆ R n s s s E E E Ω \Omega Ω [7 ] as
Per s ( E , Ω ) = L s ( E ∩ Ω , E c ∩ Ω ) + L s ( E ∩ Ω , E c ∩ Ω c ) + L s ( E ∩ Ω c , E c ∩ Ω ) = ∬ Ω ♯ χ E ( x ) χ E c ( y ) ∣ x − y ∣ n + s d x d y . \begin{aligned}
\operatorname{Per}_s(E,\Omega)=\mathcal{L}_s(E\cap\Omega,E^c\cap\Omega)+\mathcal{L}_s(E\cap\Omega,E^c\cap\Omega^c)+\mathcal{L}_s(E\cap\Omega^c,E^c\cap\Omega) \\
&=\iint_{\Omega_\sharp}\frac{\chi_E(x)\chi_{E^c}(y)}{\lvert x-y\rvert^{n+s}}\,dx\,dy.
\end{aligned} Per s ( E , Ω ) = L s ( E ∩ Ω , E c ∩ Ω ) + L s ( E ∩ Ω , E c ∩ Ω c ) + L s ( E ∩ Ω c , E c ∩ Ω ) = ∬ Ω ♯ ∣ x − y ∣ n + s χ E ( x ) χ E c ( y ) d x d y . where Ω ♯ : = R n × R n ∖ ( Ω c × Ω c ) \Omega_\sharp:=\mathbb{R}^n\times\mathbb{R}^n\setminus(\Omega^c\times\Omega^c) Ω ♯ := R n × R n ∖ ( Ω c × Ω c )
Definition 1.1. Let Ω ⊆ R n \Omega\subseteq\mathbb{R}^n Ω ⊆ R n Λ ≥ 0 \Lambda\geq0 Λ ≥ 0 γ ∈ ( 0 , s ] \gamma\in(0,s] γ ∈ ( 0 , s ] E ⊆ R n E\subseteq\mathbb{R}^n E ⊆ R n ( Λ , γ ) (\Lambda,\gamma) ( Λ , γ ) Ω \Omega Ω
Per s ( E , Ω ) ≤ Per s ( F , Ω ) + Λ r n − s + γ , (1) \operatorname{Per}_s(E,\Omega)\leq\operatorname{Per}_s(F,\Omega)+\Lambda r^{n-s+\gamma}, \tag*{(1)} Per s ( E , Ω ) ≤ Per s ( F , Ω ) + Λ r n − s + γ , (1) for any set F ⊆ R n F \subseteq\mathbb{R}^n F ⊆ R n F △ E ⊆ B ~ r F\mathbin{\triangle}E \subseteq\widetilde{B}_r F △ E ⊆ B r B ~ r ⋐ Ω \widetilde{B}_r \Subset\Omega B r ⋐ Ω Ω \Omega Ω E ⊆ R n E \subseteq\mathbb{R}^n E ⊆ R n ( Λ , γ ) (\Lambda,\gamma) ( Λ , γ ) Ω \Omega Ω ( Λ , γ ) (\Lambda,\gamma) ( Λ , γ ) Ω ′ \Omega' Ω ′ Ω ′ ⋐ Ω \Omega' \Subset\Omega Ω ′ ⋐ Ω
Remark 1.2. Let s ∈ ( 0 , 1 ) s \in(0,1) s ∈ ( 0 , 1 ) p > n s p > \frac{n}{s} p > s n H ∈ L p ( B 1 ) H \in L^p(B_1) H ∈ L p ( B 1 ) E ⊆ R n E \subseteq\mathbb{R}^n E ⊆ R n [13 ] as
J s , H ( E , B 1 ) : = Per s ( E , B 1 ) + ∫ E ∩ B 1 H ( x ) d x . \mathcal{J}_{s,H}(E,B_1) := \operatorname{Per}_s(E,B_1) + \int_{E \cap B_1} H(x)\,dx. J s , H ( E , B 1 ) := Per s ( E , B 1 ) + ∫ E ∩ B 1 H ( x ) d x . By a direct computation, we have that a minimizer of J s , H \mathcal{J}_{s,H} J s , H ( Λ , γ ) (\Lambda,\gamma) ( Λ , γ ) Λ : = ∥ H ∥ L p ( B 1 ) \Lambda:= \lVert H\rVert_{L^p(B_1)} Λ := ∥ H ∥ L p ( B 1 ) γ : = s − n p \gamma:= s - \frac{n}{p} γ := s − p n
In the wake of [13 ] , we also give the definitions of almost sub- and almost supersolution property.
Definition 1.3. Let Ω ⊆ R n \Omega\subseteq\mathbb{R}^n Ω ⊆ R n Λ > 0 \Lambda> 0 Λ > 0 γ ∈ ( 0 , s ] \gamma\in(0,s] γ ∈ ( 0 , s ] E E E ( Λ , γ ) (\Lambda,\gamma) ( Λ , γ )
Per s ( E ∪ A , Ω ) − Per s ( E , Ω ) = L s ( A , E ∖ A ) − L s ( A , E c ) ≥ − Λ r n − s + γ , (2) \operatorname{Per}_s(E \cup A,\Omega) - \operatorname{Per}_s(E,\Omega) = \mathcal{L}_s(A,E \setminus A) - \mathcal{L}_s(A,E^c) \geq-\Lambda r^{n-s+\gamma}, \tag*{(2)} Per s ( E ∪ A , Ω ) − Per s ( E , Ω ) = L s ( A , E ∖ A ) − L s ( A , E c ) ≥ − Λ r n − s + γ , (2) for every A ⊆ E ∩ B ~ r A \subseteq E \cap\widetilde{B}_r A ⊆ E ∩ B r B ~ r ⋐ Ω \widetilde{B}_r \Subset\Omega B r ⋐ Ω
Similarly, we say that a set E ⊆ R n E \subseteq\mathbb{R}^n E ⊆ R n ( Λ , γ ) (\Lambda,\gamma) ( Λ , γ ) Ω \Omega Ω
Per s ( E , Ω ) − Per s ( E ∪ A , Ω ) = L s ( A , E ) − L s ( A , E c ∖ A ) ≤ Λ r n − s + γ , (3) \operatorname{Per}_s(E,\Omega) - \operatorname{Per}_s(E \cup A,\Omega) = \mathcal{L}_s(A,E) - \mathcal{L}_s(A,E^c \setminus A) \leq\Lambda r^{n-s+\gamma}, \tag*{(3)} Per s ( E , Ω ) − Per s ( E ∪ A , Ω ) = L s ( A , E ) − L s ( A , E c ∖ A ) ≤ Λ r n − s + γ , (3) for every A ⊆ E c ∩ B ~ r A \subseteq E^c \cap\widetilde{B}_r A ⊆ E c ∩ B r B ~ r ⋐ Ω \widetilde{B}_r \Subset\Omega B r ⋐ Ω
Remark 1.4. Recalling [13 Lemma 1.3] , we have that an almost minimal set satisfies both the almost sub- and the almost supersolution property. Also, if a set satisfies both the ( Λ , γ ) (\Lambda,\gamma) ( Λ , γ ) ( Λ , γ ) (\Lambda,\gamma) ( Λ , γ ) ( 2 Λ , γ ) (2\Lambda,\gamma) ( 2Λ , γ )
We point out that, in the classical case, other definitions of almost minimality are available. Compare for instance, Definition 1.1 with the definition of almost minimality coming from capillary problems [15 Definition 1.5] (see also [13 Definition 1.1] ) or Almgren’s definition in [1 ] (see also [11 Definition 2.10] ), in which the almost minimality is encoded in a multiplicative term of lower order. This notion is also known as quasi minimality . A key advantage of our approach is that we consider a weaker definition that encompasses the others as special cases. This provides a unified framework for the topic and isolates its essential features.
Definition 1.5 (Quasi-minimality). Let K ≥ 0 K \geq0 K ≥ 0 β ∈ ( 0 , s ] \beta\in(0,s] β ∈ ( 0 , s ] E ⊂ R n E \subset\mathbb{R}^n E ⊂ R n ( K , β ) (K,\beta) ( K , β ) Ω \Omega Ω B ~ r ⋐ Ω \widetilde{B}_r \Subset\Omega B r ⋐ Ω r r r F F F F ∖ B ~ r = E ∖ B ~ r F \setminus\widetilde{B}_r = E \setminus\widetilde{B}_r F ∖ B r = E ∖ B r
Per s ( E , B ~ r ) ≤ ( 1 + K r β ) Per s ( F , B ~ r ) . (4) \operatorname{Per}_s(E,\widetilde{B}_r) \leq(1+Kr^\beta)\operatorname{Per}_s(F,\widetilde{B}_r). \tag*{(4)} Per s ( E , B r ) ≤ ( 1 + K r β ) Per s ( F , B r ) . (4) When a set E E E ( K , 0 ) (K,0) ( K , 0 ) E E E Q Q Q Q > 1 Q>1 Q > 1
We stress every quasi minimal set in the sense of Definition 1.5 is also almost minimal in the sense of Definition 1.1, according to the following:
Lemma 1.6. Let E E E ( K , β ) (K,\beta) ( K , β ) B 1 B_1 B 1 E E E ( Λ , γ ) (\Lambda,\gamma) ( Λ , γ ) B 1 B_1 B 1 γ = β \gamma=\beta γ = β Λ = Λ ( n , s , K ) \Lambda=\Lambda(n,s,K) Λ = Λ ( n , s , K )
We present a proof of Lemma 1.6 in Section A.
Regularity for almost minimal surfaces
We show that almost minimal sets have C 1 , α C^{1,\alpha} C 1 , α α ∈ ( 0 , 1 ) \alpha\in(0,1) α ∈ ( 0 , 1 )
Theorem 1.7. Let s ∈ ( 0 , 1 ) s \in(0,1) s ∈ ( 0 , 1 ) E E E ( Λ , γ ) (\Lambda,\gamma) ( Λ , γ ) s s s B 1 B_1 B 1 ε 0 > 0 \varepsilon_0>0 ε 0 > 0 η 0 = η 0 ( ε 0 ) > 0 \eta_0=\eta_0(\varepsilon_0)>0 η 0 = η 0 ( ε 0 ) > 0 n n n s s s Λ \Lambda Λ γ \gamma γ
∂ E ∩ B 1 ⊆ { ∣ x n ∣ ≤ η 0 } , \partial E \cap B_1 \subseteq\{ |x_n| \le\eta_0 \}, ∂ E ∩ B 1 ⊆ { ∣ x n ∣ ≤ η 0 } , then ∂ E ∩ B 1 / 2 \partial E \cap B_{1/2} ∂ E ∩ B 1/2 C 1 , α C^{1,\alpha} C 1 , α α ∈ ( 0 , ε 0 ) \alpha\in(0,\varepsilon_0) α ∈ ( 0 , ε 0 )
We point out that Theorem 1.7 extends1 [13 Theorem 1.4] to almost minimal sets without any further additional assumption, making the corresponding regularity theory complete and fully applicable in the setting of almost minimal sets.
The strategy to prove our result goes as follows: we first show that almost minimal sets satisfy suitable uniform density estimates (see § 2 ), namely almost minimizers of Per s \operatorname{Per}_s Per s improvement of flatness technique for almost minimal sets (see Theorem 3.2 ) to deduce Theorem 1.7. The flatness improvement argument is implemented in § 3 .
At this stage, we notice that the estimates in § 3 can be improved for C 1 , α C^{1,\alpha} C 1 , α
Corollary 1.8. Let s ∈ ( 0 , 1 ) s \in(0,1) s ∈ ( 0 , 1 ) E E E ( Λ , γ ) (\Lambda,\gamma) ( Λ , γ ) s s s B 1 B_1 B 1 η 0 > 0 \eta_0>0 η 0 > 0 n n n s s s Λ \Lambda Λ γ \gamma γ
∂ E ∩ B 1 ⊆ { ∣ x n ∣ ≤ η 0 } , \partial E \cap B_1 \subseteq\{ |x_n| \le\eta_0 \}, ∂ E ∩ B 1 ⊆ { ∣ x n ∣ ≤ η 0 } , then ∂ E ∩ B 1 / 2 \partial E \cap B_{1/2} ∂ E ∩ B 1/2 C 1 , α C^{1,\alpha} C 1 , α α ∈ ( 0 , ε ‾ ) \alpha\in(0,\overline{\varepsilon}) α ∈ ( 0 , ε )
ε ‾ = − ( n + s + 2 − 2 γ ) + [ ( n + s + 2 − 2 γ ) 2 + 4 γ ( n + s − γ ) ] 1 / 2 2 ( n + s − γ ) . \overline{\varepsilon}=\frac{-(n+s+2-2\gamma)+\left[(n+s+2-2\gamma)^2+4\gamma(n+s-\gamma)\right]^{1/2}}{2(n+s-\gamma)}. ε = 2 ( n + s − γ ) − ( n + s + 2 − 2 γ ) + [ ( n + s + 2 − 2 γ ) 2 + 4 γ ( n + s − γ ) ] 1/2 . Hausdorff dimension of the singular set
The next step in analyzing the regularity of almost minimal sets is to estimate the Hausdorff dimension of the so-called singular set. To do so, we introduce the following definitions.
Definition 1.10. Let E E E ( Λ , γ ) (\Lambda,\gamma) ( Λ , γ ) B 1 B_1 B 1 Λ ≥ 0 \Lambda\ge0 Λ ≥ 0 γ ∈ ( 0 , s ] \gamma\in(0,s] γ ∈ ( 0 , s ] x 0 ∈ B 1 ∩ ∂ E x_0 \in B_1\cap\partial E x 0 ∈ B 1 ∩ ∂ E { λ k } k \{\lambda_k\}_k { λ k } k λ k → + ∞ \lambda_k \to+\infty λ k → + ∞ k → + ∞ k \to+\infty k → + ∞
If the blow-up sequence { λ k ( E − x 0 ) } k \{\lambda_k(E-x_0)\}_k { λ k ( E − x 0 ) } k 0 0 0 C C C C C C E E E x 0 x_0 x 0 x 0 x_0 x 0 C C C x x x Σ E \Sigma_E Σ E E E E Σ E \Sigma_E Σ E R n \mathbb{R}^n R n
Adapting the result in § 3 and 4 of [13 ] to Definition 1.1, we deduce an upper bound on the Hausdorff dimension of Σ E \Sigma_E Σ E
Theorem 1.11. Let E E E ( Λ , γ ) (\Lambda,\gamma) ( Λ , γ ) B 1 B_1 B 1 dim H ( Σ E ) ≤ n − 3 \dim_{\mathrm{H}}(\Sigma_E) \le n-3 dim H ( Σ E ) ≤ n − 3
The idea to establish Theorem 1.11 relies on a suitable monotonicity formula and a blow-up argument , which allow a classification of tangent cones and ultimately yields the desired estimate of the Hausdorff dimension of Σ E \Sigma_E Σ E
In particular, let E ⊆ R n E \subseteq\mathbb{R}^n E ⊆ R n ( Λ , γ ) (\Lambda,\gamma) ( Λ , γ ) ( n + 1 ) (n+1) ( n + 1 ) u ~ \widetilde{u} u χ E − χ E c \chi_E-\chi_{E^c} χ E − χ E c [8 ] (see also [7 Section 7] ) and we define the function
Φ E ( r ) : = r s − n ∫ B r + z 1 − s ∣ ∇ u ~ ∣ 2 d x d z + n − s γ Λ ^ r γ , (5) \Phi_E(r):=r^{s-n}\int_{B_r^+}z^{1-s}|\nabla\widetilde{u}|^2\,dx\,dz+\frac{n-s}{\gamma}\widehat{\Lambda}r^\gamma, \tag*{(5)} Φ E ( r ) := r s − n ∫ B r + z 1 − s ∣∇ u ∣ 2 d x d z + γ n − s Λ r γ , (5) where ω n \omega_n ω n n n n Λ ^ \widehat{\Lambda} Λ
Proposition 1.12 (Monotonicity formula). If E E E ( Λ , γ ) (\Lambda,\gamma) ( Λ , γ ) B 1 B_1 B 1 0 ∈ ∂ E 0\in\partial E 0 ∈ ∂ E Φ E ( r ) \Phi_E(r) Φ E ( r ) r r r
A proof of Theorem 1.11 is given along with the blow-up argument in § 5 . Moreover, since the arguments involved are variations of those presented in [13 Section 3] , we postpone some details regarding the monotonicity formula to Appendix § 8 .
We now proceed with the proof of the main results.
Uniform density estimates
As claimed in the introduction, the first step to prove the regularity for almost minimal sets is to show the following density estimates:
Proposition 2.1 (Uniform density estimates). Let E E E ( Λ , γ ) (\Lambda,\gamma) ( Λ , γ ) B 1 B_1 B 1 r 0 > 0 r_0>0 r 0 > 0 c 0 ∈ ( 0 , 1 ) c_0\in(0,1) c 0 ∈ ( 0 , 1 ) n n n s s s Λ \Lambda Λ γ \gamma γ x 0 ∈ ( ∂ E ) ∩ B 1 x_0\in(\partial E)\cap B_1 x 0 ∈ ( ∂ E ) ∩ B 1 r ∈ ( 0 , min { r 0 , dist ( x 0 , ∂ B 1 ) } ) r\in(0,\min\{r_0,\operatorname{dist}(x_0,\partial B_1)\}) r ∈ ( 0 , min { r 0 , dist ( x 0 , ∂ B 1 )})
∣ E ∩ B r ( x 0 ) ∣ ≥ c 0 r n |E\cap B_r(x_0)|\ge c_0r^n ∣ E ∩ B r ( x 0 ) ∣ ≥ c 0 r n and
∣ E c ∩ B r ( x 0 ) ∣ ≥ c 0 r n . |E^c\cap B_r(x_0)|\ge c_0r^n. ∣ E c ∩ B r ( x 0 ) ∣ ≥ c 0 r n . :::`
As a consequence, we obtain that local convergence2
Corollary 2.2 (Uniform convergence for almost minimal boundaries). Let { Λ k } k \{\Lambda_k\}_k { Λ k } k [ 0 , + ∞ ) [0,+\infty) [ 0 , + ∞ ) { γ k } k \{\gamma_k\}_k { γ k } k [ 0 , s ) [0,s) [ 0 , s ) { E k } k \{E_k\}_k { E k } k ( Λ k , γ k ) (\Lambda_k,\gamma_k) ( Λ k , γ k ) B 1 B_1 B 1
Suppose that Λ k → Λ \Lambda_k \to\Lambda Λ k → Λ Λ ∈ [ 0 , + ∞ ) \Lambda\in[0,+\infty) Λ ∈ [ 0 , + ∞ ) γ k → γ \gamma_k \to\gamma γ k → γ γ ∈ ( 0 , s ] \gamma\in(0,s] γ ∈ ( 0 , s ] E k → E E_k \to E E k → E L l o c 1 ( R n ) L^1_{\mathrm{loc}}(\mathbb{R}^n) L loc 1 ( R n ) E E E s s s k → + ∞ k \to+\infty k → + ∞
Then, E E E ( Λ , γ ) (\Lambda,\gamma) ( Λ , γ ) B 1 B_1 B 1
lim k → + ∞ d H ( ∂ E k , ∂ E ) = 0. \lim_{k\to+\infty} d_H(\partial E_k,\partial E)=0. k → + ∞ lim d H ( ∂ E k , ∂ E ) = 0. Another consequence of Proposition 2.1 is the so-called clean ball condition for almost minimal sets.
Corollary 2.3 (Clean ball condition for almost minimal sets). Let E E E ( Λ , γ ) (\Lambda,\gamma) ( Λ , γ ) B 1 B_1 B 1 x ∈ ∂ E x \in\partial E x ∈ ∂ E r > 0 r>0 r > 0 B r ( x ) ⊆ B 1 B_r(x)\subseteq B_1 B r ( x ) ⊆ B 1
Then, there exist a constant c > 0 c>0 c > 0 n n n s s s Λ \Lambda Λ γ \gamma γ y 1 ∈ E y_1\in E y 1 ∈ E y 2 ∈ B 1 ∖ E y_2\in B_1\setminus E y 2 ∈ B 1 ∖ E
B c r ( y 1 ) ⊆ E ∩ B r ( x ) and B c r ( y 2 ) ⊆ E c ∩ B r ( x ) . B_{cr}(y_1)\subseteq E\cap B_r(x)
\qquad\text{and}\qquad
B_{cr}(y_2)\subseteq E^c\cap B_r(x). B cr ( y 1 ) ⊆ E ∩ B r ( x ) and B cr ( y 2 ) ⊆ E c ∩ B r ( x ) . The arguments to show Proposition 2.1, Corollary 2.2, and Corollary 2.3 are a variation3 [13 ] . For the sake of completeness, we present a proof of Proposition 2.1 in Appendix § 2.1 . For a detailed discussion on Corollaries 2.2 and 2.3, we refer to § 2 and Appendix § 7 of [13 ] . In particular, we point out that the key to show Corollary 2.2 is that the conclusion of Proposition 2.1 holds for every set E k E_k E k c 0 c_0 c 0 r 0 r_0 r 0 k k k
Proof of uniform density estimates (Proposition 2.1)
Now, we present a proof of Proposition 2.1. The argument relies on the following auxiliary results.
Lemma 2.4. Let δ 0 > 0 \delta_0>0 δ 0 > 0 E E E ( Λ , γ ) (\Lambda,\gamma) ( Λ , γ ) B 1 B_1 B 1 Λ ≤ δ 0 \Lambda\leq\delta_0 Λ ≤ δ 0
a ( 1 ) ≤ δ 0 , (6) a(1)\leq\delta_0, \tag*{(6)} a ( 1 ) ≤ δ 0 , (6) then a ( 1 / 2 ) ≤ δ 0 a(1/2)\leq\delta_0 a ( 1/2 ) ≤ δ 0
Proof. Let r ∈ ( 1 / 2 , 1 ) r\in(1/2,1) r ∈ ( 1/2 , 1 ) A r : = E ∩ B r A_r:=E\cap B_r A r := E ∩ B r μ ( r ) : = ∣ A r ∣ \mu(r):=|A_r| μ ( r ) := ∣ A r ∣ μ ′ ( r ) = H n − 1 ( E ∩ ∂ B r ) \mu'(r)=\mathcal{H}^{n-1}(E\cap\partial B_r) μ ′ ( r ) = H n − 1 ( E ∩ ∂ B r )
Now, observe that, by Sobolev embeddings, we have that
∥ χ A r ∥ L n n − s ( R n ) = ∣ A r ∣ 1 − s n ≤ C [ χ A r ] H s / 2 = C L s ( A r , A r c ) , (7) \|\chi_{A_r}\|_{L^{\frac{n}{n-s}}(\mathbb{R}^n)}
=|A_r|^{1-\frac{s}{n}}
\leq C[\chi_{A_r}]_{H^{s/2}}
=C\mathcal{L}_s(A_r,A_r^c), \tag*{(7)} ∥ χ A r ∥ L n − s n ( R n ) = ∣ A r ∣ 1 − n s ≤ C [ χ A r ] H s /2 = C L s ( A r , A r c ) , (7) for some constant C > 0 C>0 C > 0 n n n s s s
Since
L s ( A r , A r c ) = L s ( A r , E ∩ A r c ) + L s ( A r , E c ) , \mathcal{L}_s(A_r,A_r^c)
=\mathcal{L}_s(A_r,E\cap A_r^c)+\mathcal{L}_s(A_r,E^c), L s ( A r , A r c ) = L s ( A r , E ∩ A r c ) + L s ( A r , E c ) , it follows from the almost minimality of E E E
L s ( A r , A r c ) ≤ 2 L s ( A r , E ∩ A r c ) + Λ r n − s + γ ≤ 2 L s ( A r , B r c ) + Λ r n − s + γ . (8) \mathcal{L}_s(A_r,A_r^c)
\leq2\mathcal{L}_s(A_r,E\cap A_r^c)+\Lambda r^{n-s+\gamma}
\leq2\mathcal{L}_s(A_r,B_r^c)+\Lambda r^{n-s+\gamma}. \tag*{(8)} L s ( A r , A r c ) ≤ 2 L s ( A r , E ∩ A r c ) + Λ r n − s + γ ≤ 2 L s ( A r , B r c ) + Λ r n − s + γ . (8) From (2.3) and (2.4) together, we deduce that
μ ( r ) 1 − s n ≤ C L s ( A r , B r c ) + C Λ r n − s + γ . (9) \mu(r)^{1-\frac{s}{n}}
\leq C\mathcal{L}_s(A_r,B_r^c)+C\Lambda r^{n-s+\gamma}. \tag*{(9)} μ ( r ) 1 − n s ≤ C L s ( A r , B r c ) + C Λ r n − s + γ . (9) and we aim to estimate the first term on the right-hand side. To do so, observe that, by the co-area formula, we have
L s ( A r , B r c ) = ∬ A r × B r c d x d y ∣ x − y ∣ n + s ≤ C ∫ A r ( ∫ r − ∣ x ∣ + ∞ d z z s + 1 ) d x ≤ C ∫ A r d x ( r − ∣ x ∣ ) s ≤ C ∫ 0 r μ ′ ( ρ ) ( r − ρ ) s d ρ , (10) \begin{aligned}
\mathcal{L}_{s}(A_r,B_r^c)=\iint_{A_r\times B_r^c}\frac{dx\,dy}{\lvert x-y\rvert^{n+s}}\leq C\int_{A_r}\left(\int_{r-\lvert x\rvert}^{+\infty}\frac{dz}{z^{s+1}}\right)dx \\
&\leq C\int_{A_r}\frac{dx}{(r-\lvert x\rvert)^s}\leq C\int_0^r\frac{\mu'(\rho)}{(r-\rho)^s}\,d\rho, \tag*{(10)}
\end{aligned} L s ( A r , B r c ) = ∬ A r × B r c ∣ x − y ∣ n + s d x d y ≤ C ∫ A r ( ∫ r − ∣ x ∣ + ∞ z s + 1 d z ) d x ≤ C ∫ A r ( r − ∣ x ∣ ) s d x ≤ C ∫ 0 r ( r − ρ ) s μ ′ ( ρ ) d ρ , (10) for some C > 0 C>0 C > 0 n n n s s s (10) into (2.5), we find
μ ( r ) 1 − s n ≤ C ∫ 0 r μ ′ ( ρ ) ( r − ρ ) s d ρ + C Λ r n − s + γ . \mu(r)^{1-\frac{s}{n}}\leq C\int_0^r\frac{\mu'(\rho)}{(r-\rho)^s}\,d\rho+C\Lambda r^{n-s+\gamma}. μ ( r ) 1 − n s ≤ C ∫ 0 r ( r − ρ ) s μ ′ ( ρ ) d ρ + C Λ r n − s + γ . Integrating la last inequality over r ∈ ( 1 2 , 1 ) r\in\left(\frac{1}{2},1\right) r ∈ ( 2 1 , 1 )
a ( 1 / 2 ) 1 − s n ≤ C μ ( 1 / 2 ) 1 − s n ≤ C ∫ 1 / 2 1 μ ( r ) 1 − s n d r ≤ C ∫ 1 / 2 1 ( ∫ 0 r μ ′ ( ρ ) ( r − ρ ) s d ρ + Λ r n − s + γ ) d r ≤ C ∫ 0 1 / 2 ∫ 1 / 2 1 μ ′ ( ρ ) ( r − ρ ) s d r d ρ + C ∫ 1 / 2 1 ∫ ρ 1 μ ′ ( ρ ) ( r − ρ ) s d r d ρ + C Λ ∫ 1 / 2 1 r n − s + γ d r ≤ C ∫ 0 1 / 2 μ ′ ( ρ ) ( 1 − ρ ) 1 − s d ρ + C ∫ 1 / 2 1 μ ′ ( ρ ) ( 1 − ρ ) 1 − s d ρ + C Λ ≤ C μ ( 1 ) + C Λ ≤ C a ( 1 ) + C Λ , (11) \begin{aligned}
a(1/2)^{1-\frac{s}{n}}\leq C\mu(1/2)^{1-\frac{s}{n}}\leq C\int_{1/2}^1\mu(r)^{1-\frac{s}{n}}\,dr \\
&\leq C\int_{1/2}^1\left(\int_0^r\frac{\mu'(\rho)}{(r-\rho)^s}\,d\rho+\Lambda r^{n-s+\gamma}\right)\,dr \\
&\leq C\int_0^{1/2}\int_{1/2}^1\frac{\mu'(\rho)}{(r-\rho)^s}\,dr\,d\rho+C\int_{1/2}^1\int_\rho^1\frac{\mu'(\rho)}{(r-\rho)^s}\,dr\,d\rho+C\Lambda\int_{1/2}^1r^{n-s+\gamma}\,dr \\
&\leq C\int_0^{1/2}\mu'(\rho)(1-\rho)^{1-s}\,d\rho+C\int_{1/2}^1\mu'(\rho)(1-\rho)^{1-s}\,d\rho+C\Lambda\\
&\leq C\mu(1)+C\Lambda\leq Ca(1)+C\Lambda, \tag*{(11)}
\end{aligned} a ( 1/2 ) 1 − n s ≤ C μ ( 1/2 ) 1 − n s ≤ C ∫ 1/2 1 μ ( r ) 1 − n s d r ≤ C ∫ 1/2 1 ( ∫ 0 r ( r − ρ ) s μ ′ ( ρ ) d ρ + Λ r n − s + γ ) d r ≤ C ∫ 0 1/2 ∫ 1/2 1 ( r − ρ ) s μ ′ ( ρ ) d r d ρ + C ∫ 1/2 1 ∫ ρ 1 ( r − ρ ) s μ ′ ( ρ ) d r d ρ + C Λ ∫ 1/2 1 r n − s + γ d r ≤ C ∫ 0 1/2 μ ′ ( ρ ) ( 1 − ρ ) 1 − s d ρ + C ∫ 1/2 1 μ ′ ( ρ ) ( 1 − ρ ) 1 − s d ρ + C Λ ≤ C μ ( 1 ) + C Λ ≤ C a ( 1 ) + C Λ , (11) up to renaming C C C Λ ≤ δ 0 \Lambda\leq\delta_0 Λ ≤ δ 0 (11) , we conclude that
a ( 1 / 2 ) ≤ ( C a ( 1 ) + C Λ ) n n − s ≤ ( C δ 0 s / n ) n n − s δ 0 ≤ δ 0 a(1/2)\leq\bigl(Ca(1)+C\Lambda\bigr)^{\frac{n}{n-s}}\leq\bigl(C\delta_0^{s/n}\bigr)^{\frac{n}{n-s}}\delta_0\leq\delta_0 a ( 1/2 ) ≤ ( C a ( 1 ) + C Λ ) n − s n ≤ ( C δ 0 s / n ) n − s n δ 0 ≤ δ 0 whenever δ 0 \delta_0 δ 0
The other partial result needed for the proof of Proposition 2.1 is a clean-out lemma.
Lemma 2.5. Let δ 0 \delta_0 δ 0 Lemma 2.4 . Let E E E ( Λ , γ ) (\Lambda,\gamma) ( Λ , γ ) B 1 B_1 B 1 Λ ≤ δ 0 \Lambda\leq\delta_0 Λ ≤ δ 0
a ( 1 ) < δ 0 ω n 2 n . (12) a(1)<\frac{\delta_0}{\omega_n2^n}. \tag*{(12)} a ( 1 ) < ω n 2 n δ 0 . (12) Then, ∣ E ∩ B 1 / 2 ∣ = 0 \lvert E\cap B_{1/2}\rvert=0 ∣ E ∩ B 1/2 ∣ = 0
Proof. Arguing by contradiction, suppose that ∣ E ∩ B 1 / 2 ∣ > 0 \lvert E\cap B_{1/2}\rvert>0 ∣ E ∩ B 1/2 ∣ > 0 x 0 ∈ E ∩ B 1 x_0\in E\cap B_1 x 0 ∈ E ∩ B 1
lim ρ → 0 a ( ρ ) = 1. \lim_{\rho\to0}a(\rho)=1. ρ → 0 lim a ( ρ ) = 1. On the other hand, by (12) , we also have that
∣ E ∩ B 1 / 2 ( x 0 ) ∣ ∣ B 1 / 2 ∣ ≤ 2 n ∣ E ∩ B 1 ∣ ≤ δ 0 . (13) \frac{\lvert E\cap B_{1/2}(x_0)\rvert}{\lvert B_{1/2}\rvert}\leq2^n\lvert E\cap B_1\rvert\leq\delta_0. \tag*{(13)} ∣ B 1/2 ∣ ∣ E ∩ B 1/2 ( x 0 )∣ ≤ 2 n ∣ E ∩ B 1 ∣ ≤ δ 0 . (13) Without loss of generality, we assume that x 0 = 0 x_0=0 x 0 = 0 (13) reads a ( 1 / 2 ) ≤ δ 0 a(1/2)\leq\delta_0 a ( 1/2 ) ≤ δ 0 Lemma 2.4 , we also have a ( 1 / 4 ) ≤ δ 0 a(1/4)\leq\delta_0 a ( 1/4 ) ≤ δ 0 Lemma 2.4 , we infer that, for every k ∈ N k\in\mathbb{N} k ∈ N
a ( 2 − k − 1 ) ≤ ( C a ( 2 − k ) + C Λ ) n n − s ≤ ( C δ 0 s / n ) n n − s δ 0 ≤ δ 0 . a(2^{-k-1})\leq\bigl(Ca(2^{-k})+C\Lambda\bigr)^{\frac{n}{n-s}}\leq\bigl(C\delta_0^{s/n}\bigr)^{\frac{n}{n-s}}\delta_0\leq\delta_0. a ( 2 − k − 1 ) ≤ ( C a ( 2 − k ) + C Λ ) n − s n ≤ ( C δ 0 s / n ) n − s n δ 0 ≤ δ 0 . Therefore,
1 = lim k → + ∞ a ( 2 − k − 1 ) ≤ δ 0 , 1=\lim_{k\to+\infty}a(2^{-k-1})\leq\delta_0, 1 = k → + ∞ lim a ( 2 − k − 1 ) ≤ δ 0 , which is a contradiction and concludes the proof. □
Proof of Proposition 2.1. Let r 0 > 0 r_0>0 r 0 > 0 Λ r 0 γ ≤ δ 0 \Lambda r_0^\gamma\leq\delta_0 Λ r 0 γ ≤ δ 0 δ 0 \delta_0 δ 0
∣ E ∩ B ρ ∣ > 0 (14) |E\cap B_\rho|>0 \tag*{(14)} ∣ E ∩ B ρ ∣ > 0 (14) and let x 0 ∈ ( ∂ E ) ∩ B ρ x_0\in(\partial E)\cap B_\rho x 0 ∈ ( ∂ E ) ∩ B ρ ρ ∈ ( 0 , r 0 ) \rho\in(0,r_0) ρ ∈ ( 0 , r 0 ) x 0 x_0 x 0
Let us recall that, by definition, if E E E ( Λ , γ ) (\Lambda,\gamma) ( Λ , γ ) B 1 B_1 B 1
Per s ( E , B ρ ) − Per s ( F , B ρ ) = Per s ( E , B 1 ) − Per s ( F , B 1 ) ≤ Λ ρ − n + s + γ , \operatorname{Per}_s(E,B_\rho)-\operatorname{Per}_s(F,B_\rho)=\operatorname{Per}_s(E,B_1)-\operatorname{Per}_s(F,B_1)\leq\Lambda\rho^{-n+s+\gamma}, Per s ( E , B ρ ) − Per s ( F , B ρ ) = Per s ( E , B 1 ) − Per s ( F , B 1 ) ≤ Λ ρ − n + s + γ , for every F F F F Δ E ⊆ B ρ ⋐ B 1 F\mathbin{\Delta}E\subseteq B_\rho\Subset B_1 F Δ E ⊆ B ρ ⋐ B 1 ρ \rho ρ
Per s ( ρ − 1 E , B 1 ) ≤ Per s ( ρ − 1 F , B 1 ) + Λ ρ γ . (15) \operatorname{Per}_s(\rho^{-1}E,B_1)\leq\operatorname{Per}_s(\rho^{-1}F,B_1)+\Lambda\rho^\gamma. \tag*{(15)} Per s ( ρ − 1 E , B 1 ) ≤ Per s ( ρ − 1 F , B 1 ) + Λ ρ γ . (15) Thus, without loss of generality, we assume ρ = 1 \rho=1 ρ = 1 E E E ( Λ , γ ) (\Lambda,\gamma) ( Λ , γ ) B 1 B_1 B 1 Λ ≤ δ 0 \Lambda\leq\delta_0 Λ ≤ δ 0 (14) reads
∣ E ∩ B 1 ∣ > 0. (16) |E\cap B_1|>0. \tag*{(16)} ∣ E ∩ B 1 ∣ > 0. (16) We claim that
∣ E ∩ B 1 ∣ ≥ δ 0 . (17) |E\cap B_1|\geq\delta_0. \tag*{(17)} ∣ E ∩ B 1 ∣ ≥ δ 0 . (17) Arguing by contradiction, suppose that
∣ E ∩ B 1 ∣ ≤ δ 0 . |E\cap B_1|\leq\delta_0. ∣ E ∩ B 1 ∣ ≤ δ 0 . Thus, by mean of Lemma 2.5, we infer ∣ E ∩ B 1 ∣ = 0 |E\cap B_1|=0 ∣ E ∩ B 1 ∣ = 0 (16) , proving (17) . Rescaling back by ρ \rho ρ
In addition, if E E E E c E^c E c E c E^c E c E E E
Remark 2.6. We point out that, if { Λ k } k \{\Lambda_k\}_k { Λ k } k { γ k } k \{\gamma_k\}_k { γ k } k { E k } k \{E_k\}_k { E k } k Λ ≥ 0 \Lambda\geq0 Λ ≥ 0 γ ∈ ( 0 , s ] \gamma\in(0,s] γ ∈ ( 0 , s ] Λ k → Λ \Lambda_k\to\Lambda Λ k → Λ γ k → γ \gamma_k\to\gamma γ k → γ k → + ∞ k\to+\infty k → + ∞ Λ 0 ≥ 0 \Lambda_0\geq0 Λ 0 ≥ 0 γ 0 ∈ [ 0 , s ) \gamma_0\in[0,s) γ 0 ∈ [ 0 , s ) Λ k ≤ Λ 0 \Lambda_k\leq\Lambda_0 Λ k ≤ Λ 0 γ k ≤ γ 0 \gamma_k\leq\gamma_0 γ k ≤ γ 0 k k k E k E_k E k c 0 c_0 c 0 r 0 r_0 r 0 n n n s s s Λ 0 \Lambda_0 Λ 0 γ 0 \gamma_0 γ 0 c 0 c_0 c 0 r 0 r_0 r 0 k k k
Improvement of flatness
This section is devoted to implement a flatness improvement technique useful to deduce Theorem 1.7. To this purpose, we now introduce the notion of flatness for a set. For later convenience, we define, for any x 0 ∈ R n x_0\in\mathbb{R}^n x 0 ∈ R n ν ∈ S n − 1 \nu\in\mathbb{S}^{n-1} ν ∈ S n − 1 r ∈ R r\in\mathbb{R} r ∈ R
C r ν ( x 0 ) : = { ( x ′ , x ν ) ∈ R n s.t. ∣ ( x ′ − x 0 ′ ) ∣ < r } = B r n − 1 ( x 0 ) × ν R . C_r^\nu(x_0):=\{(x',x_\nu)\in\mathbb{R}^n\ \text{s.t.}\ |(x'-x_0')|<r\}=B_r^{n-1}(x_0)\times\nu\mathbb{R}. C r ν ( x 0 ) := {( x ′ , x ν ) ∈ R n s.t. ∣ ( x ′ − x 0 ′ ) ∣ < r } = B r n − 1 ( x 0 ) × ν R . When ν = e n \nu=e_n ν = e n r r r x 0 x_0 x 0 C r ( x 0 ) C_r(x_0) C r ( x 0 ) C r ν : = C r ν ( 0 ) C_r^\nu:=C_r^\nu(0) C r ν := C r ν ( 0 ) C r : = C r ( 0 ) C_r:=C_r(0) C r := C r ( 0 )
Definition 3.1 ([13 Definition 2.13] ). Let h , r > 0 h,r>0 h , r > 0 x 0 ∈ ∂ E x_0\in\partial E x 0 ∈ ∂ E E E E ( h r ) \left(\frac{h}{r}\right) ( r h ) C r ν ( x 0 ) C_r^\nu(x_0) C r ν ( x 0 )
∂ E ∩ C r ν ( x 0 ) ⊆ { ∣ ( x n − x 0 ) ⋅ ν ∣ < h } . \partial E\cap C_r^\nu(x_0)\subseteq\{ |(x_n-x_0)\cdot\nu|<h\}. ∂ E ∩ C r ν ( x 0 ) ⊆ { ∣ ( x n − x 0 ) ⋅ ν ∣ < h } . In other words, we say that a set is ( h / r ) (h/r) ( h / r ) C r ν ( x 0 ) C_r^\nu(x_0) C r ν ( x 0 ) 2 h 2h 2 h ν \nu ν
The idea of the improvement of flatness technique is to show that if an almost minimal set has controlled flatness in each ball of radius 2 − k 2^{-k} 2 − k k = 0 , … , k 0 k=0,\ldots,k_0 k = 0 , … , k 0
Theorem 3.2 (Improvement of flatness). Let α > 0 \alpha>0 α > 0 E E E ( Λ , γ ) (\Lambda,\gamma) ( Λ , γ ) Pers \operatorname{Pers} Pers B 1 B_1 B 1 x 0 ∈ ∂ E ∩ B 1 / 2 x_0\in\partial E\cap B_{1/2} x 0 ∈ ∂ E ∩ B 1/2 k 0 ∈ N k_0\in\mathbb{N} k 0 ∈ N n n n s s s Λ \Lambda Λ γ \gamma γ α \alpha α
∂ E ∩ C 2 − j ν j ⊆ { ∣ x ⋅ ν j ∣ ≤ 2 − j ( 1 + α ) } , for all j = 0 , … , k , for some k ≥ k 0 , (18) \partial E\cap C_{2^{-j}}^{\nu_j}\subseteq\left\{|x\cdot\nu_j|\leq2^{-j(1+\alpha)}\right\},\qquad\text{for all }j=0,\ldots,k,\text{ for some }k\geq k_0, \tag*{(18)} ∂ E ∩ C 2 − j ν j ⊆ { ∣ x ⋅ ν j ∣ ≤ 2 − j ( 1 + α ) } , for all j = 0 , … , k , for some k ≥ k 0 , (18) where { ν j } j = 0 k \{\nu_j\}_{j=0}^k { ν j } j = 0 k ν 0 = e n \nu_0=e_n ν 0 = e n { ν j } j ≥ k + 1 \{\nu_j\}_{j\geq k+1} { ν j } j ≥ k + 1
∂ E ∩ C 2 − j ν j ⊆ { ∣ x ⋅ ν j ∣ ≤ 2 − j ( 1 + α ) } , for all j ≥ k + 1. \partial E\cap C_{2^{-j}}^{\nu_j}\subseteq\left\{|x\cdot\nu_j|\leq2^{-j(1+\alpha)}\right\},\qquad\text{for all }j\geq k+1. ∂ E ∩ C 2 − j ν j ⊆ { ∣ x ⋅ ν j ∣ ≤ 2 − j ( 1 + α ) } , for all j ≥ k + 1. Theorem 3.2 follows easily by rescaling from the following:
Proposition 3.3 (Rescaled improvement of flatness). Let Λ ≥ 0 \Lambda\geq0 Λ ≥ 0 γ ∈ ( 0 , s ] \gamma\in(0,s] γ ∈ ( 0 , s ] α > 0 \alpha>0 α > 0 k 0 ∈ N k_0\in\mathbb{N} k 0 ∈ N n n n s s s Λ \Lambda Λ γ \gamma γ α \alpha α k ≥ k 0 k\geq k_0 k ≥ k 0 E E E ( 2 − k γ Λ , γ ) (2^{-k\gamma}\Lambda,\gamma) ( 2 − k γ Λ , γ ) B 2 k B_{2^k} B 2 k 0 ∈ ∂ E 0\in\partial E 0 ∈ ∂ E
∂ E ∩ C 2 j ⊆ { ∣ x ⋅ ν j ∣ ≤ 2 − k α 2 j ( 1 + α ) } , for all j = 0 , … , k . (19) \partial E\cap C_{2^j}\subseteq\left\{|x\cdot\nu_j|\leq2^{-k\alpha}2^{j(1+\alpha)}\right\},\qquad\text{for all }j=0,\ldots,k. \tag*{(19)} ∂ E ∩ C 2 j ⊆ { ∣ x ⋅ ν j ∣ ≤ 2 − k α 2 j ( 1 + α ) } , for all j = 0 , … , k . (19) Then there exists a unit normal ν − 1 \nu_{-1} ν − 1
∂ E ∩ C 1 / 2 ⊆ { ∣ x ⋅ ν − 1 ∣ ≤ 2 − k α − α − 1 } . (20) \partial E\cap C_{1/2}\subseteq\left\{|x\cdot\nu_{-1}|\leq2^{-k\alpha-\alpha-1}\right\}. \tag*{(20)} ∂ E ∩ C 1/2 ⊆ { ∣ x ⋅ ν − 1 ∣ ≤ 2 − k α − α − 1 } . (20) The strategy to prove Proposition 3.3 relies on a contradiction argument. Let Λ ≥ 0 \Lambda\geq0 Λ ≥ 0 γ ∈ ( 0 , s ] \gamma\in(0,s] γ ∈ ( 0 , s ] α > 0 \alpha>0 α > 0
We consider a sequence of ( 2 − k γ Λ , γ ) (2^{-k\gamma}\Lambda,\gamma) ( 2 − k γ Λ , γ ) { E k } k \{E_k\}_k { E k } k k ∈ N k\in\mathbb{N} k ∈ N a k : = 2 − k α a_k:=2^{-k\alpha} a k := 2 − k α j k ∈ N j_k\in\mathbb{N} j k ∈ N { ν j k } j = 0 j k \{\nu_j^k\}_{j=0}^{j_k} { ν j k } j = 0 j k
∂ E k ∩ C 2 j ⊆ { ∣ x ⋅ ν j k ∣ ≤ a k 2 j ( 1 + α ) } , for all j ∈ { 0 , … , j k } , (21) \partial E_k\cap C_{2^j}\subseteq\left\{|x\cdot\nu_j^k|\leq a_k2^{j(1+\alpha)}\right\},\qquad\text{for all }j\in\{0,\ldots,j_k\}, \tag*{(21)} ∂ E k ∩ C 2 j ⊆ { ∣ x ⋅ ν j k ∣ ≤ a k 2 j ( 1 + α ) } , for all j ∈ { 0 , … , j k } , (21) with ν 0 k = e n \nu_0^k=e_n ν 0 k = e n k k k
By contradiction, suppose that
∂ E k cannot fit in any cylinder of height 2 − ( k + 1 ) α in C 1 / 2 . (22) \partial E_k\text{ cannot fit in any cylinder of height }2^{-(k+1)\alpha}\text{ in }C_{1/2}. \tag*{(22)} ∂ E k cannot fit in any cylinder of height 2 − ( k + 1 ) α in C 1/2 . (22) Our goal is to show that there exists a subsequence { E k l } l \{E_{k_l}\}_l { E k l } l u u u
For later convenience, let us denote by E k ∗ E_k^* E k ∗ E k E_k E k a k a_k a k
E k ∗ : = { ( x ′ , x n ) s.t. ( x ′ , a k x n ) ∈ E k } . E_k^*:=\{(x',x_n)\text{ s.t. }(x',a_kx_n)\in E_k\}. E k ∗ := {( x ′ , x n ) s.t. ( x ′ , a k x n ) ∈ E k } . Euler-Lagrange inequalities
We remark that, as anticipated in the introduction, the main motivation for introducing almost minimizers (even in the classical case) is for being able to deal with object that are not proper minimizers of the energy functional under investigation. Therefore, such almost minimizers do not necessarily satisfy any Euler-Lagrange equation. On the other hand, we show here that one is still able to infer some Euler-Lagrange inequalities at smaller scales .
In our setting, almost minimizers for the s s s [7 Theorem 5.1] ). However, if E E E E E E
Theorem 3.4. Let E E E ( Λ , γ ) (\Lambda,\gamma) ( Λ , γ ) B 1 B_1 B 1 2 r 2r 2 r E E E x 0 ∈ ∂ E x_0 \in\partial E x 0 ∈ ∂ E ε = ε ( n , s , γ ) > 0 \varepsilon=\varepsilon(n,s,\gamma)>0 ε = ε ( n , s , γ ) > 0
H s r 1 + ε / 2 [ E ] ( x 0 ) : = ∫ B r 1 + ε / 2 c ( x 0 ) χ E ( y ) − χ E c ( y ) ∣ x r − y ∣ n + s d y ≤ c r − s + ε 1 − s 2 . (23) H_s^{r^{1+\varepsilon/2}}[E](x_0):=\int_{B_{r^{1+\varepsilon/2}}^c(x_0)}\frac{\chi_E(y)-\chi_{E^c}(y)}{|x_r-y|^{n+s}}\,dy\le cr^{-s+\varepsilon\frac{1-s}{2}}. \tag*{(23)} H s r 1 + ε /2 [ E ] ( x 0 ) := ∫ B r 1 + ε /2 c ( x 0 ) ∣ x r − y ∣ n + s χ E ( y ) − χ E c ( y ) d y ≤ c r − s + ε 2 1 − s . (23) for some constant c > 0 c>0 c > 0 n n n s s s Λ \Lambda Λ γ \gamma γ
Without loss of generality, let us suppose that x 0 = 0 ∈ ∂ E x_0=0\in\partial E x 0 = 0 ∈ ∂ E B 2 r ( − 2 r e n ) B_{2r}(-2re_n) B 2 r ( − 2 r e n ) ∂ E \partial E ∂ E 0 0 0
Proposition 3.5. Let E E E ( Λ , γ ) (\Lambda,\gamma) ( Λ , γ ) B 1 B_1 B 1 B 2 r ( − 2 r e n ) B_{2r}(-2re_n) B 2 r ( − 2 r e n ) E E E 0 ∈ ∂ E 0\in\partial E 0 ∈ ∂ E ε = ε ( n , s , γ ) > 0 \varepsilon=\varepsilon(n,s,\gamma)>0 ε = ε ( n , s , γ ) > 0 x r ∈ E c ∩ B r 1 + ε x_r\in E^c\cap B_{r^{1+\varepsilon}} x r ∈ E c ∩ B r 1 + ε
H s [ E ] ( x r ) : = ∫ R n χ E ( y ) − χ E c ( y ) ∣ x r − y ∣ n + s d y ≤ c r − s + ( γ − ε ( n + s − γ ) ) (24) H_s[E](x_r):=\int_{\mathbb{R}^n}\frac{\chi_E(y)-\chi_{E^c}(y)}{|x_r-y|^{n+s}}\,dy\le cr^{-s+(\gamma-\varepsilon(n+s-\gamma))} \tag*{(24)} H s [ E ] ( x r ) := ∫ R n ∣ x r − y ∣ n + s χ E ( y ) − χ E c ( y ) d y ≤ c r − s + ( γ − ε ( n + s − γ )) (24) for some constant c > 0 c>0 c > 0 n n n s s s Λ \Lambda Λ
Proposition 3.6. Let E E E ( Λ , γ ) (\Lambda,\gamma) ( Λ , γ ) B 1 B_1 B 1 B 2 r ( − 2 r e n ) B_{2r}(-2re_n) B 2 r ( − 2 r e n ) E E E 0 ∈ ∂ E 0\in\partial E 0 ∈ ∂ E ε = ε ( n , s , γ ) > 0 \varepsilon=\varepsilon(n,s,\gamma)>0 ε = ε ( n , s , γ ) > 0 x r ∈ E c ∩ B r 1 + ε x_r\in E^c\cap B_{r^{1+\varepsilon}} x r ∈ E c ∩ B r 1 + ε
∫ B r 1 + ε / 2 c χ E ( y ) − χ E c ( y ) ∣ x r − y ∣ n + s d y ≤ c r − s + ε 1 − s 2 . (25) \int_{B_{r^{1+\varepsilon/2}}^c}\frac{\chi_E(y)-\chi_{E^c}(y)}{|x_r-y|^{n+s}}\,dy\le cr^{-s+\varepsilon\frac{1-s}{2}}. \tag*{(25)} ∫ B r 1 + ε /2 c ∣ x r − y ∣ n + s χ E ( y ) − χ E c ( y ) d y ≤ c r − s + ε 2 1 − s . (25) for some constant c > 0 c>0 c > 0 n n n s s s Λ \Lambda Λ γ \gamma γ
Proof of Proposition 3.5. Let us take ε > 0 \varepsilon>0 ε > 0 ρ : = r 1 + ε \rho:=r^{1+\varepsilon} ρ := r 1 + ε d : = r 1 + 2 ε d:=r^{1+2\varepsilon} d := r 1 + 2 ε B r ( − r e n ) B_r(-re_n) B r ( − r e n )
A t : = E c ∩ B r ( ( t − r ) e n ) , for every t ∈ [ 0 , d ] . (26) A_t:=E^c\cap B_r((t-r)e_n),\qquad\text{for every }t\in[0,d]. \tag*{(26)} A t := E c ∩ B r (( t − r ) e n ) , for every t ∈ [ 0 , d ] . (26) We consider the set A d A_d A d E c ∩ B d ⊆ A d E^c\cap B_d\subseteq A_d E c ∩ B d ⊆ A d E E E
Per s ( E , B 1 ) − Per s ( E ∪ A d , B 1 ) ≤ Λ ρ n − s + γ = Λ r ( n − s + γ ) ( 1 + ε ) . (27) \operatorname{Per}_s(E,B_1)-\operatorname{Per}_s(E\cup A_d,B_1)\le\Lambda\rho^{n-s+\gamma}=\Lambda r^{(n-s+\gamma)(1+\varepsilon)}. \tag*{(27)} Per s ( E , B 1 ) − Per s ( E ∪ A d , B 1 ) ≤ Λ ρ n − s + γ = Λ r ( n − s + γ ) ( 1 + ε ) . (27) On the other hand, by calibration for the s s s [6 Proposition 5.2] ), we have that
Per s ( E , B 1 ) − Per s ( E ∪ A d , B 1 ) = ∫ A d H s [ E ∪ A t x ] ( x ) d x , (28) \operatorname{Per}_s(E,B_1)-\operatorname{Per}_s(E\cup A_d,B_1)=\int_{A_d}H_s[E\cup A_{t_x}](x)\,dx, \tag*{(28)} Per s ( E , B 1 ) − Per s ( E ∪ A d , B 1 ) = ∫ A d H s [ E ∪ A t x ] ( x ) d x , (28) where t x ∈ [ 0 , d ] t_x\in[0,d] t x ∈ [ 0 , d ] x ∈ ∂ ( E ∪ A t x ) x\in\partial(E\cup A_{t_x}) x ∈ ∂ ( E ∪ A t x )
Arguing by contradiction, let us suppose that, for (almost) every x ∈ A d x\in A_d x ∈ A d
H s [ E ∪ A t x ] ( x ) > Λ c 0 r − s + ( γ − ε ( n + s − γ ) ) , H_s[E\cup A_{t_x}](x)>\frac{\Lambda}{c_0}r^{-s+(\gamma-\varepsilon(n+s-\gamma))}, H s [ E ∪ A t x ] ( x ) > c 0 Λ r − s + ( γ − ε ( n + s − γ )) , where the constant c 0 > 0 c_0>0 c 0 > 0
Per s ( E , B 1 ) − Per s ( E ∪ A d , B 1 ) > Λ c 0 r − s + ( γ − ε ( n + s − γ ) ) ∣ A d ∣ ≥ Λ c 0 r − s + ( γ − ε ( n + s − γ ) ) ∣ E c ∩ B d ∣ ≥ Λ r − s + ( γ − ε ( n + s − γ ) ) d n = Λ r ( n − s + γ ) ( 1 + ε ) . \begin{aligned}
\operatorname{Per}_s(E,B_1)-\operatorname{Per}_s(E\cup A_d,B_1)>\frac{\Lambda}{c_0}r^{-s+(\gamma-\varepsilon(n+s-\gamma))}|A_d| \\
&\ge\frac{\Lambda}{c_0}r^{-s+(\gamma-\varepsilon(n+s-\gamma))}|E^c\cap B_d|\ge\Lambda r^{-s+(\gamma-\varepsilon(n+s-\gamma))}d^n=\Lambda r^{(n-s+\gamma)(1+\varepsilon)}.
\end{aligned} Per s ( E , B 1 ) − Per s ( E ∪ A d , B 1 ) > c 0 Λ r − s + ( γ − ε ( n + s − γ )) ∣ A d ∣ ≥ c 0 Λ r − s + ( γ − ε ( n + s − γ )) ∣ E c ∩ B d ∣ ≥ Λ r − s + ( γ − ε ( n + s − γ )) d n = Λ r ( n − s + γ ) ( 1 + ε ) . Plugging this into (3.10), we obtain a contradiction. Therefore, there exist t r ∈ [ 0 , d ] t_r\in[0,d] t r ∈ [ 0 , d ] x r ∈ ∂ ( E ∩ A t r ) ⊂ E c ∩ A d x_r\in\partial(E\cap A_{t_r})\subset E^c\cap A_d x r ∈ ∂ ( E ∩ A t r ) ⊂ E c ∩ A d
H s [ E ∪ A t r ] ( x r ) ≤ Λ c 0 r − s + ( γ − ε ( n + s − γ ) ) . (29) H_s[E\cup A_{t_r}](x_r)\le\frac{\Lambda}{c_0}r^{-s+(\gamma-\varepsilon(n+s-\gamma))}. \tag*{(29)} H s [ E ∪ A t r ] ( x r ) ≤ c 0 Λ r − s + ( γ − ε ( n + s − γ )) . (29) Moreover, choosing ε ∈ ( 0 , γ n + s − γ ) \varepsilon\in\left(0,\frac{\gamma}{n+s-\gamma}\right) ε ∈ ( 0 , n + s − γ γ ) γ − ε ( n + s − γ ) > 0 \gamma-\varepsilon(n+s-\gamma)>0 γ − ε ( n + s − γ ) > 0
Proof of Proposition 3.6. Let x r x_r x r t r t_r t r ξ : = r 1 + ε / 2 \xi:= r^{1+\varepsilon/2} ξ := r 1 + ε /2
∫ B ξ c χ E ( y ) − χ E c ( y ) ∣ x r − y ∣ n + s d y = H s [ E ∪ A t r ] ( x r ) − ∫ B ξ χ E ∪ A t r ( y ) − χ ( E ∪ A t r ) c ( y ) ∣ x r − y ∣ n + s d y . (30) \int_{B_\xi^c} \frac{\chi_E(y)-\chi_{E^c}(y)}{|x_r-y|^{n+s}}\,dy=H_s[E\cup A_{t_r}](x_r)-\int_{B_\xi}\frac{\chi_{E\cup A_{t_r}}(y)-\chi_{(E\cup A_{t_r})^c}(y)}{|x_r-y|^{n+s}}\,dy. \tag*{(30)} ∫ B ξ c ∣ x r − y ∣ n + s χ E ( y ) − χ E c ( y ) d y = H s [ E ∪ A t r ] ( x r ) − ∫ B ξ ∣ x r − y ∣ n + s χ E ∪ A t r ( y ) − χ ( E ∪ A t r ) c ( y ) d y . (30) Observe that the set E ∪ A t r E\cup A_{t_r} E ∪ A t r x r x_r x r − r − 1 -r^{-1} − r − 1 x r ∈ B ξ / 2 x_r\in B_{\xi/2} x r ∈ B ξ /2 r r r φ \varphi φ P : = { y n < φ ( y ′ ) } P:=\{y_n<\varphi(y')\} P := { y n < φ ( y ′ )} [3 Appendix B] (see also [10 , 2 ] ), we have that
∫ B ξ χ E ∪ A t r ( y ) − χ ( E ∪ A t r ) c ( y ) ∣ x r − y ∣ n + s d y ≥ ∫ B ξ χ P ( y ) − χ P c ( y ) ∣ x r − y ∣ n + s d y = − ∫ B ξ ′ ∫ φ ( y ′ ) − φ ( x r ′ ) ∣ x r − y ′ ∣ ∇ φ ( x r ) ⋅ ( y ′ − x r ′ ) ∣ x r − y ′ ∣ d t ( 1 + t 2 ) n + s 2 ∣ x r − y ′ ∣ n + s − 1 d y ′ ≥ − ∫ B ξ ′ ∥ φ ∥ C 2 ( B ξ ′ ) ∣ x r ′ − y ′ ∣ 2 ∣ x r ′ − y ′ ∣ ∣ x r ′ − y ′ ∣ n + s − 1 d y ′ = − ∥ φ ∥ C 2 ( B ξ ′ ) ∫ B ξ ′ d y ′ ∣ x r ′ − y ′ ∣ n + s − 2 ≥ − c r ∫ 0 3 2 ξ σ − s d σ = − c r − 1 ξ 1 − s = − c r − s + ε 2 ( 1 − s ) . (31) \begin{aligned}
\int_{B_\xi}\frac{\chi_{E\cup A_{t_r}}(y)-\chi_{(E\cup A_{t_r})^c}(y)}{|x_r-y|^{n+s}}\,dy
&\geq\int_{B_\xi}\frac{\chi_P(y)-\chi_{P^c}(y)}{|x_r-y|^{n+s}}\,dy\\
&=-\int_{B'_\xi}\int_{\frac{\varphi(y')-\varphi(x_r')}{|x_r-y'|}}^{\frac{\nabla\varphi(x_r)\cdot(y'-x_r')}{|x_r-y'|}}\frac{dt}{(1+t^2)^{\frac{n+s}{2}}|x_r-y'|^{n+s-1}}\,dy'\\
&\geq-\int_{B'_\xi}\frac{\|\varphi\|_{C^2(B'_\xi)}|x_r'-y'|^2}{|x_r'-y'|\,|x_r'-y'|^{n+s-1}}\,dy'\\
&=-\|\varphi\|_{C^2(B'_\xi)}\int_{B'_\xi}\frac{dy'}{|x_r'-y'|^{n+s-2}}\\
&\geq-\frac{c}{r}\int_0^{\frac{3}{2}\xi}\sigma^{-s}\,d\sigma
=-cr^{-1}\xi^{1-s}
=-cr^{-s+\frac{\varepsilon}{2}(1-s)}. \tag*{(31)}
\end{aligned} ∫ B ξ ∣ x r − y ∣ n + s χ E ∪ A t r ( y ) − χ ( E ∪ A t r ) c ( y ) d y ≥ ∫ B ξ ∣ x r − y ∣ n + s χ P ( y ) − χ P c ( y ) d y = − ∫ B ξ ′ ∫ ∣ x r − y ′ ∣ φ ( y ′ ) − φ ( x r ′ ) ∣ x r − y ′ ∣ ∇ φ ( x r ) ⋅ ( y ′ − x r ′ ) ( 1 + t 2 ) 2 n + s ∣ x r − y ′ ∣ n + s − 1 d t d y ′ ≥ − ∫ B ξ ′ ∣ x r ′ − y ′ ∣ ∣ x r ′ − y ′ ∣ n + s − 1 ∥ φ ∥ C 2 ( B ξ ′ ) ∣ x r ′ − y ′ ∣ 2 d y ′ = − ∥ φ ∥ C 2 ( B ξ ′ ) ∫ B ξ ′ ∣ x r ′ − y ′ ∣ n + s − 2 d y ′ ≥ − r c ∫ 0 2 3 ξ σ − s d σ = − c r − 1 ξ 1 − s = − c r − s + 2 ε ( 1 − s ) . (31) for some positive constant c c c n n n s s s
∫ B ξ c χ E ( y ) − χ E c ( y ) ∣ x r − y ∣ n + s d y ≤ c r − s + ( γ − ε ( n + s − γ ) ) + c r − s + ε 1 − s 2 , \int_{B_\xi^c}\frac{\chi_E(y)-\chi_{E^c}(y)}{|x_r-y|^{n+s}}\,dy
\leq cr^{-s+(\gamma-\varepsilon(n+s-\gamma))}+cr^{-s+\varepsilon\frac{1-s}{2}}, ∫ B ξ c ∣ x r − y ∣ n + s χ E ( y ) − χ E c ( y ) d y ≤ c r − s + ( γ − ε ( n + s − γ )) + c r − s + ε 2 1 − s , up to renaming c c c Λ \Lambda Λ γ \gamma γ ε = 2 γ 2 n + 1 + s − 2 γ ∈ ( 0 , γ n + s − γ ) \varepsilon=\frac{2\gamma}{2n+1+s-2\gamma}\in\left(0,\frac{\gamma}{n+s-\gamma}\right) ε = 2 n + 1 + s − 2 γ 2 γ ∈ ( 0 , n + s − γ γ )
γ − ε ( n + s − γ ) = ε 2 ( 1 − s ) , (32) \gamma-\varepsilon(n+s-\gamma)=\frac{\varepsilon}{2}(1-s), \tag*{(32)} γ − ε ( n + s − γ ) = 2 ε ( 1 − s ) , (32) and hence
∫ B ξ c χ E ( y ) − χ E c ( y ) ∣ x r − y ∣ n + s d y ≤ c r − s + ε 1 − s 2 . \int_{B_\xi^c}\frac{\chi_E(y)-\chi_{E^c}(y)}{|x_r-y|^{n+s}}\,dy\leq cr^{-s+\varepsilon\frac{1-s}{2}}. ∫ B ξ c ∣ x r − y ∣ n + s χ E ( y ) − χ E c ( y ) d y ≤ c r − s + ε 2 1 − s . Proof of Theorem 3.4 . Up to translations, we assume that x 0 = 0 ∈ ∂ E x_0=0\in\partial E x 0 = 0 ∈ ∂ E
ξ : = r 1 + ε 2 = ρ 1 − ε 2 ( 1 + ε ) , \xi:=r^{1+\frac{\varepsilon}{2}}=\rho^{1-\frac{\varepsilon}{2(1+\varepsilon)}}, ξ := r 1 + 2 ε = ρ 1 − 2 ( 1 + ε ) ε , so that ρ / ξ = r ε / 2 → 0 \rho/\xi=r^{\varepsilon/2}\to0 ρ / ξ = r ε /2 → 0 r → 0 r\to0 r → 0 x r x_r x r y ∈ B ξ c y\in B_\xi^c y ∈ B ξ c
1 − r ε / 2 ≤ ∣ x r − y ∣ ∣ y ∣ ≤ 1 + r ε / 2 . 1-r^{\varepsilon/2}\leq\frac{|x_r-y|}{|y|}\leq1+r^{\varepsilon/2}. 1 − r ε /2 ≤ ∣ y ∣ ∣ x r − y ∣ ≤ 1 + r ε /2 . Therefore, for a small r r r
H s ξ [ E ] ( 0 ) = ∫ B ξ c χ E ( y ) − χ E c ( y ) ∣ y ∣ n + s d y = ∫ B ξ c χ E ( y ) − χ E c ( y ) ∣ x r − y ∣ n + s ( ∣ x r − y ∣ ∣ y ∣ ) n + s d y ≤ ∫ B ξ c χ E ( y ) − χ E c ( y ) ∣ x r − y ∣ n + s d y + ( c r ε / 2 + O ( r ε ) ) ∫ B ξ c d y ∣ x r − y ∣ n + s ≤ c r − s + ε 1 − s 2 + c r ε / 2 ∫ ξ / 2 + ∞ σ − 1 − s d σ ≤ c r − s + ε 1 − s 2 , \begin{aligned}
H_s^\xi[E](0)=\int_{B_\xi^c}\frac{\chi_E(y)-\chi_{E^c}(y)}{|y|^{n+s}}\,dy=\int_{B_\xi^c}\frac{\chi_E(y)-\chi_{E^c}(y)}{|x_r-y|^{n+s}}\left(\frac{|x_r-y|}{|y|}\right)^{n+s}\,dy \\
&\leq\int_{B_\xi^c}\frac{\chi_E(y)-\chi_{E^c}(y)}{|x_r-y|^{n+s}}\,dy+\left(cr^{\varepsilon/2}+O(r^\varepsilon)\right)\int_{B_\xi^c}\frac{dy}{|x_r-y|^{n+s}} \\
&\leq cr^{-s+\varepsilon\frac{1-s}{2}}+cr^{\varepsilon/2}\int_{\xi/2}^{+\infty}\sigma^{-1-s}\,d\sigma\leq cr^{-s+\varepsilon\frac{1-s}{2}},
\end{aligned} H s ξ [ E ] ( 0 ) = ∫ B ξ c ∣ y ∣ n + s χ E ( y ) − χ E c ( y ) d y = ∫ B ξ c ∣ x r − y ∣ n + s χ E ( y ) − χ E c ( y ) ( ∣ y ∣ ∣ x r − y ∣ ) n + s d y ≤ ∫ B ξ c ∣ x r − y ∣ n + s χ E ( y ) − χ E c ( y ) d y + ( c r ε /2 + O ( r ε ) ) ∫ B ξ c ∣ x r − y ∣ n + s d y ≤ c r − s + ε 2 1 − s + c r ε /2 ∫ ξ /2 + ∞ σ − 1 − s d σ ≤ c r − s + ε 2 1 − s , for some positive constant c = c ( n , s , Λ , γ ) c=c(n,s,\Lambda,\gamma) c = c ( n , s , Λ , γ )
Whenever an almost minimal set E E E E c E^c E c
Corollary 3.7. Let E E E ( Λ , γ ) (\Lambda,\gamma) ( Λ , γ ) B 1 B_1 B 1 2 r 2r 2 r E E E x 0 ∈ ∂ E x_0\in\partial E x 0 ∈ ∂ E ε = ε ( n , s , γ ) > 0 \varepsilon=\varepsilon(n,s,\gamma)>0 ε = ε ( n , s , γ ) > 0
H s r 1 + ε / 2 [ E ] ( x 0 ) ≥ − c r − s + ε 1 − s 2 , H_s^{r^{1+\varepsilon/2}}[E](x_0)\geq-cr^{-s+\varepsilon\frac{1-s}{2}}, H s r 1 + ε /2 [ E ] ( x 0 ) ≥ − c r − s + ε 2 1 − s , for some constant c > 0 c>0 c > 0 n n n s s s Λ \Lambda Λ γ \gamma γ
Harnack-type inequalities
In this section, we present a Harnack-type result for almost-minimizers. An iterative application of this result is the main ingredient in establishing the convergence of the sequence { E k } k \{E_k\}_k { E k } k
Theorem 3.8 (Harnack-type Inequalities). Let Λ ≥ 0 \Lambda\geq0 Λ ≥ 0 γ ∈ ( 0 , s ] \gamma\in(0,s] γ ∈ ( 0 , s ] E ⊆ R n E\subseteq\mathbb{R}^n E ⊆ R n ε > 0 \varepsilon>0 ε > 0 α ∈ ( 0 , ε 2 + ε ) \alpha\in\left(0,\frac{\varepsilon}{2+\varepsilon}\right) α ∈ ( 0 , 2 + ε ε ) δ 0 \delta_0 δ 0 k 0 k_0 k 0 n n n s s s Λ \Lambda Λ γ \gamma γ α \alpha α a : = 2 − α k a:=2^{-\alpha k} a := 2 − α k k ≥ k 0 k\geq k_0 k ≥ k 0 E E E ( 2 − k γ Λ , γ ) (2^{-k\gamma}\Lambda,\gamma) ( 2 − k γ Λ , γ ) Per s \operatorname{Per}_s Per s B 2 k B_{2^k} B 2 k { ν j } j ⊆ S n − 1 \{\nu_j\}_j\subseteq\mathbb{S}^{n-1} { ν j } j ⊆ S n − 1
∂ E ∩ C 2 j ν j ⊆ { ∣ x ⋅ ν j ∣ ≤ a 2 j ( 1 + α ) } , for all j = 0 , … , k , (33) \partial E\cap C_{2^j}^{\nu_j}\subseteq\{|x\cdot\nu_j|\leq a2^{j(1+\alpha)}\},\qquad\text{for all }j=0,\ldots,k, \tag*{(33)} ∂ E ∩ C 2 j ν j ⊆ { ∣ x ⋅ ν j ∣ ≤ a 2 j ( 1 + α ) } , for all j = 0 , … , k , (33) with ν 0 = e n \nu_0=e_n ν 0 = e n
either ∂ E ∩ C δ 0 ⊆ { x n ≤ a ( 1 − δ 0 2 ) } (34) \text{either}\qquad\partial E\cap C_{\delta_0}\subseteq\{x_n\leq a(1-\delta_0^2)\} \tag*{(34)} either ∂ E ∩ C δ 0 ⊆ { x n ≤ a ( 1 − δ 0 2 )} (34) or ∂ E ∩ C δ 0 ⊆ { x n ≥ a ( δ 0 2 − 1 ) } . (35) \text{or}\qquad\partial E\cap C_{\delta_0}\subseteq\{x_n\geq a(\delta_0^2-1)\}. \tag*{(35)} or ∂ E ∩ C δ 0 ⊆ { x n ≥ a ( δ 0 2 − 1 )} . (35) We point out that, even if we have a control on the fractional mean curvature only outside a small neighborhood of a touching point, we are still able to deduce Theorem 3.8 from Theorem 3.4. The idea of the proof goes along the lines of the analogous result in [7 Lemma 6.9] (see also [13 Theorem 2.14] ). Theorem 3.8 is a direct consequence of the following:
Lemma 3.9 (Partial Harnack-type Inequality). Let Λ ≥ 0 \Lambda\geq0 Λ ≥ 0 γ ∈ ( 0 , s ] \gamma\in(0,s] γ ∈ ( 0 , s ] E ⊆ R n E\subseteq\mathbb{R}^n E ⊆ R n ε > 0 \varepsilon>0 ε > 0 α ∈ ( 0 , ε 2 + ε ) \alpha\in\left(0,\frac{\varepsilon}{2+\varepsilon}\right) α ∈ ( 0 , 2 + ε ε ) δ 0 \delta_0 δ 0 k 0 k_0 k 0 n n n s s s Λ \Lambda Λ γ \gamma γ α \alpha α a : = 2 − α k a:=2^{-\alpha k} a := 2 − α k k ≥ k 0 k\geq k_0 k ≥ k 0 E E E ( 2 − k γ Λ , γ ) (2^{-k\gamma}\Lambda,\gamma) ( 2 − k γ Λ , γ ) B 2 k B_{2^k} B 2 k
∂ E ∩ C 2 j ν j ⊆ { ∣ x ⋅ ν j ∣ ≤ a 2 j ( 1 + α ) } , for all j = 0 , … , k , (36) \partial E\cap C_{2^j}^{\nu_j}\subseteq\{|x\cdot\nu_j|\leq a2^{j(1+\alpha)}\},\qquad\text{for all }j=0,\ldots,k, \tag*{(36)} ∂ E ∩ C 2 j ν j ⊆ { ∣ x ⋅ ν j ∣ ≤ a 2 j ( 1 + α ) } , for all j = 0 , … , k , (36) with ν 0 = e n \nu_0=e_n ν 0 = e n
∣ E ∩ C δ 0 , a ∣ ≥ 1 2 ∣ C δ 0 , a ∣ = ω n − 1 δ 0 n − 1 a , (37) |E\cap C_{\delta_0,a}|\geq\frac{1}{2}|C_{\delta_0,a}|=\omega_{n-1}\delta_0^{n-1}a, \tag*{(37)} ∣ E ∩ C δ 0 , a ∣ ≥ 2 1 ∣ C δ 0 , a ∣ = ω n − 1 δ 0 n − 1 a , (37) where
C δ 0 , a : = B δ 0 n − 1 × ( − a , a ) , C_{\delta_0,a}:=B_{\delta_0}^{n-1}\times(-a,a), C δ 0 , a := B δ 0 n − 1 × ( − a , a ) , then we have that
either C δ 0 ∩ { x n < − a ( 1 − δ 0 2 ) } ⊆ E or C δ 0 ∩ { x n > a ( 1 − δ 0 2 ) } ⊆ E . (38) \begin{aligned}
&\text{either}\qquad C_{\delta_0}\cap\{x_n<-a(1-\delta_0^2)\}\subseteq E \\
&\text{or}\qquad C_{\delta_0}\cap\{x_n>a(1-\delta_0^2)\}\subseteq E. \tag*{(38)}
\end{aligned} either C δ 0 ∩ { x n < − a ( 1 − δ 0 2 )} ⊆ E or C δ 0 ∩ { x n > a ( 1 − δ 0 2 )} ⊆ E . (38) Proof. Let k 0 = k 0 ( δ 0 ) k_0=k_0(\delta_0) k 0 = k 0 ( δ 0 ) a ≤ 2 − k 0 α < δ 0 a\leq2^{-k_0\alpha}<\delta_0 a ≤ 2 − k 0 α < δ 0 (36) , we have ∂ E ∩ C 1 ⊆ { ∣ x n ∣ < a } \partial E\cap C_1\subseteq\{|x_n|<a\} ∂ E ∩ C 1 ⊆ { ∣ x n ∣ < a } { x n < − a } ∩ C 1 ⊆ E \{x_n<-a\}\cap C_1\subseteq E { x n < − a } ∩ C 1 ⊆ E (38) holds true.
Suppose to the contrary that this does not hold, namely there exists a portion of ∂ E ∩ C δ 0 \partial E\cap C_{\delta_0} ∂ E ∩ C δ 0 { − a < x n < − ( 1 − δ 0 2 ) a } \{-a<x_n<-(1-\delta_0^2)a\} { − a < x n < − ( 1 − δ 0 2 ) a } − a 2 -\frac a2 − 2 a ∂ E \partial E ∂ E C δ 0 C_{\delta_0} C δ 0 y ∈ C δ 0 y\in C_{\delta_0} y ∈ C δ 0
− a ≤ y n ≤ − ( 1 − δ 0 2 ) a . -a\leq y_n\leq-(1-\delta_0^2)a. − a ≤ y n ≤ − ( 1 − δ 0 2 ) a . Now, we estimate the fractional mean curvature of E E E y y y δ 0 < 1 / 2 \delta_0<1/2 δ 0 < 1/2
∣ ∫ B 1 / 2 c ( y ) χ E ( x ) − χ E c ( x ) ∣ x − y ∣ n + s d x ∣ ≤ ∑ j = 0 k ∣ ∫ B 2 j ( y ) ∖ B 2 j − 1 ( y ) χ E ( x ) − χ E c ( x ) ∣ x − y ∣ n + s d x ∣ + ∣ ∫ B 2 k c ( y ) χ E ( x ) − χ E c ( x ) ∣ x − y ∣ n + s d x ∣ ≤ ∑ j = 0 k ∣ ∫ B 2 j ( y ) ∖ B 2 j − 1 ( y ) χ E ( x ) − χ E c ( x ) ∣ x − y ∣ n + s d x ∣ + ω n ∫ 2 k + ∞ d r r 1 + s . (39) \begin{aligned}
\left|\int_{B^c_{1/2}(y)}\frac{\chi_E(x)-\chi_{E^c}(x)}{|x-y|^{n+s}}\,dx\right|
\leq\sum_{j=0}^{k}\left|\int_{B_{2^j}(y)\setminus B_{2^{j-1}}(y)}\frac{\chi_E(x)-\chi_{E^c}(x)}{|x-y|^{n+s}}\,dx\right|
+\left|\int_{B^c_{2^k}(y)}\frac{\chi_E(x)-\chi_{E^c}(x)}{|x-y|^{n+s}}\,dx\right| \\
&\leq\sum_{j=0}^{k}\left|\int_{B_{2^j}(y)\setminus B_{2^{j-1}}(y)}\frac{\chi_E(x)-\chi_{E^c}(x)}{|x-y|^{n+s}}\,dx\right|
+\omega_n\int_{2^k}^{+\infty}\frac{dr}{r^{1+s}}. \tag*{(39)}
\end{aligned} ∫ B 1/2 c ( y ) ∣ x − y ∣ n + s χ E ( x ) − χ E c ( x ) d x ≤ j = 0 ∑ k ∫ B 2 j ( y ) ∖ B 2 j − 1 ( y ) ∣ x − y ∣ n + s χ E ( x ) − χ E c ( x ) d x + ∫ B 2 k c ( y ) ∣ x − y ∣ n + s χ E ( x ) − χ E c ( x ) d x ≤ j = 0 ∑ k ∫ B 2 j ( y ) ∖ B 2 j − 1 ( y ) ∣ x − y ∣ n + s χ E ( x ) − χ E c ( x ) d x + ω n ∫ 2 k + ∞ r 1 + s d r . (39) Thanks to the flatness assumptions (36) , we have
∂ E ∩ B 2 j ( y ) ⊆ ∂ E ∩ C 2 j + 1 ν j ⊆ { ∣ x ⋅ ν j + 1 ∣ ≤ a 2 ( j + 1 ) ( α + 1 ) } . \partial E\cap B_{2^j}(y)\subseteq\partial E\cap C_{2j+1}^{\nu_j}\subseteq\{|x\cdot\nu_{j+1}|\leq a2^{(j+1)(\alpha+1)}\}. ∂ E ∩ B 2 j ( y ) ⊆ ∂ E ∩ C 2 j + 1 ν j ⊆ { ∣ x ⋅ ν j + 1 ∣ ≤ a 2 ( j + 1 ) ( α + 1 ) } . Moreover, y ∈ C δ 0 y\in C_{\delta_0} y ∈ C δ 0
∣ y ⋅ ν j + 1 ∣ ≤ a 2 ( j + 1 ) ( α + 1 ) . |y\cdot\nu_{j+1}|\leq a2^{(j+1)(\alpha+1)}. ∣ y ⋅ ν j + 1 ∣ ≤ a 2 ( j + 1 ) ( α + 1 ) . Combining the last two displays, we obtain
∂ E ∩ B 2 j ( y ) ⊆ { ∣ ( x − y ) ⋅ ν j + 1 ∣ ≤ 2 a 2 ( j + 1 ) ( α + 1 ) } . (40) \partial E\cap B_{2^j}(y)\subseteq\{|(x-y)\cdot\nu_{j+1}|\leq2a2^{(j+1)(\alpha+1)}\}. \tag*{(40)} ∂ E ∩ B 2 j ( y ) ⊆ { ∣ ( x − y ) ⋅ ν j + 1 ∣ ≤ 2 a 2 ( j + 1 ) ( α + 1 ) } . (40) Let us define
D : = ( B 2 j ( y ) ∖ B 2 j − 1 ( y ) ) ∖ { ∣ ( x − y ) ⋅ ν j + 1 ∣ ≤ 2 a 2 ( j + 1 ) ( α + 1 ) } . D:=(B_{2^j}(y)\setminus B_{2^{j-1}}(y))\setminus\{|(x-y)\cdot\nu_{j+1}|\leq2a2^{(j+1)(\alpha+1)}\}. D := ( B 2 j ( y ) ∖ B 2 j − 1 ( y )) ∖ { ∣ ( x − y ) ⋅ ν j + 1 ∣ ≤ 2 a 2 ( j + 1 ) ( α + 1 ) } . By symmetry, we infer that
∫ D χ E ( x ) − χ E c ( x ) ∣ x − y ∣ n + s d x = 0 , \int_D\frac{\chi_E(x)-\chi_{E^c}(x)}{|x-y|^{n+s}}\,dx=0, ∫ D ∣ x − y ∣ n + s χ E ( x ) − χ E c ( x ) d x = 0 , and hence
∣ ∫ B 2 j ( y ) ∖ B 2 j − 1 ( y ) χ E ( x ) − χ E c ( x ) ∣ x − y ∣ n + s d x ∣ ≤ ∫ B 2 j ( y ) ∖ B 2 j − 1 ( y ) χ { ∣ ( x − y ) ⋅ ν j + 1 ∣ ≤ 2 a 2 ( j + 1 ) ( α + 1 ) } ( x ) ∣ x − y ∣ n + s d x ≤ ω n ∫ 2 j − 1 2 j 4 a 2 ( j + 1 ) ( α + 1 ) r n − 2 r n + s d r = C a ∫ 2 j − 1 2 j d r r 1 + s − α , (41) \begin{aligned}
\left|\int_{B_{2^j}(y)\setminus B_{2^{j-1}}(y)}\frac{\chi_E(x)-\chi_{E^c}(x)}{|x-y|^{n+s}}\,dx\right|
\leq\int_{B_{2^j}(y)\setminus B_{2^{j-1}}(y)}\frac{\chi_{\{|(x-y)\cdot\nu_{j+1}|\leq2a2^{(j+1)(\alpha+1)}\}}(x)}{|x-y|^{n+s}}\,dx \\
&\leq\omega_n\int_{2^{j-1}}^{2^j}\frac{4a2^{(j+1)(\alpha+1)}r^{n-2}}{r^{n+s}}\,dr
=Ca\int_{2^{j-1}}^{2^j}\frac{dr}{r^{1+s-\alpha}}, \tag*{(41)}
\end{aligned} ∫ B 2 j ( y ) ∖ B 2 j − 1 ( y ) ∣ x − y ∣ n + s χ E ( x ) − χ E c ( x ) d x ≤ ∫ B 2 j ( y ) ∖ B 2 j − 1 ( y ) ∣ x − y ∣ n + s χ { ∣ ( x − y ) ⋅ ν j + 1 ∣ ≤ 2 a 2 ( j + 1 ) ( α + 1 ) } ( x ) d x ≤ ω n ∫ 2 j − 1 2 j r n + s 4 a 2 ( j + 1 ) ( α + 1 ) r n − 2 d r = C a ∫ 2 j − 1 2 j r 1 + s − α d r , (41) for some C > 0 C>0 C > 0 n n n α \alpha α (39) and (41) , we deduce that
∣ ∫ B 1 / 2 c ( y ) χ E ( x ) − χ E c ( x ) ∣ x − y ∣ n + s d x ∣ ≤ ∑ j = 0 k C a ∫ 2 j − 1 2 j d r r 1 + s − α + ω n ∫ 2 k + ∞ d r r 1 + s ≤ C a ∫ 1 / 2 + ∞ d r r 1 + s − α + ω n s 2 − s k ≤ c 1 ( a + 2 − α k ) ≤ c 1 a . (42) \begin{aligned}
\left|\int_{B^c_{1/2}(y)}\frac{\chi_E(x)-\chi_{E^c}(x)}{|x-y|^{n+s}}\,dx\right|
\leq\sum_{j=0}^{k}Ca\int_{2^{j-1}}^{2^j}\frac{dr}{r^{1+s-\alpha}}+\omega_n\int_{2^k}^{+\infty}\frac{dr}{r^{1+s}} \\
&\leq Ca\int_{1/2}^{+\infty}\frac{dr}{r^{1+s-\alpha}}+\frac{\omega_n}{s}2^{-sk}\leq c_1(a+2^{-\alpha k})\leq c_1a. \tag*{(42)}
\end{aligned} ∫ B 1/2 c ( y ) ∣ x − y ∣ n + s χ E ( x ) − χ E c ( x ) d x ≤ j = 0 ∑ k C a ∫ 2 j − 1 2 j r 1 + s − α d r + ω n ∫ 2 k + ∞ r 1 + s d r ≤ C a ∫ 1/2 + ∞ r 1 + s − α d r + s ω n 2 − s k ≤ c 1 ( a + 2 − α k ) ≤ c 1 a . (42) for some c 1 > 0 c_1 > 0 c 1 > 0 n n n s s s α \alpha α
Now, let us denote by P P P x n = y n − a 2 ∣ x ′ − y ′ ∣ 2 x_n=y_n-\frac{a}{2}|x'-y'|^2 x n = y n − 2 a ∣ x ′ − y ′ ∣ 2 ρ : = 2 − k 2 ( ε − ε α − 2 α ) \rho:=2^{-\frac{k}{2}(\varepsilon-\varepsilon\alpha-2\alpha)} ρ := 2 − 2 k ( ε − ε α − 2 α ) ε − 2 α − ε α > 0 \varepsilon-2\alpha-\varepsilon\alpha>0 ε − 2 α − ε α > 0
∫ B 1 / 2 ( y ) ∖ B ρ ( y ) χ E ( x ) − χ E c ( x ) ∣ x − y ∣ n + s d x ≥ ∫ B 1 / 2 ( y ) ∖ B ρ ( y ) χ E ∖ P ( x ) ∣ x − y ∣ n + s d x + ∫ B 1 / 2 ( y ) χ P ( x ) − χ P c ( x ) ∣ x − y ∣ n + s d x = : I 1 + I 2 . (43) \int_{B_{1/2}(y)\setminus B_\rho(y)}\frac{\chi_E(x)-\chi_{E^c}(x)}{|x-y|^{n+s}}\,dx
\geq
\int_{B_{1/2}(y)\setminus B_\rho(y)}\frac{\chi_{E\setminus P}(x)}{|x-y|^{n+s}}\,dx+
\int_{B_{1/2}(y)}\frac{\chi_P(x)-\chi_{P^c}(x)}{|x-y|^{n+s}}\,dx=:I_1+I_2. \tag*{(43)} ∫ B 1/2 ( y ) ∖ B ρ ( y ) ∣ x − y ∣ n + s χ E ( x ) − χ E c ( x ) d x ≥ ∫ B 1/2 ( y ) ∖ B ρ ( y ) ∣ x − y ∣ n + s χ E ∖ P ( x ) d x + ∫ B 1/2 ( y ) ∣ x − y ∣ n + s χ P ( x ) − χ P c ( x ) d x =: I 1 + I 2 . (43) We stress that, since P P P I 1 I_1 I 1 I 2 I_2 I 2 I 1 I_1 I 1
Since y ∈ C δ 0 ∩ { − a ≤ x n ≤ − ( 1 − δ 0 2 ) a } y\in C_{\delta_0}\cap\{-a\leq x_n\leq-(1-\delta_0^2)a\} y ∈ C δ 0 ∩ { − a ≤ x n ≤ − ( 1 − δ 0 2 ) a } δ 0 \delta_0 δ 0 a a a C δ 0 , a ⊆ B 1 / 2 ( y ) C_{\delta_0,a}\subseteq B_{1/2}(y) C δ 0 , a ⊆ B 1/2 ( y )
∣ x − y ∣ − n − s ≥ ( 3 δ 0 ) − n − s (44) |x-y|^{-n-s}\geq(3\delta_0)^{-n-s} \tag*{(44)} ∣ x − y ∣ − n − s ≥ ( 3 δ 0 ) − n − s (44) for every x ∈ C δ 0 , a x\in C_{\delta_0,a} x ∈ C δ 0 , a
By the density assumption in (3.20) and the choice of ρ \rho ρ
∣ ( E ∖ P ) ∩ B ρ ( y ) ∩ C δ 0 , a ∣ = ∣ E ∩ C δ 0 , a ∣ − ∣ P ∩ C δ 0 , a ∣ ≥ 1 2 ∣ C δ 0 , a ∣ − ∣ B ρ ( y ) ∣ − ∣ B δ 0 n − 1 × { − a ≤ x n ≤ − ( 1 − δ 0 2 ) a } ∣ = ω n − 1 δ 0 n − 1 a ( 1 − δ 0 2 ) − ω n ρ n ≥ 1 4 ω n − 1 δ 0 n − 1 a . \begin{aligned}
|(E\setminus P)\cap B_\rho(y)\cap C_{\delta_0,a}|
&=|E\cap C_{\delta_0,a}|-|P\cap C_{\delta_0,a}|\\
&\geq\frac{1}{2}|C_{\delta_0,a}|-|B_\rho(y)|-\left|B_{\delta_0^{n-1}} \times\{-a\leq x_n\leq-(1-\delta_0^2)a\}\right|\\
&=\omega_{n-1}\delta_0^{n-1}a(1-\delta_0^2)-\omega_n\rho^n
\geq\frac{1}{4}\omega_{n-1}\delta_0^{n-1}a.
\end{aligned} ∣ ( E ∖ P ) ∩ B ρ ( y ) ∩ C δ 0 , a ∣ = ∣ E ∩ C δ 0 , a ∣ − ∣ P ∩ C δ 0 , a ∣ ≥ 2 1 ∣ C δ 0 , a ∣ − ∣ B ρ ( y ) ∣ − B δ 0 n − 1 × { − a ≤ x n ≤ − ( 1 − δ 0 2 ) a } = ω n − 1 δ 0 n − 1 a ( 1 − δ 0 2 ) − ω n ρ n ≥ 4 1 ω n − 1 δ 0 n − 1 a . whenever δ 0 2 < 1 / 2 \delta_0^2<1/2 δ 0 2 < 1/2
I 1 ≥ ∫ C δ 0 , a χ E ∖ P ( x ) ∣ x − y ∣ n + s d x ≥ ∣ ( E ∖ P ) ∩ C δ 0 , a ∣ ( 3 δ 0 ) n + s ≥ c 2 a δ 0 − 1 − s , (45) I_1\geq\int_{C_{\delta_0,a}}\frac{\chi_{E\setminus P}(x)}{|x-y|^{n+s}}\,dx
\geq\frac{|(E\setminus P)\cap C_{\delta_0,a}|}{(3\delta_0)^{n+s}}
\geq c_2a\delta_0^{-1-s}, \tag*{(45)} I 1 ≥ ∫ C δ 0 , a ∣ x − y ∣ n + s χ E ∖ P ( x ) d x ≥ ( 3 δ 0 ) n + s ∣ ( E ∖ P ) ∩ C δ 0 , a ∣ ≥ c 2 a δ 0 − 1 − s , (45) for some c 2 > 0 c_2>0 c 2 > 0 n n n s s s
In addition, since P P P [10 , 5 ] )
I 2 ≥ c 3 ∫ 0 1 / 2 r n − 2 ( ∫ 0 − a 2 r 2 d t ( r 2 + t 2 ) n + s 2 ) d r ≥ − c 3 a , (46) I_2\geq c_3\int_0^{1/2}r^{n-2}\left(\int_0^{-\frac{a}{2}r^2}\frac{dt}{(r^2+t^2)^{\frac{n+s}{2}}}\right)\,dr\geq-c_3a, \tag*{(46)} I 2 ≥ c 3 ∫ 0 1/2 r n − 2 ( ∫ 0 − 2 a r 2 ( r 2 + t 2 ) 2 n + s d t ) d r ≥ − c 3 a , (46) for some c 3 > 0 c_3>0 c 3 > 0 n n n s s s
Then, plugging (3.29) and (3.30) into (3.27), we obtain
∫ B 1 / 2 ( y ) ∖ B ρ ( y ) χ E ( x ) − χ E c ( x ) ∣ x − y ∣ n + s d x ≥ ( c 2 − c 3 δ 0 1 + s ) a δ 0 − 1 − s ≥ c 4 a δ 0 − 1 − s , (47) \int_{B_{1/2}(y)\setminus B_\rho(y)}\frac{\chi_E(x)-\chi_{E^c}(x)}{|x-y|^{n+s}}\,dx
\geq(c_2-c_3\delta_0^{1+s})a\delta_0^{-1-s}
\geq c_4a\delta_0^{-1-s}, \tag*{(47)} ∫ B 1/2 ( y ) ∖ B ρ ( y ) ∣ x − y ∣ n + s χ E ( x ) − χ E c ( x ) d x ≥ ( c 2 − c 3 δ 0 1 + s ) a δ 0 − 1 − s ≥ c 4 a δ 0 − 1 − s , (47) for some c 4 > 0 c_4>0 c 4 > 0 n n n s s s α \alpha α
On the other hand, if E E E ( 2 − k γ Λ , γ ) (2^{-k\gamma}\Lambda,\gamma) ( 2 − k γ Λ , γ ) B 2 k B_{2^k} B 2 k 2 − k E 2^{-k}E 2 − k E ( Λ , γ ) (\Lambda,\gamma) ( Λ , γ ) B 1 B_1 B 1 − 2 k − 1 a = − 2 − 1 + k ( 1 − α ) -2^{k-1}a=-2^{-1+k(1-\alpha)} − 2 k − 1 a = − 2 − 1 + k ( 1 − α ) ∂ ( 2 − k E ) \partial(2^{-k}E) ∂ ( 2 − k E ) 2 − k y 2^{-k}y 2 − k y r : = 2 − k ( 1 − α ) r:=2^{-k(1-\alpha)} r := 2 − k ( 1 − α ) ζ : = r 1 + ε / 2 = 2 − k ρ \zeta:=r^{1+\varepsilon/2}=2^{-k}\rho ζ := r 1 + ε /2 = 2 − k ρ
H s ρ [ 2 − k E ] ( 2 − k y ) ≤ c r − s + ε 1 − s 2 = c 2 k ( 1 − α ) ( s − ε 1 − s 2 ) , H_s^{\rho}[2^{-k}E](2^{-k}y)\leq cr^{-s+\varepsilon\frac{1-s}{2}}=c2^{k(1-\alpha)\left(s-\varepsilon\frac{1-s}{2}\right)}, H s ρ [ 2 − k E ] ( 2 − k y ) ≤ c r − s + ε 2 1 − s = c 2 k ( 1 − α ) ( s − ε 2 1 − s ) , and hence
H s ρ [ E ] ( y ) = 2 − k s H s ρ [ 2 − k E ] ( 2 − k y ) ≤ c 2 − k 2 ( ε ( 1 − s ) − ε α ( 1 − s ) + 2 α s ) . (48) H_s^{\rho}[E](y)=2^{-ks}H_s^{\rho}[2^{-k}E](2^{-k}y)
\leq c2^{-\frac{k}{2}\left(\varepsilon(1-s)-\varepsilon\alpha(1-s)+2\alpha s\right)}. \tag*{(48)} H s ρ [ E ] ( y ) = 2 − k s H s ρ [ 2 − k E ] ( 2 − k y ) ≤ c 2 − 2 k ( ε ( 1 − s ) − ε α ( 1 − s ) + 2 α s ) . (48) Therefore, from (3.26), (3.31), and (3.32), we infer that
( c 4 − c 1 δ 0 1 + s ) a δ 0 − 1 − s = ( c 4 − c 1 δ 0 1 + s ) 2 − k α δ 0 − 1 − s ≤ c 2 − k 2 ( ε ( 1 − s ) − ε α ( 1 − s ) + 2 α s ) . (c_4-c_1\delta_0^{1+s})a\delta_0^{-1-s}
=(c_4-c_1\delta_0^{1+s})2^{-k\alpha}\delta_0^{-1-s}
\leq c2^{-\frac{k}{2}\left(\varepsilon(1-s)-\varepsilon\alpha(1-s)+2\alpha s\right)}. ( c 4 − c 1 δ 0 1 + s ) a δ 0 − 1 − s = ( c 4 − c 1 δ 0 1 + s ) 2 − k α δ 0 − 1 − s ≤ c 2 − 2 k ( ε ( 1 − s ) − ε α ( 1 − s ) + 2 α s ) . In particular, up to renaming the constants, we have
δ 0 − 1 − s ≤ c 2 − k 2 ( 1 − s ) ( ε − ε α − 2 α ) , \delta_0^{-1-s}\leq c2^{-\frac{k}{2}(1-s)(\varepsilon-\varepsilon\alpha-2\alpha)}, δ 0 − 1 − s ≤ c 2 − 2 k ( 1 − s ) ( ε − ε α − 2 α ) , with $\varepsilon-\varepsilon\alpha-2\alpha>0.
However, this leads to a contradiction whenever we take k k k δ 0 \delta_0 δ 0
Furthermore, if { x n > a } ∩ C 1 ⊆ E \{x_n>a\}\cap C_1\subseteq E { x n > a } ∩ C 1 ⊆ E P ~ \widetilde{P} P x n = y n + 1 2 ∣ x ′ − y ′ ∣ 2 x_n=y_n+\frac{1}{2} \lvert x'-y'\rvert^2 x n = y n + 2 1 ∣ x ′ − y ′ ∣ 2 E E E y y y
Proof of Theorem 3.8 . Suppose that ∣ E ∩ C δ 0 , a ∣ ≥ 1 2 ∣ C δ 0 , a ∣ \lvert E\cap C_{\delta_0,a}\rvert\geq\frac{1}{2}\lvert C_{\delta_0,a}\rvert ∣ E ∩ C δ 0 , a ∣ ≥ 2 1 ∣ C δ 0 , a ∣ E E E ( 2 − k γ Λ , γ ) (2^{-k\gamma}\Lambda,\gamma) ( 2 − k γ Λ , γ ) B 2 k B_{2^k} B 2 k C δ 0 ∩ { x n < − a ( 1 − δ 0 2 ) } ⊆ E C_{\delta_0}\cap\{x_n<-a(1-\delta_0^2)\}\subseteq E C δ 0 ∩ { x n < − a ( 1 − δ 0 2 )} ⊆ E C δ 0 ∩ { x n > a ( 1 − δ 0 2 ) } ⊆ E c C_{\delta_0}\cap\{x_n>a(1-\delta_0^2)\}\subseteq E^c C δ 0 ∩ { x n > a ( 1 − δ 0 2 )} ⊆ E c
either ∂ E ∩ C δ 0 ⊆ { x n ≥ a ( δ 0 2 − 1 ) } or ∂ E ∩ C δ 0 ⊆ { x n ≤ a ( 1 − δ 0 2 ) } , \begin{aligned}
&\text{either}\quad\partial E\cap C_{\delta_0}\subseteq\{x_n\geq a(\delta_0^2-1)\}\\
&\text{or}\quad\partial E\cap C_{\delta_0}\subseteq\{x_n\leq a(1-\delta_0^2)\},
\end{aligned} either ∂ E ∩ C δ 0 ⊆ { x n ≥ a ( δ 0 2 − 1 )} or ∂ E ∩ C δ 0 ⊆ { x n ≤ a ( 1 − δ 0 2 )} , respectively.
On the other hand, if ∣ E ∩ C δ 0 , a ∣ < 1 2 ∣ C δ 0 , a ∣ \lvert E\cap C_{\delta_0,a}\rvert<\frac{1}{2}\lvert C_{\delta_0,a}\rvert ∣ E ∩ C δ 0 , a ∣ < 2 1 ∣ C δ 0 , a ∣ ∣ E c ∩ C δ 0 , a ∣ ≥ 1 2 ∣ C δ 0 , a ∣ \lvert E^c\cap C_{\delta_0,a}\rvert\geq\frac{1}{2}\lvert C_{\delta_0,a}\rvert ∣ E c ∩ C δ 0 , a ∣ ≥ 2 1 ∣ C δ 0 , a ∣ E E E E c E^c E c E c E^c E c
either ∂ E ∩ C δ 0 ⊆ { x n ≥ a ( δ 0 2 − 1 ) } or ∂ E ∩ C δ 0 ⊆ { x n ≤ a ( 1 − δ 0 2 ) } , \begin{aligned}
&\text{either}\quad\partial E\cap C_{\delta_0}\subseteq\{x_n\geq a(\delta_0^2-1)\}\\
&\text{or}\quad\partial E\cap C_{\delta_0}\subseteq\{x_n\leq a(1-\delta_0^2)\},
\end{aligned} either ∂ E ∩ C δ 0 ⊆ { x n ≥ a ( δ 0 2 − 1 )} or ∂ E ∩ C δ 0 ⊆ { x n ≤ a ( 1 − δ 0 2 )} , as desired. □
An iterative application of Theorem 3.8 leads to the following:
Lemma 3.10 (Hölder estimates, [13 Lemma 2.20] ). Let { E k } k \{E_k\}_k { E k } k ( 2 − k γ Λ , γ ) (2^{-k\gamma}\Lambda,\gamma) ( 2 − k γ Λ , γ )
Suppose that, for every k ∈ N k\in\mathbb{N} k ∈ N h k ∈ N h_k\in\mathbb{N} h k ∈ N
∂ E k ∩ C 2 h ⊆ { ∣ x ⋅ ν h k ∣ ≤ a k 2 h ( 1 + α ) } , for all h ∈ { 0 , … , h k } , (49) \partial E_k\cap C_{2^h}\subseteq\{\lvert x\cdot\nu_h^k\rvert\leq a_k2^{h(1+\alpha)}\},\qquad\text{for all }h\in\{0,\ldots,h_k\}, \tag*{(49)} ∂ E k ∩ C 2 h ⊆ {∣ x ⋅ ν h k ∣ ≤ a k 2 h ( 1 + α ) } , for all h ∈ { 0 , … , h k } , (49) with ν 0 k = e n \nu_0^k=e_n ν 0 k = e n k k k
For any r ≥ 1 r\geq1 r ≥ 1
A k r : = { ( x ′ , x n / a k ) s.t. ( x ′ , x n ) ∈ ∂ E k ∩ C r } A_k^r:=\{(x',x_n/a_k)\ \text{s.t.}\ (x',x_n)\in\partial E_k\cap C_r\} A k r := {( x ′ , x n / a k ) s.t. ( x ′ , x n ) ∈ ∂ E k ∩ C r } for every k ∈ N k\in\mathbb{N} k ∈ N
Assume that there exists a positive constant C C C k k k A k r ⊆ { ∣ x n ∣ ≤ C } A_k^r\subseteq\{\lvert x_n\rvert\leq C\} A k r ⊆ {∣ x n ∣ ≤ C } k ∈ N k\in\mathbb{N} k ∈ N
Then, there exists k 0 ∈ N k_0\in\mathbb{N} k 0 ∈ N n n n s s s Λ \Lambda Λ γ \gamma γ α \alpha α k ≥ k 0 k\geq k_0 k ≥ k 0
A k r ∩ { ∣ x ′ ∣ < r / 2 } ⊆ subgr ( g k + ) ∖ subgr ( g k − ) , (50) A_k^r\cap\{\lvert x'\rvert<r/2\}\subseteq\operatorname{subgr}(g_k^+)\setminus\operatorname{subgr}(g_k^-), \tag*{(50)} A k r ∩ {∣ x ′ ∣ < r /2 } ⊆ subgr ( g k + ) ∖ subgr ( g k − ) , (50) where g k − g_k^- g k − g k + g_k^+ g k + θ t η \theta t^\eta θ t η η ∈ ( 0 , 1 ) \eta\in(0,1) η ∈ ( 0 , 1 ) θ > 0 \theta>0 θ > 0 C C C δ 0 \delta_0 δ 0 Theorem 3.8 .
For the convenience of the reader, a proof of Lemma 3.10 can be found in Appendix § 7 . We point out that the argument does not dependent explicitly on γ \gamma γ
Remark 3.11. By inspection of the proof of Lemma 3.10 , given any x ∈ A k r ∩ { ∣ x ′ ∣ < r 2 } x\in A_k^r\cap\{\lvert x'\rvert<\frac{r}{2}\} x ∈ A k r ∩ {∣ x ′ ∣ < 2 r }
0 < g k + ( x ′ ) − g k − ( x ′ ) ≤ 2 θ b k η , 0<g_k^+(x')-g_k^-(x')\leq2\theta b_k^\eta, 0 < g k + ( x ′ ) − g k − ( x ′ ) ≤ 2 θ b k η , for some Hölder modulus of continuity b k b_k b k b k → 0 b_k\to0 b k → 0 k → + ∞ k\to+\infty k → + ∞ g k − g_k^- g k − g k + g_k^+ g k + C 0 , η C^{0,\eta} C 0 , η u u u A k r A_k^r A k r graph ( u ) \operatorname{graph}(u) graph ( u ) { ∣ x ′ ∣ < r / 2 } \{\lvert x'\rvert<r/2\} {∣ x ′ ∣ < r /2 }
Linearized equation for the Hölder limit in Lemma 3.10
As last ingredient needed for the proof of Proposition 3.3, we establish the following partial results. We will show that the limit function u u u 1 + s 2 \frac{1+s}{2} 2 1 + s { E k } k \{E_k\}_k { E k } k k → + ∞ k \to+\infty k → + ∞ 1 + s 2 \frac{1+s}{2} 2 1 + s
Lemma 3.12 (Lemma 6.10, [7 ] ). There exists a subsequence { E k j ∗ } j \{E_{k_j}^*\}_j { E k j ∗ } j R n \mathbb{R}^n R n u u u u ( 0 ′ ) = 0 u(0') = 0 u ( 0 ′ ) = 0 c > 0 c > 0 c > 0 u ( x ′ ) ≤ c ( 1 + ∣ x ′ ∣ 1 + α ) u(x') \leq c(1 + |x'|^{1+\alpha}) u ( x ′ ) ≤ c ( 1 + ∣ x ′ ∣ 1 + α ) x ′ ∈ R n − 1 x' \in\mathbb{R}^{n-1} x ′ ∈ R n − 1
Lemma 3.13 (Lemma 6.11, [7 ] ). Let u u u { E k } k \{E_k\}_k { E k } k subgr ( u ) \operatorname{subgr}(u) subgr ( u ) R n \mathbb{R}^n R n k → + ∞ k \to+\infty k → + ∞
( − Δ ) 1 + s 2 u = 0 in R n − 1 , (-\Delta)^{\frac{1+s}{2}}u = 0 \text{ in } \mathbb{R}^{n-1}, ( − Δ ) 2 1 + s u = 0 in R n − 1 , in the viscosity sense.
Lemma 3.14 (Proposition 6.7, [7 ] ). If ( − Δ ) 1 + s 2 u = 0 (-\Delta)^{\frac{1+s}{2}}u = 0 ( − Δ ) 2 1 + s u = 0 R n − 1 \mathbb{R}^{n-1} R n − 1
∣ u ( x ) ∣ ≤ c ( 1 + ∣ x ∣ 1 + α ) , |u(x)| \leq c(1 + |x|^{1+\alpha}), ∣ u ( x ) ∣ ≤ c ( 1 + ∣ x ∣ 1 + α ) , then u u u
We refer to [7 ] and [13 ] for the proofs of Lemmata 3.12 and 3.14, respectively. On the other hand, since few modifications are needed in the proof of Lemma 3.13, we present the details here below.
We prove that the function u u u
( − Δ ) 1 + s 2 u = 0 (51) (-\Delta)^{\frac{1+s}{2}}u = 0 \tag*{(51)} ( − Δ ) 2 1 + s u = 0 (51) in the sense of [CS09, Definition 2.2]. More precisely, we show that u u u
Proof. We start showing that u u u ψ \psi ψ u u u x 0 x_0 x 0 x 0 = 0 x_0 = 0 x 0 = 0 ψ ∈ C 2 ( B ξ ) \psi\in C^2(B_\xi) ψ ∈ C 2 ( B ξ ) ξ > 0 \xi> 0 ξ > 0 ψ < u \psi< u ψ < u B ξ ∖ { 0 } B_\xi\setminus\{0\} B ξ ∖ { 0 }
We point out that, in order to prove that u u u
∫ R n − 1 u ( x ′ ) − u ( 0 ′ ) ∣ x ′ ∣ n + s d H n − 1 ( x ) ≤ 0 , (52) \int_{\mathbb{R}^{n-1}} \frac{u(x') - u(0')}{|x'|^{n+s}}\,d\mathcal{H}^{n-1}(x) \leq0, \tag*{(52)} ∫ R n − 1 ∣ x ′ ∣ n + s u ( x ′ ) − u ( 0 ′ ) d H n − 1 ( x ) ≤ 0 , (52) since
∫ R n − 1 u ( x ′ ) − u ( 0 ′ ) ∣ x ′ ∣ n + s d H n − 1 ( x ) ≥ ∫ B ξ ψ ( x ′ ) − ψ ( 0 ′ ) ∣ x ′ ∣ n + s d H n − 1 ( x ) + ∫ R n − 1 ∖ B ξ u ( x ′ ) − u ( 0 ′ ) ∣ x ′ ∣ n + s d H n − 1 ( x ) . \int_{\mathbb{R}^{n-1}} \frac{u(x') - u(0')}{|x'|^{n+s}}\,d\mathcal{H}^{n-1}(x)
\geq\int_{B_\xi} \frac{\psi(x') - \psi(0')}{|x'|^{n+s}}\,d\mathcal{H}^{n-1}(x) + \int_{\mathbb{R}^{n-1}\setminus B_\xi} \frac{u(x') - u(0')}{|x'|^{n+s}}\,d\mathcal{H}^{n-1}(x). ∫ R n − 1 ∣ x ′ ∣ n + s u ( x ′ ) − u ( 0 ′ ) d H n − 1 ( x ) ≥ ∫ B ξ ∣ x ′ ∣ n + s ψ ( x ′ ) − ψ ( 0 ′ ) d H n − 1 ( x ) + ∫ R n − 1 ∖ B ξ ∣ x ′ ∣ n + s u ( x ′ ) − u ( 0 ′ ) d H n − 1 ( x ) . Thus, we focus on proving (3.36).
Thanks to Lemma 3.12, for any given η > 0 \eta> 0 η > 0 R > 0 R > 0 R > 0 k η ∈ N k_\eta\in\mathbb{N} k η ∈ N k ≥ k η k \geq k_\eta k ≥ k η ∂ E k ∩ B R \partial E_k \cap B_R ∂ E k ∩ B R a k η a_k\eta a k η graph ( a k u ) \operatorname{graph}(a_k u) graph ( a k u ) a k : = 2 − α k a_k := 2^{-\alpha k} a k := 2 − α k
Furthermore, since u ( 0 ′ ) = 0 u(0') = 0 u ( 0 ′ ) = 0 ∣ ∇ u ( 0 ′ ) ∣ = 0 |\nabla u(0')| = 0 ∣∇ u ( 0 ′ ) ∣ = 0 − a k 2 -\frac{a_k}{2} − 2 a k ∂ E k \partial E_k ∂ E k B η B_\eta B η x k x_k x k x k → 0 x_k \to0 x k → 0 k η → + ∞ k_\eta\to+\infty k η → + ∞ η → 0 \eta\to0 η → 0
Now, for every x ˉ = ( x ˉ ′ , x ˉ n ) ∈ R n − 1 × R \bar{x} = (\bar{x}',\bar{x}_n) \in\mathbb{R}^{n-1} \times\mathbb{R} x ˉ = ( x ˉ ′ , x ˉ n ) ∈ R n − 1 × R r > 0 r > 0 r > 0
D r ( x ˉ ) : = { x ∈ R n s.t. ∣ x ′ − x ˉ ′ ∣ ≤ r , ∣ x ˉ n − x ˉ n ∣ ≤ r } . D_r(\bar{x}) := \{x \in\mathbb{R}^n \ \text{s.t. } |x' - \bar{x}'| \leq r,\ |\bar{x}_n - \bar{x}_n| \leq r\}. D r ( x ˉ ) := { x ∈ R n s.t. ∣ x ′ − x ˉ ′ ∣ ≤ r , ∣ x ˉ n − x ˉ n ∣ ≤ r } . Let also consider a small δ > 0 \delta> 0 δ > 0 R > 0 R > 0 R > 0
By the flatness assumptions in (49) , we know that
∂ E k ∩ D R ( x k ) ⊆ ∂ E k ∩ B 2 R ( x k ) ⊆ { ∣ x n − x k , n ∣ ≤ c R a k } . \partial E_k \cap D_R(x_k) \subseteq\partial E_k \cap B_{2R}(x_k) \subseteq\left\{ |x_n-x_{k,n}| \le c_Ra_k \right\}. ∂ E k ∩ D R ( x k ) ⊆ ∂ E k ∩ B 2 R ( x k ) ⊆ { ∣ x n − x k , n ∣ ≤ c R a k } . where c R c_R c R R R R k k k c R a k < δ / 2 c_Ra_k < \delta/2 c R a k < δ /2 x ∈ ( D R ( x k ) ∖ D δ ( x k ) ) ∩ { ∣ x n − x k , n ∣ ≤ c R a k } x \in\left(D_R(x_k) \setminus D_\delta(x_k)\right) \cap\left\{ |x_n-x_{k,n}| \le c_Ra_k \right\} x ∈ ( D R ( x k ) ∖ D δ ( x k ) ) ∩ { ∣ x n − x k , n ∣ ≤ c R a k }
∣ x ′ − x k ′ ∣ 2 ≥ δ 2 − ∣ x n − x k , n ∣ 2 ≥ δ 2 − c R 2 a k 2 ≥ 3 4 δ 2 . |x^\prime-x_k^\prime|^2 \ge\delta^2-|x_n-x_{k,n}|^2 \ge\delta^2-c_R^2a_k^2 \ge\frac{3}{4}\delta^2. ∣ x ′ − x k ′ ∣ 2 ≥ δ 2 − ∣ x n − x k , n ∣ 2 ≥ δ 2 − c R 2 a k 2 ≥ 4 3 δ 2 . Moreover, we observe that
∣ 1 ∣ x − x k ∣ n + s − 1 ∣ x ′ − x k ′ ∣ n + s ∣ = n + s 2 ∫ 0 1 ∣ x n − x k , n ∣ 2 ( ∣ x ′ − x k ′ ∣ 2 + t ∣ x n − x k , n ∣ 2 ) n + s + 2 2 d t ≤ n + s 2 c R 2 a k 2 ∫ 0 1 d t ∣ x ′ − x k ′ ∣ n + s + 2 . \begin{aligned}
\left|\frac{1}{|x-x_k|^{n+s}}-\frac{1}{|x^\prime-x_k^\prime|^{n+s}}\right|
=\frac{n+s}{2}\int_0^1\frac{|x_n-x_{k,n}|^2}{\left(|x^\prime-x_k^\prime|^2+t|x_n-x_{k,n}|^2\right)^{\frac{n+s+2}{2}}}\,dt \\
&\le\frac{n+s}{2}c_R^2a_k^2\int_0^1\frac{dt}{|x^\prime-x_k^\prime|^{n+s+2}}.
\end{aligned} ∣ x − x k ∣ n + s 1 − ∣ x ′ − x k ′ ∣ n + s 1 = 2 n + s ∫ 0 1 ( ∣ x ′ − x k ′ ∣ 2 + t ∣ x n − x k , n ∣ 2 ) 2 n + s + 2 ∣ x n − x k , n ∣ 2 d t ≤ 2 n + s c R 2 a k 2 ∫ 0 1 ∣ x ′ − x k ′ ∣ n + s + 2 d t . It follows from the last two displays that
∣ 1 ∣ x − x k ∣ n + s − 1 ∣ x ′ − x k ′ ∣ n + s ∣ ≤ C a k 2 , \left|\frac{1}{|x-x_k|^{n+s}}-\frac{1}{|x^\prime-x_k^\prime|^{n+s}}\right|\le Ca_k^2, ∣ x − x k ∣ n + s 1 − ∣ x ′ − x k ′ ∣ n + s 1 ≤ C a k 2 , for some positive constant C C C n n n s s s R R R δ \delta δ
Using this and the fact that ∂ E k \partial E_k ∂ E k a k η a_k\eta a k η a k graph ( u ) a_k\operatorname{graph}(u) a k graph ( u )
∫ D R ( x k ) ∖ D δ ( x k ) χ E k ( x ) − χ E k c ( x ) ∣ x − x k ∣ n + s d x = 2 a k ∫ B R n − 1 ( x k ′ ) ∖ B δ n − 1 ( x k ′ ) u ( x ′ ) − u ( x k ′ ) ∣ x ′ − x k ′ ∣ n + s d H n − 1 ( x ) + O ( a k 3 ) = 2 a k ∫ B R n − 1 ( x k ′ ) ∖ B δ n − 1 ( x k ′ ) u ( x ′ ) − u ( x k ′ ) ∣ x ′ − x k ′ ∣ n + s d H n − 1 ( x ) + O ( η a k ) + O ( a k 3 ) . (53) \begin{aligned}
\int_{D_R(x_k)\setminus D_\delta(x_k)}
\frac{\chi_{E_k}(x)-\chi_{E_k^c}(x)}{|x-x_k|^{n+s}}\,dx
=2a_k\int_{B_R^{n-1}(x_k^\prime)\setminus B_\delta^{n-1}(x_k^\prime)}
\frac{u(x^\prime)-u(x_k^\prime)}{|x^\prime-x_k^\prime|^{n+s}}\,d\mathcal{H}^{n-1}(x)+O(a_k^3) \\
&=2a_k\int_{B_R^{n-1}(x_k^\prime)\setminus B_\delta^{n-1}(x_k^\prime)}
\frac{u(x^\prime)-u(x_k^\prime)}{|x^\prime-x_k^\prime|^{n+s}}\,d\mathcal{H}^{n-1}(x)+O(\eta a_k)+O(a_k^3). \tag*{(53)}
\end{aligned} ∫ D R ( x k ) ∖ D δ ( x k ) ∣ x − x k ∣ n + s χ E k ( x ) − χ E k c ( x ) d x = 2 a k ∫ B R n − 1 ( x k ′ ) ∖ B δ n − 1 ( x k ′ ) ∣ x ′ − x k ′ ∣ n + s u ( x ′ ) − u ( x k ′ ) d H n − 1 ( x ) + O ( a k 3 ) = 2 a k ∫ B R n − 1 ( x k ′ ) ∖ B δ n − 1 ( x k ′ ) ∣ x ′ − x k ′ ∣ n + s u ( x ′ ) − u ( x k ′ ) d H n − 1 ( x ) + O ( η a k ) + O ( a k 3 ) . (53) We now write the set D R ( x k ) ∖ D δ ( x k ) D_R(x_k)\setminus D_\delta(x_k) D R ( x k ) ∖ D δ ( x k )
B δ c ( x k ) ∖ [ D R ( x k ) c ∪ ( D δ ( x k ) ∩ B δ c ( x k ) ) ] . B_\delta^c(x_k)\setminus\left[D_R(x_k)^c\cup\left(D_\delta(x_k)\cap B_\delta^c(x_k)\right)\right]. B δ c ( x k ) ∖ [ D R ( x k ) c ∪ ( D δ ( x k ) ∩ B δ c ( x k ) ) ] . Thus, we have
∫ D R ( x k ) ∖ D δ ( x k ) χ E k ( x ) − χ E k c ( x ) ∣ x − x k ∣ n + s d x = ∫ B ρ c ( x k ) χ E k ( x ) − χ E k c ( x ) ∣ x − x k ∣ n + s d x − ∫ D R c ( x k ) χ E k ( x ) − χ E k c ( x ) ∣ x − x k ∣ n + s d x − ∫ D δ ( x k ) ∖ B ρ ( x k ) χ E k ( x ) − χ E k c ( x ) ∣ x − x k ∣ n + s d x , (54) \int_{D_R(x_k)\setminus D_\delta(x_k)}
\frac{\chi_{E_k}(x)-\chi_{E_k^c}(x)}{|x-x_k|^{n+s}}\,dx
=
\int_{B_\rho^c(x_k)}
\frac{\chi_{E_k}(x)-\chi_{E_k^c}(x)}{|x-x_k|^{n+s}}\,dx
-\int_{D_R^c(x_k)}
\frac{\chi_{E_k}(x)-\chi_{E_k^c}(x)}{|x-x_k|^{n+s}}\,dx
-\int_{D_\delta(x_k)\setminus B_\rho(x_k)}
\frac{\chi_{E_k}(x)-\chi_{E_k^c}(x)}{|x-x_k|^{n+s}}\,dx, \tag*{(54)} ∫ D R ( x k ) ∖ D δ ( x k ) ∣ x − x k ∣ n + s χ E k ( x ) − χ E k c ( x ) d x = ∫ B ρ c ( x k ) ∣ x − x k ∣ n + s χ E k ( x ) − χ E k c ( x ) d x − ∫ D R c ( x k ) ∣ x − x k ∣ n + s χ E k ( x ) − χ E k c ( x ) d x − ∫ D δ ( x k ) ∖ B ρ ( x k ) ∣ x − x k ∣ n + s χ E k ( x ) − χ E k c ( x ) d x , (54) and we prove suitable estimates for the right-hand side.
To this end, recalling the estimates in (42) , we have
∣ ∫ D R c ( x k ) χ E k ( x ) − χ E k c ( x ) ∣ x − x k ∣ n + s d x ∣ ≤ C ′ a k ∫ R + ∞ ρ α − 1 − s d ρ + ω n s 2 − k s ≤ C ′ ( a k R α − s + a k 1 + μ ) . (55) \begin{aligned}
\left|\int_{D_R^c(x_k)}
\frac{\chi_{E_k}(x)-\chi_{E_k^c}(x)}{|x-x_k|^{n+s}}\,dx\right|
\le C^\prime a_k\int_R^{+\infty}\rho^{\alpha-1-s}\,d\rho+\frac{\omega_n}{s}2^{-ks} \\
&\le C^\prime\left(a_kR^{\alpha-s}+a_k^{1+\mu}\right). \tag*{(55)}
\end{aligned} ∫ D R c ( x k ) ∣ x − x k ∣ n + s χ E k ( x ) − χ E k c ( x ) d x ≤ C ′ a k ∫ R + ∞ ρ α − 1 − s d ρ + s ω n 2 − k s ≤ C ′ ( a k R α − s + a k 1 + μ ) . (55) for some positive exponent μ \mu μ C ′ C^\prime C ′ n n n s s s α \alpha α
To estimate the term involving D δ ( x k ) ∖ B ρ ( x k ) D_\delta(x_k)\setminus B_\rho(x_k) D δ ( x k ) ∖ B ρ ( x k ) P k P_k P k x k , n − a k 2 ∣ x ′ − x k ′ ∣ 2 x_{k,n}-\frac{a_k}{2}|x^\prime-x_k^\prime|^2 x k , n − 2 a k ∣ x ′ − x k ′ ∣ 2 P k P_k P k P k P_k P k x k x_k x k k k k ρ : = 2 − k 2 ( ε − ε α − 2 α ) \rho:=2^{-\frac{k}{2}(\varepsilon-\varepsilon^\alpha-2\alpha)} ρ := 2 − 2 k ( ε − ε α − 2 α ) k k k ρ < δ \rho< \delta ρ < δ (46) , we obtain
∫ D δ ( x k ) ∖ B ρ ( x k ) χ E k ( x ) − χ E k c ( x ) ∣ x − x k ∣ n + s d x ≥ ∫ D δ ( x k ) ∖ B ρ ( x k ) χ P k ( x ) − χ P k c ( x ) ∣ x − x k ∣ n + s d x ≥ − ∫ 0 δ C a k r n r n + s d r = − C a k δ 1 − s , (56) \begin{aligned}
\int_{D_{\delta}(x_k)\setminus B_{\rho}(x_k)} \frac{\chi_{E_k}(x)-\chi_{E_k^c}(x)}{|x-x_k|^{n+s}}\,dx \ge\int_{D_{\delta}(x_k)\setminus B_{\rho}(x_k)} \frac{\chi_{P_k}(x)-\chi_{P_k^c}(x)}{|x-x_k|^{n+s}}\,dx \\
&\ge-\int_0^\delta\frac{C a_k r^n}{r^{n+s}}\,dr=-C a_k\delta^{1-s}, \tag*{(56)}
\end{aligned} ∫ D δ ( x k ) ∖ B ρ ( x k ) ∣ x − x k ∣ n + s χ E k ( x ) − χ E k c ( x ) d x ≥ ∫ D δ ( x k ) ∖ B ρ ( x k ) ∣ x − x k ∣ n + s χ P k ( x ) − χ P k c ( x ) d x ≥ − ∫ 0 δ r n + s C a k r n d r = − C a k δ 1 − s , (56) for a positive constant C C C k k k
Therefore, using (54) , (55) and (56) in (53) , we obtain that
∫ B R n − 1 ( x k ) ∖ B δ n − 1 ( x k ) u ( x ′ ) − u ( x k ′ ) ∣ x ′ − x k ′ ∣ n + s d H n − 1 ( x ) = 1 2 a k ∫ D R ( x k ) ∖ D δ ( x k ) χ E k ( x ) − χ E k c ( x ) ∣ x − x k ∣ n + s d x + O ( η ) + O ( a k 2 ) ≤ 1 2 a k ∫ B ρ c ( x k ) χ E k ( x ) − χ E k c ( x ) ∣ x − x k ∣ n + s d x + C δ 1 − s + O ( η ) + O ( a k μ ) + O ( R α − s ) . (57) \begin{aligned}
\int_{B_R^{n-1}(x_k)\setminus B_\delta^{n-1}(x_k)}\frac{u(x')-u(x'_k)}{|x'-x'_k|^{n+s}}\,d\mathcal{H}^{n-1}(x)
&=\frac{1}{2a_k}\int_{D_R(x_k)\setminus D_\delta(x_k)}\frac{\chi_{E_k}(x)-\chi_{E_k^c}(x)}{|x-x_k|^{n+s}}\,dx+O(\eta)+O(a_k^2)\\
&\le\frac{1}{2a_k}\int_{B_\rho^c(x_k)}\frac{\chi_{E_k}(x)-\chi_{E_k^c}(x)}{|x-x_k|^{n+s}}\,dx+C\delta^{1-s}+O(\eta)+O(a_k^\mu)+O(R^{\alpha-s}). \tag*{(57)}
\end{aligned} ∫ B R n − 1 ( x k ) ∖ B δ n − 1 ( x k ) ∣ x ′ − x k ′ ∣ n + s u ( x ′ ) − u ( x k ′ ) d H n − 1 ( x ) = 2 a k 1 ∫ D R ( x k ) ∖ D δ ( x k ) ∣ x − x k ∣ n + s χ E k ( x ) − χ E k c ( x ) d x + O ( η ) + O ( a k 2 ) ≤ 2 a k 1 ∫ B ρ c ( x k ) ∣ x − x k ∣ n + s χ E k ( x ) − χ E k c ( x ) d x + C δ 1 − s + O ( η ) + O ( a k μ ) + O ( R α − s ) . (57) Thanks to the assumptions on E k E_k E k (48) ), we know that
1 2 a k ∫ B ρ c ( x k ) χ E k ( x ) − χ E k c ( x ) ∣ x − x k ∣ n + s d x ≤ C a k − 1 2 − k 2 ( ε ( 1 − s ) − ε α ( 1 − s ) − 2 α s ) = C 2 − k 2 ( 1 − s ) ( ε − ε α − 2 α ) . \frac{1}{2a_k}\int_{B_\rho^c(x_k)}\frac{\chi_{E_k}(x)-\chi_{E_k^c}(x)}{|x-x_k|^{n+s}}\,dx\le C a_k^{-1}2^{-\frac{k}{2}(\varepsilon(1-s)-\varepsilon\alpha(1-s)-2\alpha s)}=C2^{-\frac{k}{2(1-s)}(\varepsilon-\varepsilon\alpha-2\alpha)}. 2 a k 1 ∫ B ρ c ( x k ) ∣ x − x k ∣ n + s χ E k ( x ) − χ E k c ( x ) d x ≤ C a k − 1 2 − 2 k ( ε ( 1 − s ) − ε α ( 1 − s ) − 2 α s ) = C 2 − 2 ( 1 − s ) k ( ε − ε α − 2 α ) . It thereby follows that
∫ B R n − 1 ( x k ) ∖ B δ n − 1 ( x k ) u ( x ′ ) − u ( x k ′ ) ∣ x ′ − x k ′ ∣ n + s d H n − 1 ( x ) ≤ C 2 − k 2 ( 1 − s ) ( ε − ε α − 2 α ) + C δ 1 − s + O ( η ) + O ( a k μ ) + O ( R α − s ) . (58) \begin{aligned}
\int_{B_R^{n-1}(x_k)\setminus B_\delta^{n-1}(x_k)}\frac{u(x')-u(x'_k)}{|x'-x'_k|^{n+s}}\,d\mathcal{H}^{n-1}(x)
\le C2^{-\frac{k}{2(1-s)}(\varepsilon-\varepsilon\alpha-2\alpha)}+C\delta^{1-s}+O(\eta)+O(a_k^\mu)+O(R^{\alpha-s}). \tag*{(58)}
\end{aligned} ∫ B R n − 1 ( x k ) ∖ B δ n − 1 ( x k ) ∣ x ′ − x k ′ ∣ n + s u ( x ′ ) − u ( x k ′ ) d H n − 1 ( x ) ≤ C 2 − 2 ( 1 − s ) k ( ε − ε α − 2 α ) + C δ 1 − s + O ( η ) + O ( a k μ ) + O ( R α − s ) . (58) Since α ∈ ( 0 , ε 2 + ε ) \alpha\in\left(0,\frac{\varepsilon}{2+\varepsilon}\right) α ∈ ( 0 , 2 + ε ε ) η → 0 \eta\to0 η → 0 k → + ∞ k\to+\infty k → + ∞ (58) , we infer that
∫ B R n − 1 ∖ B δ n − 1 u ( x ′ ) − u ( 0 ′ ) ∣ x ′ ∣ n + s d H n − 1 ( x ) ≤ C δ 1 − s + O ( R α − s ) . \int_{B_R^{n-1}\setminus B_\delta^{n-1}}\frac{u(x')-u(0')}{|x'|^{n+s}}\,d\mathcal{H}^{n-1}(x)\le C\delta^{1-s}+O(R^{\alpha-s}). ∫ B R n − 1 ∖ B δ n − 1 ∣ x ′ ∣ n + s u ( x ′ ) − u ( 0 ′ ) d H n − 1 ( x ) ≤ C δ 1 − s + O ( R α − s ) . Hence, taking the limits as δ → 0 \delta\to0 δ → 0 R → + ∞ R\to+\infty R → + ∞ (52) .
A similar argument to the one above shows that u u u (51) .
Let φ \varphi φ C 2 C^2 C 2 u u u 0 0 0 (54) with a tangent parabola of opening a k 2 \frac{a_k}{2} 2 a k
∫ D δ ( x k ) ∖ B ρ ( x k ) χ E k ( x ) − χ E k c ( x ) ∣ x − x k ∣ n + s d x ≤ C a k δ 1 − s . \int_{D_\delta(x_k)\setminus B_\rho(x_k)}\frac{\chi_{E_k}(x)-\chi_{E_k^c}(x)}{|x-x_k|^{n+s}}\,dx\le C a_k\delta^{1-s}. ∫ D δ ( x k ) ∖ B ρ ( x k ) ∣ x − x k ∣ n + s χ E k ( x ) − χ E k c ( x ) d x ≤ C a k δ 1 − s . This, together with (53) , (55) , and Corollary 3.7, entails
∫ B R n − 1 ( x k ) ∖ B δ n − 1 ( x k ) u ( x ′ ) − u ( x k ′ ) ∣ x ′ − x k ′ ∣ n + s d H n − 1 ( x ) ≥ 1 2 a k ∫ B ρ c ( x k ) χ E k ( x ) − χ E k c ( x ) ∣ x − x k ∣ n + s d x − C δ 1 − s + O ( η ) + O ( a k μ ) + O ( R α − s ) ≥ − C 2 − k 2 ( 1 − s ) ( ε − ε α − 2 α ) − C δ 1 − s + O ( η ) + O ( a k μ ) + O ( R α − s ) . (59) \begin{aligned}
\int_{B_R^{n-1}(x_k)\setminus B_\delta^{n-1}(x_k)}\frac{u(x')-u(x'_k)}{|x'-x'_k|^{n+s}}\,d\mathcal{H}^{n-1}(x)
&\ge\frac{1}{2a_k}\int_{B_\rho^c(x_k)}\frac{\chi_{E_k}(x)-\chi_{E_k^c}(x)}{|x-x_k|^{n+s}}\,dx-C\delta^{1-s}+O(\eta)+O(a_k^\mu)+O(R^{\alpha-s})\\
&\ge-C2^{-\frac{k}{2(1-s)}(\varepsilon-\varepsilon\alpha-2\alpha)}-C\delta^{1-s}+O(\eta)+O(a_k^\mu)+O(R^{\alpha-s}). \tag*{(59)}
\end{aligned} ∫ B R n − 1 ( x k ) ∖ B δ n − 1 ( x k ) ∣ x ′ − x k ′ ∣ n + s u ( x ′ ) − u ( x k ′ ) d H n − 1 ( x ) ≥ 2 a k 1 ∫ B ρ c ( x k ) ∣ x − x k ∣ n + s χ E k ( x ) − χ E k c ( x ) d x − C δ 1 − s + O ( η ) + O ( a k μ ) + O ( R α − s ) ≥ − C 2 − 2 ( 1 − s ) k ( ε − ε α − 2 α ) − C δ 1 − s + O ( η ) + O ( a k μ ) + O ( R α − s ) . (59) Taking η → 0 \eta\to0 η → 0 k → + ∞ k\to+\infty k → + ∞ (59) , we obtain that
∫ B R n − 1 ∖ B δ n − 1 u ( x ′ ) − u ( 0 ′ ) ∣ x ′ ∣ n + s d H n − 1 ( x ) ≥ − C δ 1 − s + O ( R α − s ) . \int_{B_R^{n-1}\setminus B_\delta^{n-1}}\frac{u(x')-u(0')}{|x'|^{n+s}}\,d\mathcal{H}^{n-1}(x)\ge-C\delta^{1-s}+O(R^{\alpha-s}). ∫ B R n − 1 ∖ B δ n − 1 ∣ x ′ ∣ n + s u ( x ′ ) − u ( 0 ′ ) d H n − 1 ( x ) ≥ − C δ 1 − s + O ( R α − s ) . Finally, taking the limits as δ → 0 \delta\to0 δ → 0 R → + ∞ R\to+\infty R → + ∞ u u u (51) . □
Proof of the flatness improvement (Proposition 3.3)
Arguing by contradiction, suppose that the conclusion in (3.3) does not hold true. Then, we construct a sequence { E k } k \{E_k\}_k { E k } k
∂ E k ∩ B 2 j ⊆ { ∣ x ⋅ ν j ∣ ≤ 2 j + α ( j − k ) } , for all j ∈ { 0 , … , j k } , \partial E_k \cap B_{2^j} \subseteq\{ |x \cdot\nu_j| \le2^{j+\alpha(j-k)} \}, \qquad\text{for all } j \in\{0,\ldots,j_k\}, ∂ E k ∩ B 2 j ⊆ { ∣ x ⋅ ν j ∣ ≤ 2 j + α ( j − k ) } , for all j ∈ { 0 , … , j k } , but ∂ E k ∩ B 1 / 2 ⊈ { ∣ x ⋅ ν − 1 ∣ ≤ 2 − 1 − α − α k } \partial E_k \cap B_{1/2} \not\subseteq\{ |x \cdot\nu_{-1}| \le2^{-1-\alpha-\alpha k} \} ∂ E k ∩ B 1/2 ⊆ { ∣ x ⋅ ν − 1 ∣ ≤ 2 − 1 − α − α k }
Thanks to Lemma 3.12, we have that { E k } k \{E_k\}_k { E k } k B 1 B_1 B 1 u u u u u u u ( 0 ′ ) = 0 u(0')=0 u ( 0 ′ ) = 0 ν ∈ S n − 1 \nu\in\mathbb{S}^{n-1} ν ∈ S n − 1
E : = subgr ( u ) = { x ⋅ ν ≤ 0 } . E := \operatorname{subgr}(u) = \{x \cdot\nu\le0\}. E := subgr ( u ) = { x ⋅ ν ≤ 0 } . Since the sequence { E k } k \{E_k\}_k { E k } k E k E_k E k η \eta η E E E B 1 / 2 B_{1/2} B 1/2 k k k η \eta η k ˉ \bar{k} k ˉ
∂ E k ˉ ∩ B 1 / 2 ⊆ { ∣ x ⋅ ν ∣ ≤ 2 − 1 − α − α k ˉ } . \partial E_{\bar{k}} \cap B_{1/2} \subseteq\{ |x \cdot\nu| \le2^{-1-\alpha-\alpha\bar{k}} \}. ∂ E k ˉ ∩ B 1/2 ⊆ { ∣ x ⋅ ν ∣ ≤ 2 − 1 − α − α k ˉ } . However, this contradicts (3.44), concluding the proof.
Hölder regularity for almost minimal sets
Proof of Theorem 1.7
Now, we show that every almost minimizer in B 1 B_1 B 1 C 1 , α C^{1,\alpha} C 1 , α B 1 / 2 B_{1/2} B 1/2 α ∈ ( 0 , ε 0 ) \alpha\in(0,\varepsilon_0) α ∈ ( 0 , ε 0 ) ε 0 : = ε 2 + ε \varepsilon_0 := \frac{\varepsilon}{2+\varepsilon} ε 0 := 2 + ε ε ε \varepsilon ε
To this purpose, consider x 0 ∈ ∂ E ∩ B 1 / 2 x_0 \in\partial E \cap B_{1/2} x 0 ∈ ∂ E ∩ B 1/2 k 0 k_0 k 0 η 0 < 2 − k 0 ( 1 + α ) \eta_0 < 2^{-k_0(1+\alpha)} η 0 < 2 − k 0 ( 1 + α ) E − x 0 E-x_0 E − x 0 { ν j } j \{\nu_j\}_j { ν j } j
∂ E ∩ B 2 − j ( x 0 ) ⊆ { ∣ ( x − x 0 ) ⋅ ν j ∣ ≤ 2 − j ( 1 + α ) } , (60) \partial E \cap B_{2^{-j}}(x_0) \subseteq\{ |(x-x_0)\cdot\nu_j| \le2^{-j(1+\alpha)} \}, \tag*{(60)} ∂ E ∩ B 2 − j ( x 0 ) ⊆ { ∣ ( x − x 0 ) ⋅ ν j ∣ ≤ 2 − j ( 1 + α ) } , (60) for every j ∈ { 0 , … , k 0 } j \in\{0,\ldots,k_0\} j ∈ { 0 , … , k 0 } ν 0 = e n \nu_0=e_n ν 0 = e n [13 ] ), we obtain the estimate
∣ ν j − ν j + 1 ∣ ≤ c 2 − j α . (61) |\nu_j-\nu_{j+1}| \le c2^{-j\alpha}. \tag*{(61)} ∣ ν j − ν j + 1 ∣ ≤ c 2 − j α . (61) Thus, the sequence { ν j } j \{\nu_j\}_j { ν j } j ν ( x 0 ) \nu(x_0) ν ( x 0 )
∣ ν ( x 0 ) − ν j ∣ = lim k → + ∞ ∣ ν j + k − ν j ∣ ≤ lim k → + ∞ ∑ q = 0 k − 1 ∣ ν j + q + 1 − ν j + q ∣ ≤ lim k → + ∞ ∑ q = 0 k − 1 2 − ( j + q ) α ≤ c 2 − j α . \begin{aligned}
|\nu(x_0)-\nu_j|
&= \lim_{k\to+\infty}|\nu_{j+k}-\nu_j|
\le\lim_{k\to+\infty}\sum_{q=0}^{k-1}|\nu_{j+q+1}-\nu_{j+q}| \\
&\le\lim_{k\to+\infty}\sum_{q=0}^{k-1}2^{-(j+q)\alpha}
\le c2^{-j\alpha}.
\end{aligned} ∣ ν ( x 0 ) − ν j ∣ = k → + ∞ lim ∣ ν j + k − ν j ∣ ≤ k → + ∞ lim q = 0 ∑ k − 1 ∣ ν j + q + 1 − ν j + q ∣ ≤ k → + ∞ lim q = 0 ∑ k − 1 2 − ( j + q ) α ≤ c 2 − j α . In view of this, we obtain that
∣ ( x − x 0 ) ⋅ ν ( x 0 ) ∣ ≤ ∣ ( x − x 0 ) ⋅ ν j ∣ + ∣ x − x 0 ∣ ∣ ν ( x 0 ) − ν j ∣ ≤ c 2 − j ( 1 + α ) , |(x-x_0)\cdot\nu(x_0)| \le|(x-x_0)\cdot\nu_j|+|x-x_0||\nu(x_0)-\nu_j| \le c2^{-j(1+\alpha)}, ∣ ( x − x 0 ) ⋅ ν ( x 0 ) ∣ ≤ ∣ ( x − x 0 ) ⋅ ν j ∣ + ∣ x − x 0 ∣∣ ν ( x 0 ) − ν j ∣ ≤ c 2 − j ( 1 + α ) , for every x ∈ ∂ E ∩ B 2 − j ( x 0 ) x \in\partial E \cap B_{2^{-j}}(x_0) x ∈ ∂ E ∩ B 2 − j ( x 0 )
∂ E ∩ B 2 − j ( x 0 ) ⊆ { ∣ ( x − x 0 ) ⋅ ν ( x 0 ) ∣ < c 2 − j ( 1 + α ) } . (62) \partial E \cap B_{2^{-j}}(x_0) \subseteq\{ |(x-x_0)\cdot\nu(x_0)| < c2^{-j(1+\alpha)} \}. \tag*{(62)} ∂ E ∩ B 2 − j ( x 0 ) ⊆ { ∣ ( x − x 0 ) ⋅ ν ( x 0 ) ∣ < c 2 − j ( 1 + α ) } . (62) Now, for every points x 0 , y 0 ∈ ∂ E ∩ B 1 / 2 x_0,y_0 \in\partial E \cap B_{1/2} x 0 , y 0 ∈ ∂ E ∩ B 1/2 x 0 ∈ B 2 − j ( y 0 ) ∖ B 2 − j − 1 ( y 0 ) x_0 \in B_{2^{-j}}(y_0)\setminus B_{2^{-j-1}}(y_0) x 0 ∈ B 2 − j ( y 0 ) ∖ B 2 − j − 1 ( y 0 ) j ∈ N j \in\mathbb{N} j ∈ N ν ( x 0 ) \nu(x_0) ν ( x 0 ) ν ( y 0 ) \nu(y_0) ν ( y 0 )
either { ( x − y 0 ) ⋅ ν ( x 0 ) > − c 2 − j ( 1 + α ) } ⊆ { x ⋅ ν ( y 0 ) < c 2 − j ( 1 + α ) } \text{either } \{(x-y_0)\cdot\nu(x_0)>-c2^{-j(1+\alpha)}\}\subseteq\{x\cdot\nu(y_0)<c2^{-j(1+\alpha)}\} either {( x − y 0 ) ⋅ ν ( x 0 ) > − c 2 − j ( 1 + α ) } ⊆ { x ⋅ ν ( y 0 ) < c 2 − j ( 1 + α ) } or
{ x ⋅ ν ( y 0 ) > − c 2 − j ( 1 + α ) } ⊆ { ( x − y 0 ) ⋅ ν ( x 0 ) < c 2 − j ( 1 + α ) } \{x\cdot\nu(y_0)>-c2^{-j(1+\alpha)}\}\subseteq\{(x-y_0)\cdot\nu(x_0)<c2^{-j(1+\alpha)}\} { x ⋅ ν ( y 0 ) > − c 2 − j ( 1 + α ) } ⊆ {( x − y 0 ) ⋅ ν ( x 0 ) < c 2 − j ( 1 + α ) } in B 2 − j ( y 0 ) B_{2^{-j}}(y_0) B 2 − j ( y 0 )
∣ ν ( x 0 ) − ν ( y 0 ) ∣ ≤ C 2 − j α ≤ 2 C ∣ x 0 − y 0 ∣ α . |\nu(x_0)-\nu(y_0)| \le C2^{-j\alpha} \le2C|x_0-y_0|^\alpha. ∣ ν ( x 0 ) − ν ( y 0 ) ∣ ≤ C 2 − j α ≤ 2 C ∣ x 0 − y 0 ∣ α . Hence, ν \nu ν ∂ E ∩ B 1 / 2 \partial E \cap B_{1/2} ∂ E ∩ B 1/2 C 1 , α C^{1,\alpha} C 1 , α [14 Theorem 5.8] ).
Proof of Corollary 1.8
Now, we show that with Theorem 1.7 in hand, we are able to improve the estimate in Theorem 3.4, and consequently we infer a better Hölder regularity for almost minimal sets. The main tool of our strategy is the following technical lemma, which provide sharper uniform density estimates for C 1 , α C^{1,\alpha} C 1 , α
Lemma 4.1. Let E E E C 1 , α C^{1,\alpha} C 1 , α R n \mathbb{R}^n R n α ∈ ( 0 , ε 0 ) \alpha\in(0,\varepsilon_0) α ∈ ( 0 , ε 0 ) ε 0 \varepsilon_0 ε 0 B 2 r ( − 2 r e n ) B_{2r}(-2r e_n) B 2 r ( − 2 r e n ) ∂ E \partial E ∂ E r > 0 r>0 r > 0 A d A_d A d d = r 1 + 2 ε d=r^{1+2\varepsilon} d = r 1 + 2 ε ε > 0 \varepsilon>0 ε > 0
Then, there exists c 0 c_0 c 0 n n n s s s Λ \Lambda Λ γ \gamma γ
∣ A d ∣ ≥ c 0 r n ( 1 + 2 ε ) − α ( 1 + α ) ( n − 1 ) (63) |A_d| \geq c_0 r^{n(1+2\varepsilon)-\frac{\alpha}{(1+\alpha)(n-1)}} \tag*{(63)} ∣ A d ∣ ≥ c 0 r n ( 1 + 2 ε ) − ( 1 + α ) ( n − 1 ) α (63) Arguing as in Section 3, and employing Lemma 4.1 in place of Proposition 2.1 in the proof of Theorem 3.4, we infer the following:
Corollary 4.2. Let Λ ≥ 0 \Lambda\geq0 Λ ≥ 0 γ \gamma γ E E E C 1 , α C^{1,\alpha} C 1 , α ( Λ , γ ) (\Lambda,\gamma) ( Λ , γ ) B 1 B_1 B 1 α ∈ ( 0 , ε 0 ) \alpha\in(0,\varepsilon_0) α ∈ ( 0 , ε 0 ) ε 0 ∈ ( 0 , s ) \varepsilon_0 \in(0,s) ε 0 ∈ ( 0 , s )
Then, E E E C 1 , α C^{1,\alpha} C 1 , α α ∈ ( 0 , ε 1 ) \alpha\in(0,\varepsilon_1) α ∈ ( 0 , ε 1 )
ε 1 = γ + ε 0 ( n + γ − 1 ) 2 n + 1 + s − γ + ε 0 ( n + s − γ ) . \varepsilon_1=\frac{\gamma+\varepsilon_0(n+\gamma-1)}{2n+1+s-\gamma+\varepsilon_0(n+s-\gamma)}. ε 1 = 2 n + 1 + s − γ + ε 0 ( n + s − γ ) γ + ε 0 ( n + γ − 1 ) . Proof of Lemma 4.1. Without loss of generality, we can suppose that E E E
{ ( x ′ , x n ) ∈ R n s.t. x n < ∣ x ′ ∣ 1 + α } \left\{(x',x_n)\in\mathbb{R}^n \ \text{s.t.}\ x_n<|x'|^{1+\alpha}\right\} { ( x ′ , x n ) ∈ R n s.t. x n < ∣ x ′ ∣ 1 + α } around the origin. Then, we estimate the volume of the set A d A_d A d ∂ E \partial E ∂ E
Figure 1. The set A d A_d A d E = { x n < ∣ x ′ ∣ 1 + α } E=\{x_n<|x'|^{1+\alpha}\} E = { x n < ∣ x ′ ∣ 1 + α }
To find the intersection between ∂ E \partial E ∂ E ∂ A d \partial A_d ∂ A d
d − ∣ x ′ ∣ 2 2 r = ∣ x ′ ∣ 1 + α . (64) d-\frac{|x'|^2}{2r}=|x'|^{1+\alpha}. \tag*{(64)} d − 2 r ∣ x ′ ∣ 2 = ∣ x ′ ∣ 1 + α . (64) Let us assume that a solution of (4.5) has the form ∣ x ′ ∣ = r β |x'|=r^\beta ∣ x ′ ∣ = r β β ≥ 0 \beta\geq0 β ≥ 0 d = r 1 + 2 ε d=r^{1+2\varepsilon} d = r 1 + 2 ε β \beta β
r 2 β + r 2 β ( 1 + α ) = 2 r 2 + 2 ε . (65) r^{2\beta}+r^{2\beta(1+\alpha)}=2r^{2+2\varepsilon}. \tag*{(65)} r 2 β + r 2 β ( 1 + α ) = 2 r 2 + 2 ε . (65) Since r < 1 r<1 r < 1 2 + 2 ε 2+2\varepsilon 2 + 2 ε r 2 β ≥ r 1 + β ( 1 + α ) r^{2\beta}\ge r^{1+\beta(1+\alpha)} r 2 β ≥ r 1 + β ( 1 + α ) β = 1 + ε \beta=1+\varepsilon β = 1 + ε α < ε 2 + ε \alpha<\frac{\varepsilon}{2+\varepsilon} α < 2 + ε ε
1 + β ( 1 + α ) = 2 + ε + α + α ε < 2 + 2 ε , 1+\beta(1+\alpha)=2+\varepsilon+\alpha+\alpha\varepsilon<2+2\varepsilon, 1 + β ( 1 + α ) = 2 + ε + α + α ε < 2 + 2 ε , against r 2 β ≥ r 1 + β ( 1 + α ) r^{2\beta}\ge r^{1+\beta(1+\alpha)} r 2 β ≥ r 1 + β ( 1 + α ) r 2 β ≤ r 1 + β ( 1 + α ) r^{2\beta}\le r^{1+\beta(1+\alpha)} r 2 β ≤ r 1 + β ( 1 + α )
β = 1 + 2 ε 1 + α . \beta=\frac{1+2\varepsilon}{1+\alpha}. β = 1 + α 1 + 2 ε . Therefore, we conclude that
∣ A d ∣ ≥ ∫ B r β n − 1 ( d − ∣ x ′ ∣ 2 2 r − ∣ x ′ ∣ 1 + α ) d x ′ = ω n ( d r β − 1 2 ( n − 1 ) r ( n + 1 ) β − 1 − 1 n + α r ( n + α ) β ) ≥ n + α − 2 2 ( n + α ) r ( 1 + 2 ε ) n + α 1 + α = : c 0 r n ( 1 + 2 ε ) − α 1 + α ( 1 + ε ) ( n − 1 ) . \begin{aligned}
|A_d|&\ge\int_{B_{r^\beta}^{n-1}}\left(d-\frac{|x'|^2}{2r}-|x'|^{1+\alpha}\right)\,dx'\\
&=\omega_n\left(dr^\beta-\frac{1}{2(n-1)}r^{(n+1)\beta-1}-\frac{1}{n+\alpha}r^{(n+\alpha)\beta}\right)\\
&\ge\frac{n+\alpha-2}{2(n+\alpha)}r^{(1+2\varepsilon)\frac{n+\alpha}{1+\alpha}}=:c_0r^{n(1+2\varepsilon)-\frac{\alpha}{1+\alpha}(1+\varepsilon)(n-1)}.
\end{aligned} ∣ A d ∣ ≥ ∫ B r β n − 1 ( d − 2 r ∣ x ′ ∣ 2 − ∣ x ′ ∣ 1 + α ) d x ′ = ω n ( d r β − 2 ( n − 1 ) 1 r ( n + 1 ) β − 1 − n + α 1 r ( n + α ) β ) ≥ 2 ( n + α ) n + α − 2 r ( 1 + 2 ε ) 1 + α n + α =: c 0 r n ( 1 + 2 ε ) − 1 + α α ( 1 + ε ) ( n − 1 ) . as desired.
Proof of Corollary 1.8. Thanks to Theorem 1.7, we know that E E E C 1 , α C^{1,\alpha} C 1 , α α ≤ ε 0 \alpha\le\varepsilon_0 α ≤ ε 0 E E E
ε 1 = γ + ε 0 ( n + γ − 1 ) 2 n + 1 + s − γ + ε 0 ( n + s − γ ) . \varepsilon_1=\frac{\gamma+\varepsilon_0(n+\gamma-1)}{2n+1+s-\gamma+\varepsilon_0(n+s-\gamma)}. ε 1 = 2 n + 1 + s − γ + ε 0 ( n + s − γ ) γ + ε 0 ( n + γ − 1 ) . Iterating this procedure, we improve the regularity for E E E { ε j } j \{\varepsilon_j\}_j { ε j } j
ε j + 1 = γ + ε j ( n + γ − 1 ) 2 n + 1 + s − γ + ε j ( n + s − γ ) . \varepsilon_{j+1}=\frac{\gamma+\varepsilon_j(n+\gamma-1)}{2n+1+s-\gamma+\varepsilon_j(n+s-\gamma)}. ε j + 1 = 2 n + 1 + s − γ + ε j ( n + s − γ ) γ + ε j ( n + γ − 1 ) . Since the sequence { ε j } j \{\varepsilon_j\}_j { ε j } j j j j
ε ‾ : = lim j → + ∞ ε j , \overline{\varepsilon}:=\lim_{j\to+\infty}\varepsilon_j, ε := j → + ∞ lim ε j , and it satisfies
ε ‾ = γ + ε ‾ ( n + γ − 1 ) 2 n + 1 + s − γ + ε ‾ ( n + s − γ ) . \overline{\varepsilon}=\frac{\gamma+\overline{\varepsilon}(n+\gamma-1)}{2n+1+s-\gamma+\overline{\varepsilon}(n+s-\gamma)}. ε = 2 n + 1 + s − γ + ε ( n + s − γ ) γ + ε ( n + γ − 1 ) . In particular, we have
ε ‾ = − ( n + s + 2 − 2 γ ) + [ ( n + s + 2 − 2 γ ) 2 + 4 γ ( n + s − γ ) ] 1 / 2 2 ( n + s − γ ) . \overline{\varepsilon}=\frac{-(n+s+2-2\gamma)+\left[(n+s+2-2\gamma)^2+4\gamma(n+s-\gamma)\right]^{1/2}}{2(n+s-\gamma)}. ε = 2 ( n + s − γ ) − ( n + s + 2 − 2 γ ) + [ ( n + s + 2 − 2 γ ) 2 + 4 γ ( n + s − γ ) ] 1/2 . Hausdorff dimension of the singular set (Theorem 1.11)
By mean of to the monotonicity formula from Proposition 1.12, we are able to study regular and singular points (in the sense of Definition 1.10) of E E E E E E B 1 B_1 B 1 0 ∈ ∂ E 0\in\partial E 0 ∈ ∂ E λ E \lambda E λ E λ → + ∞ \lambda\to+\infty λ → + ∞ s s s
Definition 5.1. We say that a set C C C s s s 0 0 0 s s s C C C Per s ( ⋅ , B 1 ) \operatorname{Per}_s(\cdot,B_1) Per s ( ⋅ , B 1 ) λ C = C \lambda C=C λ C = C λ > 0 \lambda>0 λ > 0
The main tool in our proof is the following:
Lemma 5.2 ([7 Proposition 9.1] ). Let { λ k } k \{\lambda_k\}_k { λ k } k { Λ k } k \{\Lambda_k\}_k { Λ k } k λ k → + ∞ \lambda_k \to+\infty λ k → + ∞ Λ k → Λ ∈ [ 0 , + ∞ ) \Lambda_k \to\Lambda\in[0,+\infty) Λ k → Λ ∈ [ 0 , + ∞ ) k → + ∞ k \to+\infty k → + ∞ γ ∈ ( 0 , s ] \gamma\in(0,s] γ ∈ ( 0 , s ] { E k } k \{E_k\}_k { E k } k ( Λ k , γ ) (\Lambda_k,\gamma) ( Λ k , γ ) B λ k B_{\lambda_k} B λ k E k → E E_k \to E E k → E L l o c 1 L^1_{\mathrm{loc}} L loc 1 E E E k → + ∞ k \to+\infty k → + ∞ u k : = χ E k − χ E k c u_k := \chi_{E_k} - \chi_{E_k^c} u k := χ E k − χ E k c u : = χ E − χ E c u := \chi_E - \chi_{E^c} u := χ E − χ E c u ~ k \widetilde{u}_k u k u ~ \widetilde{u} u
(i) u ~ k → u ~ \widetilde{u}_k \to\widetilde{u} u k → u K ⋐ R + n + 1 K \Subset\mathbb{R}^{n+1}_+ K ⋐ R + n + 1
(ii) ∇ u ~ k → ∇ u ~ \nabla\widetilde{u}_k \to\nabla\widetilde{u} ∇ u k → ∇ u L l o c 2 ( R + n + 1 , z 1 − s d x d z ) L^2_{\mathrm{loc}}(\mathbb{R}^{n+1}_+,z^{1-s}\,dx\,dz) L loc 2 ( R + n + 1 , z 1 − s d x d z )
In particular, Φ E k → Φ E \Phi_{E_k} \to\Phi_E Φ E k → Φ E
The proof of Lemma 5.2 is a variation of the proof of [7 Proposition 9.1] . See also [13 Appendix H] for a proof in the setting of ( Λ , 0 ) (\Lambda,0) ( Λ , 0 )
We point out that E E E ( n + 1 ) (n+1) ( n + 1 ) χ E − χ E c \chi_E-\chi_{E^c} χ E − χ E c 0 0 0 Ξ E \Xi_E Ξ E [7 Corollary 8.2] ).
By Lemma 5.2, we infer that the limit set of a convergent blow-up sequence { E k : = λ k E } k \{E_k := \lambda_k E\}_k { E k := λ k E } k
Theorem 5.3 (Blow up limit). Let E E E ( Λ , γ ) (\Lambda,\gamma) ( Λ , γ ) B 1 B_1 B 1 0 ∈ ∂ E 0 \in\partial E 0 ∈ ∂ E { λ k } k \{\lambda_k\}_k { λ k } k λ k → + ∞ \lambda_k \to+\infty λ k → + ∞ k → + ∞ k \to+\infty k → + ∞ C C C λ k E → C \lambda_k E \to C λ k E → C L l o c 1 ( R n ) L^1_{\mathrm{loc}}(\mathbb{R}^n) L loc 1 ( R n ) C C C E E E
Proof. Notice that if E E E ( Λ , γ ) (\Lambda,\gamma) ( Λ , γ ) B 1 B_1 B 1 λ k E \lambda_k E λ k E ( λ k − γ Λ , γ ) (\lambda_k^{-\gamma}\Lambda,\gamma) ( λ k − γ Λ , γ ) B λ k B_{\lambda_k} B λ k k k k λ k E \lambda_k E λ k E B 1 B_1 B 1
Let us consider the set A k 1 : = B 1 ∪ ( λ k E ∩ B 1 c ) A_k^1 := B_1 \cup(\lambda_k E \cap B_1^c) A k 1 := B 1 ∪ ( λ k E ∩ B 1 c ) A k 1 △ λ k E ⊆ B 1 A_k^1 \mathbin{\triangle} \lambda_k E \subseteq B_1 A k 1 △ λ k E ⊆ B 1 λ k E \lambda_k E λ k E
∥ χ λ k E ∥ W 1 , s ( B 1 ) = ∥ χ λ k E ∥ L 1 ( B 1 ) + [ χ λ k E ] W 1 , s ( B 1 ) ≤ ∣ B 1 ∣ + 2 Pers ( λ k E , B 1 ) ≤ ∣ B 1 ∣ + 2 Pers ( A k 1 , B 1 ) + 2 λ k − γ Λ ≤ ∣ B 1 ∣ + 2 Pers ( B 1 , R n ) + 2 Λ , \begin{aligned}
\|\chi_{\lambda_k E}\|_{W^{1,s}(B_1)}
&= \|\chi_{\lambda_k E}\|_{L^1(B_1)} + [\chi_{\lambda_k E}]_{W^{1,s}(B_1)} \\
&\le|B_1| + 2\operatorname{Pers}(\lambda_k E,B_1) \le|B_1| + 2\operatorname{Pers}(A_k^1,B_1) + 2\lambda_k^{-\gamma}\Lambda\\
&\le|B_1| + 2\operatorname{Pers}(B_1,\mathbb{R}^n) + 2\Lambda,
\end{aligned} ∥ χ λ k E ∥ W 1 , s ( B 1 ) = ∥ χ λ k E ∥ L 1 ( B 1 ) + [ χ λ k E ] W 1 , s ( B 1 ) ≤ ∣ B 1 ∣ + 2 Pers ( λ k E , B 1 ) ≤ ∣ B 1 ∣ + 2 Pers ( A k 1 , B 1 ) + 2 λ k − γ Λ ≤ ∣ B 1 ∣ + 2 Pers ( B 1 , R n ) + 2Λ , provided that k k k
Using the Sobolev embedding W 1 , s ( B 1 ) ↪ ↪ L 1 ( B 1 ) W^{1,s}(B_1) \hookrightarrow\hookrightarrow L^1(B_1) W 1 , s ( B 1 ) ↪↪ L 1 ( B 1 ) λ k 1 \lambda_k^1 λ k 1
( λ k 1 E ) ∩ B 1 → E 1 in L 1 ( B 1 ) (\lambda_k^1 E)\cap B_1 \to E_1 \text{ in } L^1(B_1) ( λ k 1 E ) ∩ B 1 → E 1 in L 1 ( B 1 ) for some set E 1 ⊆ B 1 E_1 \subseteq B_1 E 1 ⊆ B 1 { λ k 1 E } k \{\lambda_k^1 E\}_k { λ k 1 E } k B 2 B_2 B 2 { λ k 2 } k ⊆ { λ k 1 } k \{\lambda_k^2\}_k \subseteq\{\lambda_k^1\}_k { λ k 2 } k ⊆ { λ k 1 } k E 2 ⊆ B 2 E_2 \subseteq B_2 E 2 ⊆ B 2
( λ k 2 E ) ∩ B 2 → E 2 in L 1 ( B 2 ) . (\lambda_k^2 E)\cap B_2 \to E_2 \text{ in } L^1(B_2). ( λ k 2 E ) ∩ B 2 → E 2 in L 1 ( B 2 ) . Moreover, by the uniqueness of the limit, we have that E 2 ∩ B 1 = E 1 E_2 \cap B_1 = E_1 E 2 ∩ B 1 = E 1
Iterating this procedure m m m { λ k m } k \{\lambda_k^m\}_k { λ k m } k ( λ k m E ) ∩ B m → E m (\lambda_k^m E)\cap B_m \to E_m ( λ k m E ) ∩ B m → E m L 1 ( B m ) L^1(B_m) L 1 ( B m ) k → + ∞ k \to+\infty k → + ∞ E m ∩ B j = E j E_m \cap B_j = E_j E m ∩ B j = E j j = 1 , … , m − 1 j=1,\ldots,m-1 j = 1 , … , m − 1
Using a diagonal argument, we extract a subsequence { λ k k } k \{\lambda_k^k\}_k { λ k k } k
λ k k E → ⋃ m = 1 + ∞ E m = : C in L l o c 1 ( R n ) . (66) \lambda_k^k E \to\bigcup_{m=1}^{+\infty} E_m =: C \text{ in } L^1_{\mathrm{loc}}(\mathbb{R}^n). \tag*{(66)} λ k k E → m = 1 ⋃ + ∞ E m =: C in L loc 1 ( R n ) . (66) Furthermore, recall that λ k E \lambda_k E λ k E ( λ k − γ Λ , γ ) (\lambda_k^{-\gamma}\Lambda,\gamma) ( λ k − γ Λ , γ ) B λ k B_{\lambda_k} B λ k
lim k → + ∞ Φ λ k E ( r ) = Φ C ( r ) . \lim_{k\to+\infty}\Phi_{\lambda_k E}(r)=\Phi_C(r). k → + ∞ lim Φ λ k E ( r ) = Φ C ( r ) . As a consequence, recalling the notation in (C.4), we obtain that
Φ C ( r ) = lim k → + ∞ Φ λ k E ( r ) = lim k → + ∞ Ξ λ k E ( r ) + n − s γ Λ ^ r γ = lim k → + ∞ Ξ E ( r λ k ) + n − s γ Λ ^ r γ . \Phi_C(r) = \lim_{k \to+\infty} \Phi_{\lambda_k E}(r) = \lim_{k \to+\infty} \Xi_{\lambda_k E}(r) + \frac{n-s}{\gamma}\widehat{\Lambda}r^\gamma
= \lim_{k \to+\infty} \Xi_E\left(\frac{r}{\lambda_k}\right) + \frac{n-s}{\gamma}\widehat{\Lambda}r^\gamma. Φ C ( r ) = k → + ∞ lim Φ λ k E ( r ) = k → + ∞ lim Ξ λ k E ( r ) + γ n − s Λ r γ = k → + ∞ lim Ξ E ( λ k r ) + γ n − s Λ r γ . Now, since Φ E \Phi_E Φ E Φ E ( r / λ k ) \Phi_E(r/\lambda_k) Φ E ( r / λ k ) k → + ∞ k \to+\infty k → + ∞ γ < s \gamma< s γ < s
lim k → + ∞ Ξ E ( r λ k ) = lim k → + ∞ Φ E ( r λ k ) − n − s γ Λ ^ ( r λ k ) γ = Φ E ( 0 ) = Ξ E ( 0 ) . \lim_{k \to+\infty} \Xi_E\left(\frac{r}{\lambda_k}\right) = \lim_{k \to+\infty} \Phi_E\left(\frac{r}{\lambda_k}\right) - \frac{n-s}{\gamma}\widehat{\Lambda}\left(\frac{r}{\lambda_k}\right)^\gamma= \Phi_E(0) = \Xi_E(0). k → + ∞ lim Ξ E ( λ k r ) = k → + ∞ lim Φ E ( λ k r ) − γ n − s Λ ( λ k r ) γ = Φ E ( 0 ) = Ξ E ( 0 ) . As a result, we find that
Φ C ( r ) = Ξ E ( 0 ) + n − s γ Λ ^ r γ . \Phi_C(r) = \Xi_E(0) + \frac{n-s}{\gamma}\widehat{\Lambda}r^\gamma. Φ C ( r ) = Ξ E ( 0 ) + γ n − s Λ r γ . which reads
Ξ C ( r ) = Φ C ( r ) − n − s γ Λ ^ r γ = Ξ E ( 0 ) . \Xi_C(r) = \Phi_C(r) - \frac{n-s}{\gamma}\widehat{\Lambda}r^\gamma= \Xi_E(0). Ξ C ( r ) = Φ C ( r ) − γ n − s Λ r γ = Ξ E ( 0 ) . This shows that Ξ C \Xi_C Ξ C C C C
Classification of tangent cones
Now, we show how the classification of the tangent cones to an almost minimal set allows to deduce regularity.
Theorem 5.4. Let E E E ( Λ , γ ) (\Lambda,\gamma) ( Λ , γ ) B 1 B_1 B 1 0 ∈ ∂ E 0 \in\partial E 0 ∈ ∂ E E E E 0 0 0
Then, there exists δ > 0 \delta> 0 δ > 0 ∂ E ∩ B δ \partial E \cap B_\delta ∂ E ∩ B δ C 1 , α C^{1,\alpha} C 1 , α
The proof of Theorem 5.4 relies on following auxiliary result and will be presented here below.
Theorem 5.5 (Energy gap). Let Π : = { x 1 > 0 } \Pi:= \{x_1 > 0\} Π := { x 1 > 0 } C C C
(5.2)
Ξ Π ≤ Ξ C (67) \Xi_\Pi\leq\Xi_C \tag*{(67)} Ξ Π ≤ Ξ C (67) and equality holds if and only if C C C
In particular, if C C C δ 0 > 0 \delta_0 > 0 δ 0 > 0 n n n s s s
(5.3)
Ξ Π ≤ Ξ C − δ 0 . (68) \Xi_\Pi\leq\Xi_C - \delta_0. \tag*{(68)} Ξ Π ≤ Ξ C − δ 0 . (68) In other words, Theorem 5.5 states that half-spaces are the tangent cones with least local energy. Moreover, there exists an energy gap between regular and singular tangent cones. We remark that the same argument as the one employed to prove [7 Theorem 9.6] shows Theorem 5.5, up to using the clean ball condition for almost minimizers (see Corollary 2.3).
Proof of Theorem 5.4. Let { λ k } k \{\lambda_k\}_k { λ k } k λ k → + ∞ \lambda_k \to+\infty λ k → + ∞ k → + ∞ k \to+\infty k → + ∞ { λ k E } k \{\lambda_k E\}_k { λ k E } k λ k E → Π \lambda_k E \to\Pi λ k E → Π L l o c 1 L^1_{\mathrm{loc}} L loc 1 Π \Pi Π λ k E → Π \lambda_k E \to\Pi λ k E → Π
Moreover, up to rotations, we can always assume that Π = { x n < 0 } \Pi= \{x_n < 0\} Π = { x n < 0 } k ˉ ∈ N \bar{k} \in\mathbb{N} k ˉ ∈ N k ≥ k ˉ k \geq\bar{k} k ≥ k ˉ
∂ ( λ k E ) ∩ B 1 ⊆ { ∣ x n ∣ < η 0 } , \partial(\lambda_k E) \cap B_1 \subseteq\{|x_n| < \eta_0\}, ∂ ( λ k E ) ∩ B 1 ⊆ { ∣ x n ∣ < η 0 } , where η 0 \eta_0 η 0
In particular, thanks to Theorem 1.7, we infer that ∂ ( λ k E ) \partial(\lambda_k E) ∂ ( λ k E ) C 1 , α C^{1,\alpha} C 1 , α B 1 / 2 B_{1/2} B 1/2 ∂ E \partial E ∂ E C 1 , α C^{1,\alpha} C 1 , α B 1 / ( 2 λ k ) B_{1/(2\lambda_k)} B 1/ ( 2 λ k )
Quasi-minimality implies almost minimality (Lemma 1.6)
This section is devoted to proving Lemma 1.6.
First, observe that, if E E E ( K , β ) (K,\beta) ( K , β ) B 1 B_1 B 1 B ~ r ⋐ B 1 \widetilde{B}_r \Subset B_1 B r ⋐ B 1
Per s ( E , B ~ r ) ≤ K ~ r n − s , (69) \operatorname{Per}_s(E,\widetilde{B}_r) \leq\widetilde{K}r^{n-s}, \tag*{(69)} Per s ( E , B r ) ≤ K r n − s , (69) for some K ~ > 0 \widetilde{K}>0 K > 0 n n n s s s F r : = E ∖ B ~ r F_r:=E\setminus\widetilde{B}_r F r := E ∖ B r
Per s ( E , B ~ r ) ≤ ( 1 + K r β ) Per s ( F r , B ~ r ) ≤ K ~ L s ( B ~ r , B ~ r c ) , \operatorname{Per}_s(E,\widetilde{B}_r) \leq(1+Kr^\beta)\operatorname{Per}_s(F_r,\widetilde{B}_r) \leq\widetilde{K}\mathcal{L}_s(\widetilde{B}_r,\widetilde{B}_r^c), Per s ( E , B r ) ≤ ( 1 + K r β ) Per s ( F r , B r ) ≤ K L s ( B r , B r c ) , up to renaming K ~ \widetilde{K} K
Let now F ⊆ R n F\subseteq\mathbb{R}^n F ⊆ R n E Δ F ⊆ B ~ r E\mathbin{\Delta}F\subseteq\widetilde{B}_r E Δ F ⊆ B r B ~ r ⋐ B 1 \widetilde{B}_r\Subset B_1 B r ⋐ B 1
1 1 + K r β Per s ( E , B ~ r ) ≤ Per s ( F , B ~ r ) \frac{1}{1+Kr^\beta}\operatorname{Per}_s(E,\widetilde{B}_r)\leq\operatorname{Per}_s(F,\widetilde{B}_r) 1 + K r β 1 Per s ( E , B r ) ≤ Per s ( F , B r ) which, together with (A.1), reads
Per s ( E , B ~ r ) ≤ Per s ( F , B ~ r ) + K r β Per s ( E , B ~ r ) ≤ Per s ( F , B ~ r ) + K ~ r n − s + β , \operatorname{Per}_s(E,\widetilde{B}_r)\leq\operatorname{Per}_s(F,\widetilde{B}_r)+Kr^\beta\operatorname{Per}_s(E,\widetilde{B}_r)\leq\operatorname{Per}_s(F,\widetilde{B}_r)+\widetilde{K}r^{n-s+\beta}, Per s ( E , B r ) ≤ Per s ( F , B r ) + K r β Per s ( E , B r ) ≤ Per s ( F , B r ) + K r n − s + β , establishing the desired result.
Proof of the Hölder estimates (Lemma 3.10)
In this section, we present a proof of Lemma 3.10. We refer to [13 Lemma 2.20] for more details (when γ = 0 \gamma=0 γ = 0 γ \gamma γ
Let k ∈ N k\in\mathbb{N} k ∈ N x 0 ∈ A k r ∩ { ∣ x ′ ∣ < r / 2 } x_0\in A_k^r\cap\{|x'|<r/2\} x 0 ∈ A k r ∩ { ∣ x ′ ∣ < r /2 }
Σ k , j : = A k r ∩ { ∣ x ′ − x 0 ′ ∣ < 1 2 ( δ 0 2 ) j } . \Sigma_{k,j}:=A_k^r\cap\left\{|x'-x'_0|<\frac{1}{2}\left(\frac{\delta_0}{2}\right)^j\right\}. Σ k , j := A k r ∩ { ∣ x ′ − x 0 ′ ∣ < 2 1 ( 2 δ 0 ) j } . Notice that, since
∂ E k ∗ ∩ C 2 h ⊆ { ∣ x ⋅ ν h k ∣ ≤ 2 h ( 1 + α ) } , for all h ∈ { 0 , … , h k } , \partial E_k^*\cap C_{2^h}\subseteq\left\{|x\cdot\nu_h^k|\leq2^{h(1+\alpha)}\right\},\qquad\text{for all }h\in\{0,\ldots,h_k\}, ∂ E k ∗ ∩ C 2 h ⊆ { ∣ x ⋅ ν h k ∣ ≤ 2 h ( 1 + α ) } , for all h ∈ { 0 , … , h k } , then also
Σ k , j = Σ k , j ∩ C 2 h ⊆ { ∣ x ⋅ ν h k ∣ ≤ 2 h ( 1 + α ) } , for all h ∈ { 0 , … , h k } . \Sigma_{k,j}=\Sigma_{k,j}\cap C_{2^h}\subseteq\left\{|x\cdot\nu_h^k|\leq2^{h(1+\alpha)}\right\},\qquad\text{for all }h\in\{0,\ldots,h_k\}. Σ k , j = Σ k , j ∩ C 2 h ⊆ { ∣ x ⋅ ν h k ∣ ≤ 2 h ( 1 + α ) } , for all h ∈ { 0 , … , h k } . Therefore, applying iteratively Harnack’s Inequality in Theorem 3.8 (see [13 Corollary 2.17] ), we infer that
Σ k , j ⊆ { ∣ x n − z j ∣ < 2 2 + α ( 1 − δ 0 2 2 ) j } , for all j ∈ { 0 , … , j k } , (70) \Sigma_{k,j}\subseteq\left\{|x_n-z_j|<2^{2+\alpha}\left(1-\frac{\delta_0^2}{2}\right)^j\right\},\qquad\text{for all }j\in\{0,\ldots,j_k\}, \tag*{(70)} Σ k , j ⊆ { ∣ x n − z j ∣ < 2 2 + α ( 1 − 2 δ 0 2 ) j } , for all j ∈ { 0 , … , j k } , (70) for some z j = z j ( x 0 ) z_j=z_j(x_0) z j = z j ( x 0 ) ∣ z j ∣ ≤ 1 |z_j|\leq1 ∣ z j ∣ ≤ 1
z j − ( 1 − δ 0 2 2 ) j ≤ z j + 1 − ( 1 − δ 0 2 2 ) j ≤ z j + 1 + ( 1 − δ 0 2 2 ) j ≤ z j + ( 1 − δ 0 2 2 ) j , (71) z_j-\left(1-\frac{\delta_0^2}{2}\right)^j\leq z_{j+1}-\left(1-\frac{\delta_0^2}{2}\right)^j\leq z_{j+1}+\left(1-\frac{\delta_0^2}{2}\right)^j\leq z_j+\left(1-\frac{\delta_0^2}{2}\right)^j, \tag*{(71)} z j − ( 1 − 2 δ 0 2 ) j ≤ z j + 1 − ( 1 − 2 δ 0 2 ) j ≤ z j + 1 + ( 1 − 2 δ 0 2 ) j ≤ z j + ( 1 − 2 δ 0 2 ) j , (71) for all j j j k k k j k → + ∞ j_k\to+\infty j k → + ∞ k → + ∞ k\to+\infty k → + ∞
We claim that, for every x ∈ A k r ∩ { ∣ x ′ ∣ < r / 2 } x\in A_k^r\cap\{|x'|<r/2\} x ∈ A k r ∩ { ∣ x ′ ∣ < r /2 }
∣ x n − z j k ∣ < θ max { ∣ x ′ − x 0 ′ ∣ , b k } η , (72) |x_n-z_{j_k}|<\theta\max\{|x'-x'_0|,b_k\}^{\eta}, \tag*{(72)} ∣ x n − z j k ∣ < θ max { ∣ x ′ − x 0 ′ ∣ , b k } η , (72) where b k b_k b k
Let us first observe that, given x ∈ A k r ∩ { ∣ x ′ ∣ < r / 2 } x\in A_k^r\cap\{|x'|<r/2\} x ∈ A k r ∩ { ∣ x ′ ∣ < r /2 } ∣ x ′ − x 0 ′ ∣ < r |x'-x'_0|<r ∣ x ′ − x 0 ′ ∣ < r ∣ x ′ − x 0 ′ ∣ |x'-x'_0| ∣ x ′ − x 0 ′ ∣ r r r 1 2 \frac{1}{2} 2 1 1 2 \frac{1}{2} 2 1 1 2 ( δ 0 2 ) j k \frac{1}{2}\left(\frac{\delta_0}{2}\right)^{j_k} 2 1 ( 2 δ 0 ) j k 1 2 ( δ 0 2 ) j k \frac{1}{2}\left(\frac{\delta_0}{2}\right)^{j_k} 2 1 ( 2 δ 0 ) j k 1 2 ( δ 0 2 ) j k < ∣ x ′ − x 0 ′ ∣ < 1 2 \frac{1}{2}\left(\frac{\delta_0}{2}\right)^{j_k}<|x'-x'_0|<\frac{1}{2} 2 1 ( 2 δ 0 ) j k < ∣ x ′ − x 0 ′ ∣ < 2 1 j ≤ j k j\leq j_k j ≤ j k N \mathbb{N} N
1 2 ( δ 0 2 ) j + 1 ≤ ∣ x ′ − x 0 ′ ∣ < 1 2 ( δ 0 2 ) j . \frac{1}{2}\left(\frac{\delta_0}{2}\right)^{j+1}\leq|x'-x'_0|<\frac{1}{2}\left(\frac{\delta_0}{2}\right)^j. 2 1 ( 2 δ 0 ) j + 1 ≤ ∣ x ′ − x 0 ′ ∣ < 2 1 ( 2 δ 0 ) j . Rearranging the latter, we obtain
j < log ( 2 ∣ x ′ − x 0 ′ ∣ ) log ( δ 0 / 2 ) ≤ j + 1. j<\frac{\log(2|x'-x'_0|)}{\log(\delta_0/2)}\leq j+1. j < log ( δ 0 /2 ) log ( 2∣ x ′ − x 0 ′ ∣ ) ≤ j + 1. Moreover, (B.1) yields ∣ x n − z j ∣ ≤ 2 2 + α ( 1 − δ 0 2 2 ) j |x_n-z_j|\leq2^{2+\alpha}\left(1-\frac{\delta_0^2}{2}\right)^j ∣ x n − z j ∣ ≤ 2 2 + α ( 1 − 2 δ 0 2 ) j
η : = log ( 1 − δ 0 2 2 ) log ( δ 0 / 2 ) ∈ ( 0 , 1 ) , (73) \eta:=\frac{\log\left(1-\frac{\delta_0^2}{2}\right)}{\log(\delta_0/2)}\in(0,1), \tag*{(73)} η := log ( δ 0 /2 ) log ( 1 − 2 δ 0 2 ) ∈ ( 0 , 1 ) , (73) we obtain that
( 1 − δ 0 2 2 ) j ≤ ( 1 − δ 0 2 2 ) log ( 2 ∣ x ′ − x 0 ′ ∣ ) log ( δ 0 / 2 ) − 1 = ( 1 − δ 0 2 2 ) − 1 2 η ∣ x ′ − x 0 ′ ∣ η . \left(1-\frac{\delta_0^2}{2}\right)^j\leq\left(1-\frac{\delta_0^2}{2}\right)^{\frac{\log(2|x'-x'_0|)}{\log(\delta_0/2)}-1}=\left(1-\frac{\delta_0^2}{2}\right)^{-1}2^\eta|x'-x'_0|^\eta. ( 1 − 2 δ 0 2 ) j ≤ ( 1 − 2 δ 0 2 ) l o g ( δ 0 /2 ) l o g ( 2∣ x ′ − x 0 ′ ∣ ) − 1 = ( 1 − 2 δ 0 2 ) − 1 2 η ∣ x ′ − x 0 ′ ∣ η . Thus, defining θ : = 2 η ( 1 + 2 2 + α ) ( 1 − δ 0 2 2 ) − 1 \theta:=2^\eta(1+2^{2+\alpha})\left(1-\frac{\delta_0^2}{2}\right)^{-1} θ := 2 η ( 1 + 2 2 + α ) ( 1 − 2 δ 0 2 ) − 1
∣ x n − z j k ∣ ≤ ∣ x n − z j ∣ + ∣ z j − z j k ∣ ≤ ( 1 + 2 2 + α ) ( 1 − δ 0 2 2 ) j ≤ θ ∣ x ′ − x 0 ′ ∣ η , |x_n-z_{j_k}|\leq|x_n-z_j|+|z_j-z_{j_k}|\leq(1+2^{2+\alpha})\left(1-\frac{\delta_0^2}{2}\right)^j\leq\theta|x'-x'_0|^\eta, ∣ x n − z j k ∣ ≤ ∣ x n − z j ∣ + ∣ z j − z j k ∣ ≤ ( 1 + 2 2 + α ) ( 1 − 2 δ 0 2 ) j ≤ θ ∣ x ′ − x 0 ′ ∣ η , which gives (B.3) in this case.
Now, if 1 2 ≤ ∣ x ′ − x 0 ′ ∣ < r \frac{1}{2}\leq|x'-x'_0|<r 2 1 ≤ ∣ x ′ − x 0 ′ ∣ < r ∣ x ′ − x 0 ′ ∣ η ≥ 2 − η |x'-x'_0|^\eta\geq2^{-\eta} ∣ x ′ − x 0 ′ ∣ η ≥ 2 − η η \eta η
∣ x n − z j k ∣ ≤ ∣ x n ∣ + ∣ z j k ∣ ≤ 2 C . |x_n-z_{j_k}|\leq|x_n|+|z_{j_k}|\leq2C. ∣ x n − z j k ∣ ≤ ∣ x n ∣ + ∣ z j k ∣ ≤ 2 C . Thus,
∣ x n − z j k ∣ ≤ θ ∣ x ′ − x 0 ′ ∣ η , |x_n-z_{j_k}|\leq\theta|x'-x'_0|^\eta, ∣ x n − z j k ∣ ≤ θ ∣ x ′ − x 0 ′ ∣ η , provided that θ ≥ 2 1 + η C \theta\geq2^{1+\eta}C θ ≥ 2 1 + η C
As last, we deal with the case ∣ x ′ − x 0 ′ ∣ ≤ 1 2 ( δ 0 2 ) j k |x'-x'_0|\leq\frac{1}{2}\left(\frac{\delta_0}{2}\right)^{j_k} ∣ x ′ − x 0 ′ ∣ ≤ 2 1 ( 2 δ 0 ) j k Σ k , j k ⊆ Σ k , j k − 1 \Sigma_{k,j_k}\subseteq\Sigma_{k,j_k-1} Σ k , j k ⊆ Σ k , j k − 1
∣ x n − z j k ∣ < 2 2 + α ( 1 − δ 0 2 2 ) j k − 1 + ∣ z j k − 1 − z j k ∣ ≤ ( 1 + 2 2 + α ) ( 1 − δ 0 2 2 ) j k − 1 . |x_n-z_{j_k}|<2^{2+\alpha}\left(1-\frac{\delta_0^2}{2}\right)^{j_k-1}+|z_{j_k-1}-z_{j_k}|\leq(1+2^{2+\alpha})\left(1-\frac{\delta_0^2}{2}\right)^{j_k-1}. ∣ x n − z j k ∣ < 2 2 + α ( 1 − 2 δ 0 2 ) j k − 1 + ∣ z j k − 1 − z j k ∣ ≤ ( 1 + 2 2 + α ) ( 1 − 2 δ 0 2 ) j k − 1 . Defining b k : = ( 1 − δ 0 2 2 ) j k b_k:=\left(1-\frac{\delta_0^2}{2}\right)^{j_k} b k := ( 1 − 2 δ 0 2 ) j k b k → 0 b_k\to0 b k → 0 k → + ∞ k\to+\infty k → + ∞
∣ x n − z j k ∣ < ( 1 + 2 2 + α ) ( 1 − δ 0 2 2 ) − 1 b k ≤ θ b k η , |x_n-z_{j_k}|<(1+2^{2+\alpha})\left(1-\frac{\delta_0^2}{2}\right)^{-1}b_k\leq\theta b_k^\eta, ∣ x n − z j k ∣ < ( 1 + 2 2 + α ) ( 1 − 2 δ 0 2 ) − 1 b k ≤ θ b k η , which completes the proof of (B.3).
From (B.3), we infer that A k r ∩ { ∣ x ′ − x 0 ′ ∣ < r 2 } A_k^r\cap\{|x'-x'_0|<\frac{r}{2}\} A k r ∩ { ∣ x ′ − x 0 ′ ∣ < 2 r }
f k − [ x 0 ] ( x ′ ) : = z j k − θ max { b k , ∣ x ′ − x 0 ′ ∣ } η , f_k^-[x_0](x'):=z_{j_k}-\theta\max\{b_k,|x'-x'_0|\}^\eta, f k − [ x 0 ] ( x ′ ) := z j k − θ max { b k , ∣ x ′ − x 0 ′ ∣ } η , and below the graph of the Hölder continuous function
f k + [ x 0 ] ( x ′ ) : = z j k + θ max { b k , ∣ x ′ − x 0 ′ ∣ } η . f_k^+[x_0](x'):=z_{j_k}+\theta\max\{b_k,|x'-x'_0|\}^\eta. f k + [ x 0 ] ( x ′ ) := z j k + θ max { b k , ∣ x ′ − x 0 ′ ∣ } η . Taking the infimum and the supremum respectively over all x 0 ∈ A k r ∩ { ∣ x ′ − x 0 ′ ∣ < r 2 } x_0\in A_k^r\cap\{|x'-x'_0|<\frac{r}{2}\} x 0 ∈ A k r ∩ { ∣ x ′ − x 0 ′ ∣ < 2 r } A k r ∩ { ∣ x ′ − x 0 ′ ∣ < r 2 } A_k^r\cap\{|x'-x'_0|<\frac{r}{2}\} A k r ∩ { ∣ x ′ − x 0 ′ ∣ < 2 r }
g k − : = inf x 0 ∈ A k r ∩ { ∣ x ′ − x 0 ′ ∣ < r 2 } f k − [ x 0 ] g_k^-:=\inf_{x_0\in A_k^r\cap\{|x'-x'_0|<\frac{r}{2}\}}f_k^-[x_0] g k − := x 0 ∈ A k r ∩ { ∣ x ′ − x 0 ′ ∣ < 2 r } inf f k − [ x 0 ] and
g k + : = sup x 0 ∈ A k r ∩ { ∣ x ′ − x 0 ′ ∣ < r 2 } f k + [ x 0 ] , g_k^+:=\sup_{x_0\in A_k^r\cap\{|x'-x'_0|<\frac{r}{2}\}}f_k^+[x_0], g k + := x 0 ∈ A k r ∩ { ∣ x ′ − x 0 ′ ∣ < 2 r } sup f k + [ x 0 ] , concluding the proof.
Monotonicity formula (Proposition 1.12)
In this section, we prove the monotonicity formula stated in Proposition 1.12. To this purpose, we recall the extension technique presented in [8 ] . Let
R + n + 1 : = { ( x , z ) ∈ R n + 1 s.t. x ∈ R n , z ∈ [ 0 , + ∞ ) } . \mathbb{R}^{n+1}_{+}:=\left\{(x,z)\in\mathbb{R}^{n+1}\ \text{s.t.}\ x\in\mathbb{R}^{n},\ z\in[0,+\infty)\right\}. R + n + 1 := { ( x , z ) ∈ R n + 1 s.t. x ∈ R n , z ∈ [ 0 , + ∞ ) } . the ( n + 1 ) (n+1) ( n + 1 ) Ω ⊆ R n + 1 \Omega\subseteq\mathbb{R}^{n+1} Ω ⊆ R n + 1
Ω 0 : = Ω ∩ { z = 0 } and Ω + : = Ω ∩ { z > 0 } . \Omega_0:=\Omega\cap\{z=0\}\qquad\text{and}\qquad\Omega_+:=\Omega\cap\{z>0\}. Ω 0 := Ω ∩ { z = 0 } and Ω + := Ω ∩ { z > 0 } . For any set E ⊆ R n × { 0 } E\subseteq\mathbb{R}^{n}\times\{0\} E ⊆ R n × { 0 } ( n + 1 ) (n+1) ( n + 1 ) u : = χ E − χ E c u:=\chi_E-\chi_{E^c} u := χ E − χ E c
u ~ ( x , z ) : = u ∗ P ( x , z ) , (74) \widetilde{u}(x,z):=u*P(x,z), \tag*{(74)} u ( x , z ) := u ∗ P ( x , z ) , (74) where
P ( x , z ) : = C P z s ( x 2 + z 2 ) n + s 2 , P(x,z):=C_P\frac{z^s}{(x^2+z^2)^{\frac{n+s}{2}}}, P ( x , z ) := C P ( x 2 + z 2 ) 2 n + s z s , and
C P : = ( ∫ R n d x ( 1 + x 2 ) ( n + 2 ) / 2 ) − 1 . C_P:=\left(\int_{\mathbb{R}^n}\frac{dx}{(1+x^2)^{(n+2)/2}}\right)^{-1}. C P := ( ∫ R n ( 1 + x 2 ) ( n + 2 ) /2 d x ) − 1 . Then, we have the following equivalence:
Proposition C.1 ([13 Proposition 1.7] ). E E E ( Λ , γ ) (\Lambda,\gamma) ( Λ , γ ) B 1 B_1 B 1 c ~ n , s \widetilde{c}_{n,s} c n , s u ~ \widetilde{u} u u = χ E − χ E c u=\chi_E-\chi_{E^c} u = χ E − χ E c R + n + 1 \mathbb{R}^{n+1}_+ R + n + 1
∫ Ω + z 1 − s ∣ ∇ u ~ ∣ 2 d x d z ≤ ∫ ( B ~ r n + 1 ) + z 1 − s ∣ ∇ v ‾ ∣ 2 d x d z + 8 c ~ n , s Λ r n − s + γ (75) \int_{\Omega_+}z^{1-s}|\nabla\widetilde{u}|^2\,dx\,dz\leq\int_{(\widetilde{B}_r^{n+1})_+}z^{1-s}|\nabla\overline{v}|^2\,dx\,dz+8\widetilde{c}_{n,s}\Lambda r^{n-s+\gamma} \tag*{(75)} ∫ Ω + z 1 − s ∣∇ u ∣ 2 d x d z ≤ ∫ ( B r n + 1 ) + z 1 − s ∣∇ v ∣ 2 d x d z + 8 c n , s Λ r n − s + γ (75) for every Ω ⊆ R n + 1 \Omega\subseteq\mathbb{R}^{n+1} Ω ⊆ R n + 1 Ω 0 ⊆ B ~ r × { 0 } ⋐ B 1 × { 0 } \Omega_0\subseteq\widetilde{B}_r\times\{0\}\Subset B_1\times\{0\} Ω 0 ⊆ B r × { 0 } ⋐ B 1 × { 0 } r > 0 r>0 r > 0 v ‾ : R + n + 1 → R \overline{v}:\mathbb{R}^{n+1}_+\to\mathbb{R} v : R + n + 1 → R supp ( v ‾ − u ~ ) ⊆ Ω \operatorname{supp}(\overline{v}-\widetilde{u})\subseteq\Omega supp ( v − u ) ⊆ Ω ∣ v ‾ ∣ = 1 |\overline{v}|=1 ∣ v ∣ = 1 Ω 0 \Omega_0 Ω 0
In other words, Proposition C.1 states that a set E E E u ~ \widetilde{u} u
Remark C.2. We point out that the constant c ~ n , s \widetilde{c}_{n,s} c n , s [7 Lemma 7.2] . Since it will appear several times in the forthcoming discussion, given Λ ≥ 0 \Lambda\geq0 Λ ≥ 0
Λ ^ : = 8 c ~ n , s Λ . \widehat{\Lambda}:=8\widetilde{c}_{n,s}\Lambda. Λ := 8 c n , s Λ. Now, we consider a set E ⊆ R n E\subseteq\mathbb{R}^n E ⊆ R n R > 0 R>0 R > 0 r ∈ ( 0 , R ) r\in(0,R) r ∈ ( 0 , R )
Φ E ( r ) : = r s − n ∫ B r + z 1 − s ∣ ∇ u ~ ∣ 2 d x d z + n − s γ Λ ^ r γ . (76) \Phi_E(r):=r^{s-n}\int_{B_r^+}z^{1-s}|\nabla\widetilde{u}|^2\,dx\,dz+\frac{n-s}{\gamma}\widehat{\Lambda}r^\gamma. \tag*{(76)} Φ E ( r ) := r s − n ∫ B r + z 1 − s ∣∇ u ∣ 2 d x d z + γ n − s Λ r γ . (76) For later convenience, we also define
Ξ E ( r ) : = r s − n ∫ B r + z 1 − s ∣ ∇ u ~ ∣ 2 d x d z . (77) \Xi_E(r):=r^{s-n}\int_{B_r^+}z^{1-s}|\nabla\widetilde{u}|^2\,dx\,dz. \tag*{(77)} Ξ E ( r ) := r s − n ∫ B r + z 1 − s ∣∇ u ∣ 2 d x d z . (77) The proof of Proposition 1.12 is a modification of the one for the case γ = s \gamma=s γ = s [13 Section 3] (as well as a proof of the analogue of Proposition C.1). For the convenience of the reader, we sketch the main steps below. The result is a consequence of the following intermediate results.
Proposition C.3. Let 0 < r 1 < r 2 < R 0<r_1<r_2<R 0 < r 1 < r 2 < R Φ E \Phi_E Φ E [ r 1 , r 2 ] [r_1,r_2] [ r 1 , r 2 ]
Proof. It follows immediately from the absolute continuity of the Lebesgue integral and the smoothness of r s − γ r^{s-\gamma} r s − γ
Proposition C.4. The function Φ E \Phi_E Φ E r ∈ ( 0 , R ) r \in(0,R) r ∈ ( 0 , R )
Φ E ′ ( r ) ≥ 0 , for a.e. r ∈ ( 0 , R ) . \Phi'_E(r) \ge0,\qquad\text{for a.e. }r \in(0,R). Φ E ′ ( r ) ≥ 0 , for a.e. r ∈ ( 0 , R ) . Proof. Thanks to Proposition C.3 and the Fundamental Theorem of Calculus for the Lebesgue integral (see e.g. [16 Theorem 7.20] ), Φ E \Phi_E Φ E r ∈ ( 0 , R ) r \in(0,R) r ∈ ( 0 , R )
Φ E ′ ( r ) = r s − n ∫ ∂ B r + z 1 − s ∣ ∇ u ~ ( x , z ) ∣ 2 d H n ( x , z ) − ( n − s ) r s − n − 1 ∫ B r + z 1 − s ∣ ∇ u ~ ( x , z ) ∣ 2 d x d z + ( n − s ) Λ ^ r γ − 1 = ( n − s ) r s − n − 1 ( r n − s ∫ ∂ B r + z 1 − s ∣ ∇ u ~ ( x , z ) ∣ 2 d H n ( x , z ) − ∫ B r + z 1 − s ∣ ∇ u ~ ( x , z ) ∣ 2 d x d z + Λ ^ r n − s + γ ) . \begin{aligned}
\Phi'_E(r)={}&r^{s-n}\int_{\partial B_r^+}z^{1-s}|\nabla\widetilde{u}(x,z)|^2\,d\mathcal{H}^n(x,z)\\
&-(n-s)r^{s-n-1}\int_{B_r^+}z^{1-s}|\nabla\widetilde{u}(x,z)|^2\,dx\,dz+(n-s)\widehat{\Lambda}r^{\gamma-1}\\
={}&(n-s)r^{s-n-1}\left(\frac{r}{n-s}\int_{\partial B_r^+}z^{1-s}|\nabla\widetilde{u}(x,z)|^2\,d\mathcal{H}^n(x,z)\right.\\
&\left.-\int_{B_r^+}z^{1-s}|\nabla\widetilde{u}(x,z)|^2\,dx\,dz+\widehat{\Lambda}r^{n-s+\gamma}\right).
\end{aligned} Φ E ′ ( r ) = = r s − n ∫ ∂ B r + z 1 − s ∣∇ u ( x , z ) ∣ 2 d H n ( x , z ) − ( n − s ) r s − n − 1 ∫ B r + z 1 − s ∣∇ u ( x , z ) ∣ 2 d x d z + ( n − s ) Λ r γ − 1 ( n − s ) r s − n − 1 ( n − s r ∫ ∂ B r + z 1 − s ∣∇ u ( x , z ) ∣ 2 d H n ( x , z ) − ∫ B r + z 1 − s ∣∇ u ( x , z ) ∣ 2 d x d z + Λ r n − s + γ ) . Therefore, the positivity of Φ E ′ \Phi'_E Φ E ′
r n − s ∫ ∂ B r + z 1 − s ∣ ∇ u ~ ( x , z ) ∣ 2 d H n ( x , z ) − ∫ B r + z 1 − s ∣ ∇ u ~ ( x , z ) ∣ 2 d x d z + Λ ^ r n − s + γ − 1 ≥ 0. (78) \frac{r}{n-s}\int_{\partial B_r^+}z^{1-s}|\nabla\widetilde{u}(x,z)|^2\,d\mathcal{H}^n(x,z)-\int_{B_r^+}z^{1-s}|\nabla\widetilde{u}(x,z)|^2\,dx\,dz+\widehat{\Lambda}r^{n-s+\gamma-1}\ge0. \tag*{(78)} n − s r ∫ ∂ B r + z 1 − s ∣∇ u ( x , z ) ∣ 2 d H n ( x , z ) − ∫ B r + z 1 − s ∣∇ u ( x , z ) ∣ 2 d x d z + Λ r n − s + γ − 1 ≥ 0. (78) With this aim, we consider the function
v ~ r ( x , z ) : = { u ~ ( r x ∣ ( x , z ) ∣ , r z ∣ ( x , z ) ∣ ) , if 0 < ∣ x ∣ 2 + z 2 ≤ r 2 , u ~ ( x , z ) , if ∣ x ∣ 2 + z 2 > r 2 . \widetilde{v}_r(x,z):=
\begin{cases}
\widetilde{u}\left(\dfrac{rx}{|(x,z)|},\dfrac{rz}{|(x,z)|}\right), & \text{if }0<|x|^2+z^2\le r^2,\\
\widetilde{u}(x,z), & \text{if }|x|^2+z^2>r^2.
\end{cases} v r ( x , z ) := ⎩ ⎨ ⎧ u ( ∣ ( x , z ) ∣ r x , ∣ ( x , z ) ∣ r z ) , u ( x , z ) , if 0 < ∣ x ∣ 2 + z 2 ≤ r 2 , if ∣ x ∣ 2 + z 2 > r 2 . We claim that there exists a sets W ⊆ ( 0 , + ∞ ) \mathcal{W}\subseteq(0,+\infty) W ⊆ ( 0 , + ∞ ) H 1 ( ( 0 , + ∞ ) ∖ W ) = 0 \mathcal{H}^1((0,+\infty)\setminus\mathcal{W})=0 H 1 (( 0 , + ∞ ) ∖ W ) = 0 r ∈ W r\in\mathcal{W} r ∈ W G r ⊆ R n G_r\subseteq\mathbb{R}^n G r ⊆ R n R n \mathbb{R}^n R n x ∈ G r x\in G_r x ∈ G r
v ~ r ( x , 0 ) = { u ~ ( r x ∣ x ∣ , 0 ) = χ E ( r x ∣ x ∣ ) − χ E c ( r x ∣ x ∣ ) , if 0 < ∣ x ∣ ≤ r , u ~ ( x , 0 ) = χ E ( x ) − χ E c ( x ) , if ∣ x ∣ > r , \widetilde{v}_r(x,0)=
\begin{cases}
\widetilde{u}\left(\dfrac{rx}{|x|},0\right)=\chi_E\left(\dfrac{rx}{|x|}\right)-\chi_{E^c}\left(\dfrac{rx}{|x|}\right), & \text{if }0<|x|\le r,\\
\widetilde{u}(x,0)=\chi_E(x)-\chi_{E^c}(x), & \text{if }|x|>r,
\end{cases} v r ( x , 0 ) = ⎩ ⎨ ⎧ u ( ∣ x ∣ r x , 0 ) = χ E ( ∣ x ∣ r x ) − χ E c ( ∣ x ∣ r x ) , u ( x , 0 ) = χ E ( x ) − χ E c ( x ) , if 0 < ∣ x ∣ ≤ r , if ∣ x ∣ > r , is well-defined (and thus v ~ r ( x , 0 ) ∈ { − 1 , 1 } \widetilde{v}_r(x,0)\in\{-1,1\} v r ( x , 0 ) ∈ { − 1 , 1 } W \mathcal{W} W G r G_r G r v ~ r \widetilde{v}_r v r [13 ] .
In what follows, we pick r ∈ ( 0 , R ) ∩ W r\in(0,R)\cap\mathcal{W} r ∈ ( 0 , R ) ∩ W v ~ : = v ~ r \widetilde{v}:=\widetilde{v}_r v := v r F : = F r F:=F_r F := F r
∫ B r + z 1 − s ∣ ∇ v ~ ( x , z ) ∣ 2 d x d z ≥ ∫ B r + z 1 − s ∣ ∇ u ~ ( x , z ) ∣ 2 d x d z − Λ ^ r n − s + γ . (79) \int_{B_r^+}z^{1-s}|\nabla\widetilde{v}(x,z)|^2\,dx\,dz\ge\int_{B_r^+}z^{1-s}|\nabla\widetilde{u}(x,z)|^2\,dx\,dz-\widehat{\Lambda}r^{n-s+\gamma}. \tag*{(79)} ∫ B r + z 1 − s ∣∇ v ( x , z ) ∣ 2 d x d z ≥ ∫ B r + z 1 − s ∣∇ u ( x , z ) ∣ 2 d x d z − Λ r n − s + γ . (79) Since v ~ \widetilde{v} v B r ∖ { 0 } B_r\setminus\{0\} B r ∖ { 0 }
∇ v ~ ( x , z ) = r ∣ ( x , z ) ∣ ∇ T u ~ ( r ∣ ( x , z ) ∣ x , r ∣ ( x , z ) ∣ z ) , in B r ∖ { 0 } , \nabla\widetilde{v}(x,z)=\frac{r}{|(x,z)|}\nabla^T\widetilde{u}\left(\frac{r}{|(x,z)|}x,\frac{r}{|(x,z)|}z\right),\qquad\text{in }B_r\setminus\{0\}, ∇ v ( x , z ) = ∣ ( x , z ) ∣ r ∇ T u ( ∣ ( x , z ) ∣ r x , ∣ ( x , z ) ∣ r z ) , in B r ∖ { 0 } , where ∇ T \nabla^T ∇ T ∇ \nabla ∇
∫ B r + z 1 − s ∣ ∇ v ~ ( x , z ) ∣ 2 d x d z = ∫ 0 r ∫ ∂ B t + z 1 − s ∣ ∇ v ~ ( x , z ) ∣ 2 d H n ( x , z ) d t = ∫ 0 r ∫ ∂ B t + r 2 t 2 z 1 − s ∣ ∇ T u ~ ( r t x , r t z ) ∣ 2 d H n ( x , z ) d t . \begin{aligned}
\int_{B_r^+}z^{1-s}|\nabla\widetilde{v}(x,z)|^2\,dx\,dz
&=\int_0^r\int_{\partial B_t^+}z^{1-s}|\nabla\widetilde{v}(x,z)|^2\,d\mathcal{H}^n(x,z)\,dt\\
&=\int_0^r\int_{\partial B_t^+}\frac{r^2}{t^2}z^{1-s}\left|\nabla^T\widetilde{u}\left(\frac{r}{t}x,\frac{r}{t}z\right)\right|^2\,d\mathcal{H}^n(x,z)\,dt.
\end{aligned} ∫ B r + z 1 − s ∣∇ v ( x , z ) ∣ 2 d x d z = ∫ 0 r ∫ ∂ B t + z 1 − s ∣∇ v ( x , z ) ∣ 2 d H n ( x , z ) d t = ∫ 0 r ∫ ∂ B t + t 2 r 2 z 1 − s ∇ T u ( t r x , t r z ) 2 d H n ( x , z ) d t . Adopting the change of variable ( x ^ , z ^ ) : = ( t r x , t r z ) (\hat{x},\hat{z}) := \left(\frac{t}{r}x,\frac{t}{r}z\right) ( x ^ , z ^ ) := ( r t x , r t z )
∫ B r + z 1 − s ∣ ∇ v ~ ( x , z ) ∣ 2 d x d z = ∫ 0 r ∫ ∂ B r + ( t r ) n − 2 ( t r z ^ ) 1 − s ∣ ∇ T u ~ ( x ^ , z ^ ) ∣ 2 d H n ( x ^ , z ^ ) d t = ∫ 0 r ( t r ) n − 1 − s d t ∫ ∂ B r + z ^ 1 − s ∣ ∇ T u ~ ( x ^ , z ^ ) ∣ 2 d H n ( x ^ , z ^ ) = r n − s ∫ ∂ B r + z ^ 1 − s ∣ ∇ T u ~ ( x ^ , z ^ ) ∣ 2 d H n ( x ^ , z ^ ) . (80) \begin{aligned}
\int_{B_r^+} z^{1-s}|\nabla\tilde{v}(x,z)|^2\,dx\,dz
&= \int_0^r \int_{\partial B_r^+} \left(\frac{t}{r}\right)^{n-2}\left(\frac{t}{r}\hat{z}\right)^{1-s}|\nabla^T\tilde{u}(\hat{x},\hat{z})|^2\,d\mathcal{H}^n(\hat{x},\hat{z})\,dt \\
&= \int_0^r \left(\frac{t}{r}\right)^{n-1-s}\,dt \int_{\partial B_r^+} \hat{z}^{1-s}|\nabla^T\tilde{u}(\hat{x},\hat{z})|^2\,d\mathcal{H}^n(\hat{x},\hat{z}) \\
&= \frac{r}{n-s}\int_{\partial B_r^+}\hat{z}^{1-s}|\nabla^T\tilde{u}(\hat{x},\hat{z})|^2\,d\mathcal{H}^n(\hat{x},\hat{z}). \tag*{(80)}
\end{aligned} ∫ B r + z 1 − s ∣∇ v ~ ( x , z ) ∣ 2 d x d z = ∫ 0 r ∫ ∂ B r + ( r t ) n − 2 ( r t z ^ ) 1 − s ∣ ∇ T u ~ ( x ^ , z ^ ) ∣ 2 d H n ( x ^ , z ^ ) d t = ∫ 0 r ( r t ) n − 1 − s d t ∫ ∂ B r + z ^ 1 − s ∣ ∇ T u ~ ( x ^ , z ^ ) ∣ 2 d H n ( x ^ , z ^ ) = n − s r ∫ ∂ B r + z ^ 1 − s ∣ ∇ T u ~ ( x ^ , z ^ ) ∣ 2 d H n ( x ^ , z ^ ) . (80) Plugging (C.7) into (C.6), we obtain that
r n − s ∫ ∂ B r + z 1 − s ∣ ∇ u ~ ( x , z ) ∣ 2 d H n ( x , z ) ≥ r n − s ∫ ∂ B r + z 1 − s ∣ ∇ T u ~ ( x , z ) ∣ 2 d H n ( x , z ) ≥ ∫ B r + z 1 − s ∣ ∇ u ~ ∣ 2 d x d z − Λ ^ r n − s + γ , \begin{aligned}
\frac{r}{n-s}\int_{\partial B_r^+}z^{1-s}|\nabla\tilde{u}(x,z)|^2\,d\mathcal{H}^n(x,z)
\geq\frac{r}{n-s}\int_{\partial B_r^+}z^{1-s}|\nabla^T\tilde{u}(x,z)|^2\,d\mathcal{H}^n(x,z) \\
&\geq\int_{B_r^+}z^{1-s}|\nabla\tilde{u}|^2\,dx\,dz-\widehat{\Lambda}r^{n-s+\gamma},
\end{aligned} n − s r ∫ ∂ B r + z 1 − s ∣∇ u ~ ( x , z ) ∣ 2 d H n ( x , z ) ≥ n − s r ∫ ∂ B r + z 1 − s ∣ ∇ T u ~ ( x , z ) ∣ 2 d H n ( x , z ) ≥ ∫ B r + z 1 − s ∣∇ u ~ ∣ 2 d x d z − Λ r n − s + γ , which gives (C.5), as desired. □
Appendix D. A note on Q Q Q
In this section we focus on Q Q Q Q > 1 Q>1 Q > 1 Q Q Q s s s γ > 0 \gamma>0 γ > 0
Proposition D.1 (Uniform density estimates). Let E E E Q Q Q B 1 B_1 B 1 Q > 1 Q>1 Q > 1 r 0 > 0 r_0>0 r 0 > 0 c 0 ∈ ( 0 , 1 ) c_0\in(0,1) c 0 ∈ ( 0 , 1 ) n n n s s s Λ \Lambda Λ γ \gamma γ x 0 ∈ ( ∂ E ) ∩ B 1 x_0\in(\partial E)\cap B_1 x 0 ∈ ( ∂ E ) ∩ B 1 r ∈ ( 0 , min { r 0 , dist ( x 0 , ∂ B 1 ) } ) r\in(0,\min\{r_0,\operatorname{dist}(x_0,\partial B_1)\}) r ∈ ( 0 , min { r 0 , dist ( x 0 , ∂ B 1 )})
∣ E ∩ B r ( x 0 ) ∣ ≥ c 0 r n |E\cap B_r(x_0)|\geq c_0r^n ∣ E ∩ B r ( x 0 ) ∣ ≥ c 0 r n and
∣ E c ∩ B r ( x 0 ) ∣ ≥ c 0 r n . |E^c\cap B_r(x_0)|\geq c_0r^n. ∣ E c ∩ B r ( x 0 ) ∣ ≥ c 0 r n . We recall that, for any set E E E E E E r r r
a ( r ) : = ∣ E ∩ B r ∣ ∣ B r ∣ . a(r):=\frac{|E\cap B_r|}{|B_r|}. a ( r ) := ∣ B r ∣ ∣ E ∩ B r ∣ . The proof of Proposition D.1 follows the same structure as the one of Proposition 2.1. However, in the setting of Q Q Q
Lemma D.2. Let δ 0 > 0 \delta_0>0 δ 0 > 0 E E E Q Q Q B 2 k ˉ B_{2^{\bar{k}}} B 2 k ˉ k ˉ ∈ N \bar{k}\in\mathbb{N} k ˉ ∈ N 2 − k ˉ s ≤ δ 0 2^{-\bar{k}s}\leq\delta_0 2 − k ˉ s ≤ δ 0
a ( 2 k ) ≤ δ 0 , for all k ∈ { 0 , … , k ˉ } , a(2^k)\leq\delta_0,\qquad\text{for all }k\in\{0,\ldots,\bar{k}\}, a ( 2 k ) ≤ δ 0 , for all k ∈ { 0 , … , k ˉ } , then a ( 1 / 2 ) ≤ δ 0 a(1/2)\leq\delta_0 a ( 1/2 ) ≤ δ 0
Proof. Let r ∈ ( 1 2 , 1 ) r\in\left(\frac{1}{2},1\right) r ∈ ( 2 1 , 1 ) A r : = E ∩ B r A_r:=E\cap B_r A r := E ∩ B r μ ( r ) : = ∣ A r ∣ \mu(r):=|A_r| μ ( r ) := ∣ A r ∣ μ ′ ( r ) = H n − 1 ( E ∩ ∂ B r ) \mu'(r)=\mathcal{H}^{n-1}(E\cap\partial B_r) μ ′ ( r ) = H n − 1 ( E ∩ ∂ B r )
Now, notice that, by Sobolev embeddings, we have that
∥ χ A r ∥ L n n − s ( R n ) = ∣ A r ∣ 1 − s n ≤ C [ χ A r ] H s / 2 = C L s ( A r , A r c ) , (81) \|\chi_{A_r}\|_{L^{\frac{n}{n-s}}(\mathbb{R}^n)}=|A_r|^{1-\frac{s}{n}}\leq C[\chi_{A_r}]_{H^{s/2}}=C\mathcal{L}_s(A_r,A_r^c), \tag*{(81)} ∥ χ A r ∥ L n − s n ( R n ) = ∣ A r ∣ 1 − n s ≤ C [ χ A r ] H s /2 = C L s ( A r , A r c ) , (81) for some constant C > 0 C>0 C > 0 n n n s s s
Since
L s ( A r , A r c ) = L s ( A r , E ∩ A r c ) + L s ( A r , E c ) , \mathcal{L}_s(A_r,A_r^c)=\mathcal{L}_s(A_r,E\cap A_r^c)+\mathcal{L}_s(A_r,E^c), L s ( A r , A r c ) = L s ( A r , E ∩ A r c ) + L s ( A r , E c ) , it follows from the Q Q Q E E E
L s ( A r , A r c ) ≤ 2 L s ( A r , E ∩ A r c ) + Q Per s ( E ∖ B r , B r ) ≤ 2 L s ( A r , B r c ) + Q L s ( E ∩ B r c , B r ) . (82) \begin{aligned}
\mathcal{L}_s(A_r,A_r^c) &\le2\mathcal{L}_s(A_r,E\cap A_r^c)+Q\operatorname{Per}_s(E\setminus B_r,B_r)\\
&\le2\mathcal{L}_s(A_r,B_r^c)+Q\mathcal{L}_s(E\cap B_r^c,B_r). \tag*{(82)}
\end{aligned} L s ( A r , A r c ) ≤ 2 L s ( A r , E ∩ A r c ) + Q Per s ( E ∖ B r , B r ) ≤ 2 L s ( A r , B r c ) + Q L s ( E ∩ B r c , B r ) . (82) From (D.3) and (D.4) together, we deduce that
μ ( r ) 1 − s n ≤ C L s ( A r , B r c ) + Q L s ( E ∩ B r c , B r ) . (83) \mu(r)^{1-\frac{s}{n}}\le C\mathcal{L}_s(A_r,B_r^c)+Q\mathcal{L}_s(E\cap B_r^c,B_r). \tag*{(83)} μ ( r ) 1 − n s ≤ C L s ( A r , B r c ) + Q L s ( E ∩ B r c , B r ) . (83) and we aim to estimate the two terms on the right-hand side.
To do so, observe that, by the co-area formula, we have that
L s ( A r , B r c ) = ∬ A r × B r c d x d y ∣ x − y ∣ n + s ≤ C ∫ A r ( ∫ r − ∣ x ∣ + ∞ d z z s + 1 ) d x ≤ C ∫ A r d x ( r − ∣ x ∣ ) s ≤ C ∫ 0 r μ ′ ( ρ ) ( r − ρ ) s d ρ . (84) \begin{aligned}
\mathcal{L}_s(A_r,B_r^c)&=\iint_{A_r\times B_r^c}\frac{dx\,dy}{\lvert x-y\rvert^{n+s}}\le C\int_{A_r}\left(\int_{r-\lvert x\rvert}^{+\infty}\frac{dz}{z^{s+1}}\right)dx\\
&\le C\int_{A_r}\frac{dx}{(r-\lvert x\rvert)^s}\le C\int_0^r\frac{\mu'(\rho)}{(r-\rho)^s}\,d\rho. \tag*{(84)}
\end{aligned} L s ( A r , B r c ) = ∬ A r × B r c ∣ x − y ∣ n + s d x d y ≤ C ∫ A r ( ∫ r − ∣ x ∣ + ∞ z s + 1 d z ) d x ≤ C ∫ A r ( r − ∣ x ∣ ) s d x ≤ C ∫ 0 r ( r − ρ ) s μ ′ ( ρ ) d ρ . (84) for some C > 0 C>0 C > 0 n n n s s s
Then, we provide an estimate for
L s ( E ∩ B r c , B r ) = ∬ B r × E ∩ ( B 1 ∖ B r ) d x d y ∣ x − y ∣ n + s + ∑ k = 0 k ˉ ∬ B r × E ∩ C k d x d y ∣ x − y ∣ n + s + ∬ B r × E ∩ B 2 k ˉ c d x d y ∣ x − y ∣ n + s , \begin{aligned}
\mathcal{L}_s(E\cap B_r^c,B_r)
&=\iint_{B_r\times E\cap(B_1\setminus B_r)}\frac{dx\,dy}{\lvert x-y\rvert^{n+s}}
+\sum_{k=0}^{\bar{k}}\iint_{B_r\times E\cap C_k}\frac{dx\,dy}{\lvert x-y\rvert^{n+s}}\\
&\qquad+\iint_{B_r\times E\cap B_{2^{\bar{k}}}^c}\frac{dx\,dy}{\lvert x-y\rvert^{n+s}},
\end{aligned} L s ( E ∩ B r c , B r ) = ∬ B r × E ∩ ( B 1 ∖ B r ) ∣ x − y ∣ n + s d x d y + k = 0 ∑ k ˉ ∬ B r × E ∩ C k ∣ x − y ∣ n + s d x d y + ∬ B r × E ∩ B 2 k ˉ c ∣ x − y ∣ n + s d x d y , where C k : = B 2 k + 1 ∖ B 2 k C_k:=B_{2^{k+1}}\setminus B_{2^k} C k := B 2 k + 1 ∖ B 2 k
First, using co-area formula, we deduce that
∬ B r × E ∩ ( B 1 ∖ B r ) d x d y ∣ x − y ∣ n + s ≤ ∫ E ∩ ( B 1 ∖ B r ) ∫ B ∣ x ∣ ( x ) ∖ B ∣ x ∣ − r ( x ) d x d y ∣ x − y ∣ n + s ≤ C ∫ E ∩ ( B 1 ∖ B r ) ∫ ∣ x ∣ − r ∣ x ∣ d x d z ∣ z ∣ n + s ≤ C ∫ E ∩ ( B 1 ∖ B r ) d x ( ∣ x ∣ − r ) s = C ∫ r 1 μ ′ ( ρ ) ( ρ − r ) s d ρ . (85) \begin{aligned}
\iint_{B_r\times E\cap(B_1\setminus B_r)}\frac{dx\,dy}{\lvert x-y\rvert^{n+s}}
&\le\int_{E\cap(B_1\setminus B_r)}\int_{B_{\lvert x\rvert}(x)\setminus B_{\lvert x\rvert-r}(x)}\frac{dx\,dy}{\lvert x-y\rvert^{n+s}}\\
&\le C\int_{E\cap(B_1\setminus B_r)}\int_{\lvert x\rvert-r}^{\lvert x\rvert}\frac{dx\,dz}{\lvert z\rvert^{n+s}}
\le C\int_{E\cap(B_1\setminus B_r)}\frac{dx}{(\lvert x\rvert-r)^s}\\
&=C\int_r^1\frac{\mu'(\rho)}{(\rho-r)^s}\,d\rho. \tag*{(85)}
\end{aligned} ∬ B r × E ∩ ( B 1 ∖ B r ) ∣ x − y ∣ n + s d x d y ≤ ∫ E ∩ ( B 1 ∖ B r ) ∫ B ∣ x ∣ ( x ) ∖ B ∣ x ∣ − r ( x ) ∣ x − y ∣ n + s d x d y ≤ C ∫ E ∩ ( B 1 ∖ B r ) ∫ ∣ x ∣ − r ∣ x ∣ ∣ z ∣ n + s d x d z ≤ C ∫ E ∩ ( B 1 ∖ B r ) (∣ x ∣ − r ) s d x = C ∫ r 1 ( ρ − r ) s μ ′ ( ρ ) d ρ . (85) Moreover, for every k ∈ { 0 , … , k ˉ − 1 } k\in\{0,\ldots,\bar{k}-1\} k ∈ { 0 , … , k ˉ − 1 }
∬ B r × E ∩ C k d x d y ∣ x − y ∣ n + s ≤ ∬ B r × E ∩ C k d x d y ( 2 k − r ) n + s ≤ C 2 − k ( n + s ) ∣ E ∩ C k ∣ ≤ C 2 − k s ∣ E ∩ B 2 k + 1 ∣ 2 n ( k + 1 ) ≤ C 2 − k s a ( 2 k + 1 ) , (86) \begin{aligned}
\iint_{B_r\times E\cap C_k}\frac{dx\,dy}{\lvert x-y\rvert^{n+s}}
&\le\iint_{B_r\times E\cap C_k}\frac{dx\,dy}{(2^k-r)^{n+s}}\\
&\le C2^{-k(n+s)}\lvert E\cap C_k\rvert
\le C2^{-ks}\frac{\lvert E\cap B_{2^{k+1}}\rvert}{2^{n(k+1)}}\\
&\le C2^{-ks}a(2^{k+1}), \tag*{(86)}
\end{aligned} ∬ B r × E ∩ C k ∣ x − y ∣ n + s d x d y ≤ ∬ B r × E ∩ C k ( 2 k − r ) n + s d x d y ≤ C 2 − k ( n + s ) ∣ E ∩ C k ∣ ≤ C 2 − k s 2 n ( k + 1 ) ∣ E ∩ B 2 k + 1 ∣ ≤ C 2 − k s a ( 2 k + 1 ) , (86) for some constant C > 0 C>0 C > 0 n n n s s s
As last, since ∣ x − y ∣ ≥ ∣ x ∣ 2 \lvert x-y\rvert\ge\frac{\lvert x\rvert}{2} ∣ x − y ∣ ≥ 2 ∣ x ∣ x ∈ E ∩ B 2 k ˉ c x\in E\cap B_{2^{\bar{k}}}^c x ∈ E ∩ B 2 k ˉ c y ∈ B r y\in B_r y ∈ B r
∬ B r × E ∩ B 2 k ˉ c d x d y ∣ x − y ∣ n + s ≤ ∬ B r × E ∩ B 2 k ˉ c d x d y ( ∣ x ∣ − r ) n + s ≤ C ∫ B 2 k ˉ c d x ∣ x ∣ n + s ≤ C 2 − k ˉ s . (87) \begin{aligned}
\iint_{B_r\times E\cap B_{2^{\bar{k}}}^c}\frac{dx\,dy}{\lvert x-y\rvert^{n+s}}
&\le\iint_{B_r\times E\cap B_{2^{\bar{k}}}^c}\frac{dx\,dy}{(\lvert x\rvert-r)^{n+s}}\\
&\le C\int_{B_{2^{\bar{k}}}^c}\frac{dx}{\lvert x\rvert^{n+s}}\le C2^{-\bar{k}s}. \tag*{(87)}
\end{aligned} ∬ B r × E ∩ B 2 k ˉ c ∣ x − y ∣ n + s d x d y ≤ ∬ B r × E ∩ B 2 k ˉ c (∣ x ∣ − r ) n + s d x d y ≤ C ∫ B 2 k ˉ c ∣ x ∣ n + s d x ≤ C 2 − k ˉ s . (87) up to relabeling C C C
Now, plugging (D.6), (D.7), (D.8), and (D.9) into (D.5), we find
μ ( r ) 1 − s n ≤ ∫ 0 r μ ′ ( ρ ) ( r − ρ ) s d ρ + ∫ r 1 μ ′ ( ρ ) ( ρ − r ) s d ρ + ∑ k = 1 k ˉ 2 − k s a ( 2 k ) + C Q 2 − k ˉ s . \mu(r)^{1-\frac{s}{n}}\le\int_0^r\frac{\mu'(\rho)}{(r-\rho)^s}\,d\rho+\int_r^1\frac{\mu'(\rho)}{(\rho-r)^s}\,d\rho+\sum_{k=1}^{\bar{k}}2^{-ks}a(2^k)+CQ2^{-\bar{k}s}. μ ( r ) 1 − n s ≤ ∫ 0 r ( r − ρ ) s μ ′ ( ρ ) d ρ + ∫ r 1 ( ρ − r ) s μ ′ ( ρ ) d ρ + k = 1 ∑ k ˉ 2 − k s a ( 2 k ) + C Q 2 − k ˉ s . Now, we integrate la last inequality over r ∈ ( 1 2 , 1 ) r \in\left(\frac{1}{2},1\right) r ∈ ( 2 1 , 1 )
a ( 1 / 2 ) 1 − s n ≤ C μ ( 1 / 2 ) 1 − s n ≤ C ∫ 1 / 2 1 μ ( r ) 1 − s n d r ≤ C ∫ 1 / 2 1 ( ∫ 0 r μ ′ ( ρ ) ( r − ρ ) s d ρ + ∫ r 1 μ ′ ( ρ ) ( ρ − r ) s d ρ + ∑ k = 1 k ˉ 2 − k s a ( 2 k ) + Q 2 − k ˉ ) d r (88) \begin{aligned}
a(1/2)^{1-\frac{s}{n}} \le C\mu(1/2)^{1-\frac{s}{n}} \le C\int_{1/2}^{1}\mu(r)^{1-\frac{s}{n}}\,\mathrm{d}r \\
&\le C\int_{1/2}^{1}\left(\int_{0}^{r}\frac{\mu'(\rho)}{(r-\rho)^{s}}\,\mathrm{d}\rho+\int_{r}^{1}\frac{\mu'(\rho)}{(\rho-r)^{s}}\,\mathrm{d}\rho+\sum_{k=1}^{\bar{k}}2^{-ks}a(2^{k})+Q2^{-\bar{k}}\right)\mathrm{d}r \tag*{(88)}
\end{aligned} a ( 1/2 ) 1 − n s ≤ C μ ( 1/2 ) 1 − n s ≤ C ∫ 1/2 1 μ ( r ) 1 − n s d r ≤ C ∫ 1/2 1 ∫ 0 r ( r − ρ ) s μ ′ ( ρ ) d ρ + ∫ r 1 ( ρ − r ) s μ ′ ( ρ ) d ρ + k = 1 ∑ k ˉ 2 − k s a ( 2 k ) + Q 2 − k ˉ d r (88) The first term on the right-hand side gives
∫ 1 / 2 1 ∫ 0 r μ ′ ( ρ ) ( r − ρ ) s d ρ d r = ∫ 0 1 / 2 ∫ 1 / 2 1 μ ′ ( ρ ) ( r − ρ ) s d r d ρ + ∫ 1 / 2 1 ∫ ρ 1 μ ′ ( ρ ) ( r − ρ ) s d r d ρ ≤ C ∫ 0 1 / 2 μ ′ ( ρ ) ( 1 − ρ ) 1 − s d ρ + C ∫ 1 / 2 1 μ ′ ( ρ ) ( 1 − ρ ) 1 − s d ρ ≤ C μ ( 1 ) ≤ C a ( 1 ) . (89) \begin{aligned}
\int_{1/2}^{1}\int_{0}^{r}\frac{\mu'(\rho)}{(r-\rho)^{s}}\,\mathrm{d}\rho\,\mathrm{d}r=\int_{0}^{1/2}\int_{1/2}^{1}\frac{\mu'(\rho)}{(r-\rho)^{s}}\,\mathrm{d}r\,\mathrm{d}\rho+\int_{1/2}^{1}\int_{\rho}^{1}\frac{\mu'(\rho)}{(r-\rho)^{s}}\,\mathrm{d}r\,\mathrm{d}\rho\\
&\le C\int_{0}^{1/2}\mu'(\rho)(1-\rho)^{1-s}\,\mathrm{d}\rho+C\int_{1/2}^{1}\mu'(\rho)(1-\rho)^{1-s}\,\mathrm{d}\rho\\
&\le C\mu(1)\le Ca(1). \tag*{(89)}
\end{aligned} ∫ 1/2 1 ∫ 0 r ( r − ρ ) s μ ′ ( ρ ) d ρ d r = ∫ 0 1/2 ∫ 1/2 1 ( r − ρ ) s μ ′ ( ρ ) d r d ρ + ∫ 1/2 1 ∫ ρ 1 ( r − ρ ) s μ ′ ( ρ ) d r d ρ ≤ C ∫ 0 1/2 μ ′ ( ρ ) ( 1 − ρ ) 1 − s d ρ + C ∫ 1/2 1 μ ′ ( ρ ) ( 1 − ρ ) 1 − s d ρ ≤ C μ ( 1 ) ≤ C a ( 1 ) . (89) up to renaming C > 0 C>0 C > 0
Integrating the second term on the right-hand side of (D.10), we obtain
∫ 1 / 2 1 ∫ r 1 μ ′ ( ρ ) ( ρ − r ) s d ρ d r = ∫ 1 / 2 1 ∫ 1 / 2 ρ μ ′ ( ρ ) ( ρ − r ) s d r d ρ ≤ C ∫ 1 / 2 1 μ ′ ( ρ ) ( ρ − 1 2 ) 1 − s d ρ ≤ C μ ( 1 ) ≤ C a ( 1 ) . (90) \begin{aligned}
\int_{1/2}^{1}\int_{r}^{1}\frac{\mu'(\rho)}{(\rho-r)^{s}}\,\mathrm{d}\rho\,\mathrm{d}r=\int_{1/2}^{1}\int_{1/2}^{\rho}\frac{\mu'(\rho)}{(\rho-r)^{s}}\,\mathrm{d}r\,\mathrm{d}\rho\\
&\le C\int_{1/2}^{1}\mu'(\rho)\left(\rho-\frac{1}{2}\right)^{1-s}\,\mathrm{d}\rho\le C\mu(1)\le Ca(1). \tag*{(90)}
\end{aligned} ∫ 1/2 1 ∫ r 1 ( ρ − r ) s μ ′ ( ρ ) d ρ d r = ∫ 1/2 1 ∫ 1/2 ρ ( ρ − r ) s μ ′ ( ρ ) d r d ρ ≤ C ∫ 1/2 1 μ ′ ( ρ ) ( ρ − 2 1 ) 1 − s d ρ ≤ C μ ( 1 ) ≤ C a ( 1 ) . (90) for some C > 0 C>0 C > 0 n n n s s s
Moreover, integrating the last two term on the right-hand side of (D.10), we infer that
∫ 1 / 2 1 ( ∑ k = 1 k ˉ 2 − k s a ( 2 k ) + Q 2 − k ˉ s ) d r ≤ 1 2 ∑ k = 1 k ˉ 2 − k s a ( 2 k ) + 1 2 Q 2 − k ˉ s . (91) \int_{1/2}^{1}\left(\sum_{k=1}^{\bar{k}}2^{-ks}a(2^{k})+Q2^{-\bar{k}s}\right)\,\mathrm{d}r\le\frac{1}{2}\sum_{k=1}^{\bar{k}}2^{-ks}a(2^{k})+\frac{1}{2}Q2^{-\bar{k}s}. \tag*{(91)} ∫ 1/2 1 k = 1 ∑ k ˉ 2 − k s a ( 2 k ) + Q 2 − k ˉ s d r ≤ 2 1 k = 1 ∑ k ˉ 2 − k s a ( 2 k ) + 2 1 Q 2 − k ˉ s . (91) From (D.11), (D.12), and (D.13) together with (D.10), we deduce that
a ( 1 / 2 ) 1 − s n ≤ C a ( 1 ) + C ∑ k = 1 k ˉ 2 − k s a ( 2 k ) + C Q 2 − k ˉ s ≤ C ∑ k = 0 k ˉ 2 − k s a ( 2 k ) + C Q 2 − k ˉ s . (92) \begin{aligned}
a(1/2)^{1-\frac{s}{n}}\le Ca(1)+C\sum_{k=1}^{\bar{k}}2^{-ks}a(2^{k})+CQ2^{-\bar{k}s} \\
&\le C\sum_{k=0}^{\bar{k}}2^{-ks}a(2^{k})+CQ2^{-\bar{k}s}. \tag*{(92)}
\end{aligned} a ( 1/2 ) 1 − n s ≤ C a ( 1 ) + C k = 1 ∑ k ˉ 2 − k s a ( 2 k ) + C Q 2 − k ˉ s ≤ C k = 0 ∑ k ˉ 2 − k s a ( 2 k ) + C Q 2 − k ˉ s . (92) up to renaming C C C
Using (D.2) and that 2 − k ˉ s ≤ δ 0 2^{-\bar{k}s}\le\delta_0 2 − k ˉ s ≤ δ 0
a ( 1 / 2 ) ≤ ( C ∑ k = 0 k ˉ 2 − k s a ( 2 k ) + C Q 2 − k ˉ s ) n n − s ≤ ( C k ˉ δ 0 + C Q δ 0 ) n n − s ≤ [ C ( k ˉ + Q ) δ 0 s / n ] n n − s δ 0 ≤ δ 0 , \begin{aligned}
a(1/2)\le\left(C\sum_{k=0}^{\bar{k}}2^{-ks}a(2^{k})+CQ2^{-\bar{k}s}\right)^{\frac{n}{n-s}}\le\left(C\bar{k}\delta_0+CQ\delta_0\right)^{\frac{n}{n-s}} \\
&\le\left[C(\bar{k}+Q)\delta_0^{s/n}\right]^{\frac{n}{n-s}}\delta_0\le\delta_0,
\end{aligned} a ( 1/2 ) ≤ C k = 0 ∑ k ˉ 2 − k s a ( 2 k ) + C Q 2 − k ˉ s n − s n ≤ ( C k ˉ δ 0 + C Q δ 0 ) n − s n ≤ [ C ( k ˉ + Q ) δ 0 s / n ] n − s n δ 0 ≤ δ 0 , whenever δ 0 \delta_0 δ 0
Corollary D.3. Let δ 0 > 0 \delta_0>0 δ 0 > 0 E E E Q Q Q B 2 k ˉ B_{2^{\bar{k}}} B 2 k ˉ k ˉ ∈ N \bar{k}\in\mathbb{N} k ˉ ∈ N 2 − k ˉ s ≤ δ 0 2^{-\bar{k}s}\le\delta_0 2 − k ˉ s ≤ δ 0 a ( 2 − k ˉ ) ≤ 2 − k ˉ n δ 0 a(2^{-\bar{k}})\le2^{-\bar{k}n}\delta_0 a ( 2 − k ˉ ) ≤ 2 − k ˉ n δ 0 a ( 1 / 2 ) ≤ δ 0 a(1/2)\le\delta_0 a ( 1/2 ) ≤ δ 0
The other partial result needed for the proof of Proposition D.1 is a clean-out lemma.
Lemma D.4. Let δ 0 \delta_0 δ 0 k ˉ \bar{k} k ˉ Corollary D.3 . Let E E E Q Q Q B 2 k ˉ B_{2\bar{k}} B 2 k ˉ
a ( 2 k ˉ ) ≤ δ 0 ω n 2 n ( k ˉ + 1 ) . (93) a(2^{\bar{k}}) \le\frac{\delta_0}{\omega_n 2^{n(\bar{k}+1)}}. \tag*{(93)} a ( 2 k ˉ ) ≤ ω n 2 n ( k ˉ + 1 ) δ 0 . (93) Then, ∣ E ∩ B 1 / 2 ∣ = 0 |E \cap B_{1/2}| = 0 ∣ E ∩ B 1/2 ∣ = 0
Proof. Arguing by contradiction, suppose that ∣ E ∩ B 1 / 2 ∣ > 0 |E \cap B_{1/2}| > 0 ∣ E ∩ B 1/2 ∣ > 0 x 0 ∈ E ∩ B 1 x_0 \in E \cap B_1 x 0 ∈ E ∩ B 1
lim ρ → 0 a ( ρ ) = 1. \lim_{\rho\to0} a(\rho) = 1. ρ → 0 lim a ( ρ ) = 1. On the other hand, by (D.15), we have that also
∣ E ∩ B 2 k ˉ − 1 ( x 0 ) ∣ ∣ B 2 k ˉ − 1 ∣ ≤ ω n 2 n ∣ E ∩ B 2 k ˉ ∣ ≤ 2 − n k ˉ δ 0 . (94) \frac{|E \cap B_{2\bar{k}-1}(x_0)|}{|B_{2\bar{k}-1}|} \le\omega_n 2^n |E \cap B_{2\bar{k}}| \le2^{-n\bar{k}}\delta_0. \tag*{(94)} ∣ B 2 k ˉ − 1 ∣ ∣ E ∩ B 2 k ˉ − 1 ( x 0 ) ∣ ≤ ω n 2 n ∣ E ∩ B 2 k ˉ ∣ ≤ 2 − n k ˉ δ 0 . (94) Without loss of generality, we assume that x 0 = 0 x_0 = 0 x 0 = 0 a ( 2 k ˉ − 1 ) ≤ 2 − n k ˉ δ 0 a(2^{\bar{k}-1}) \le2^{-n\bar{k}}\delta_0 a ( 2 k ˉ − 1 ) ≤ 2 − n k ˉ δ 0 a ( 1 / 4 ) ≤ δ 0 a(1/4) \le\delta_0 a ( 1/4 ) ≤ δ 0 k ∈ N k \in\mathbb{N} k ∈ N
a ( 2 − k − 2 ) ≤ ( C ∑ j = 0 n k ˉ a ( 2 − k + j ) + C Q 2 − k ˉ s ) n n − s ≤ [ C ( 1 + Q ) δ 0 s / n ] n n − s δ 0 ≤ δ 0 . a(2^{-k-2}) \le\left(C \sum_{j=0}^{\frac{n}{\bar{k}}} a(2^{-k+j}) + CQ2^{-\bar{k}s}\right)^{\frac{n}{n-s}} \le\left[C(1+Q)\delta_0^{s/n}\right]^{\frac{n}{n-s}}\delta_0 \le\delta_0. a ( 2 − k − 2 ) ≤ C j = 0 ∑ k ˉ n a ( 2 − k + j ) + C Q 2 − k ˉ s n − s n ≤ [ C ( 1 + Q ) δ 0 s / n ] n − s n δ 0 ≤ δ 0 . Therefore,
1 = lim k → + ∞ a ( 2 − k − 2 ) ≤ δ 0 , 1 = \lim_{k \to+\infty} a(2^{-k-2}) \le\delta_0, 1 = k → + ∞ lim a ( 2 − k − 2 ) ≤ δ 0 , which is a contradiction and concludes the proof. □
Proof of Proposition D.1. Suppose that
∣ E ∩ B ρ ∣ > 0 , (95) |E \cap B_\rho| > 0, \tag*{(95)} ∣ E ∩ B ρ ∣ > 0 , (95) and let x 0 ∈ ( ∂ E ) ∩ B ρ x_0 \in(\partial E) \cap B_\rho x 0 ∈ ( ∂ E ) ∩ B ρ ρ > 0 \rho> 0 ρ > 0 x 0 x_0 x 0
Let us recall that, by definition, if E E E Q Q Q B 1 B_1 B 1 Q ≥ 1 Q \ge1 Q ≥ 1
Per s ( E , B ρ ) ≤ Q Per s ( F , B ρ ) , \operatorname{Per}_s(E,B_\rho) \le Q\operatorname{Per}_s(F,B_\rho), Per s ( E , B ρ ) ≤ Q Per s ( F , B ρ ) , for every F F F F △ E ⊆ B ρ ⋐ B 1 F \mathbin{\triangle} E \subseteq B_\rho\Subset B_1 F △ E ⊆ B ρ ⋐ B 1
Per s ( ρ − 1 E , B 1 ) ≤ Q Per s ( ρ − 1 F , B 1 ) , (96) \operatorname{Per}_s(\rho^{-1}E,B_1) \le Q\operatorname{Per}_s(\rho^{-1}F,B_1), \tag*{(96)} Per s ( ρ − 1 E , B 1 ) ≤ Q Per s ( ρ − 1 F , B 1 ) , (96) Thus, without loss of generality, we assume ρ = 1 \rho= 1 ρ = 1 E E E Q Q Q B 2 k ˉ B_{2\bar{k}} B 2 k ˉ k ˉ \bar{k} k ˉ
∣ E ∩ B 1 ∣ > 0. (97) |E \cap B_1| > 0. \tag*{(97)} ∣ E ∩ B 1 ∣ > 0. (97) We claim that
∣ E ∩ B 1 ∣ ≥ δ 0 , (98) |E \cap B_1| \ge\delta_0, \tag*{(98)} ∣ E ∩ B 1 ∣ ≥ δ 0 , (98) where δ 0 \delta_0 δ 0
Arguing by contradiction, suppose that
∣ E ∩ B 1 ∣ ≤ δ 0 . |E \cap B_1| \le\delta_0. ∣ E ∩ B 1 ∣ ≤ δ 0 . Thus, by mean of Lemma D.4, we infer ∣ E ∩ B 1 ∣ = 0 |E \cap B_1| = 0 ∣ E ∩ B 1 ∣ = 0 ρ \rho ρ
In addition, if E E E E c E^c E c E c E^c E c E E E
Optimality of Lipschitz-regularity of Q Q Q
Now, we exhibit a counterexample to Hölder regularity of Q Q Q ε > 0 \varepsilon>0 ε > 0 E ε E_\varepsilon E ε Q ε Q_\varepsilon Q ε B 1 B_1 B 1 Q ε → 1 Q_\varepsilon\to1 Q ε → 1 ε → 0 \varepsilon\to0 ε → 0
Lemma D.5. Let Ω ⊆ R n \Omega\subseteq\mathbb{R}^n Ω ⊆ R n open bounded domain . Let ε ∈ ( 0 , 1 ) \varepsilon\in(0,1) ε ∈ ( 0 , 1 ) Φ ε : R n → R n \Phi_\varepsilon:\mathbb{R}^n\to\mathbb{R}^n Φ ε : R n → R n bi-Lipschitz transformation such that
1 1 + ε ≤ ∣ det J Φ ε ∣ ≤ 1 + ε . (99) \frac{1}{1+\varepsilon}\leq\lvert\det J\Phi_\varepsilon\rvert\leq1+\varepsilon. \tag*{(99)} 1 + ε 1 ≤ ∣ det J Φ ε ∣ ≤ 1 + ε . (99) Then, for any set F ⊆ R n F\subseteq\mathbb{R}^n F ⊆ R n
Per s ( F , Ω ) ≤ ( 1 + K ε ) Per s ( Φ ε ( F ) , Φ ε ( Ω ) ) , \operatorname{Per}_s(F,\Omega)\leq(1+K\varepsilon)\operatorname{Per}_s(\Phi_\varepsilon(F),\Phi_\varepsilon(\Omega)), Per s ( F , Ω ) ≤ ( 1 + K ε ) Per s ( Φ ε ( F ) , Φ ε ( Ω )) , for some K > 0 K>0 K > 0 n n n s s s
Proof. Recall that
Per s ( F , Ω ) = L s ( F ∩ Ω , F c ∩ Ω ) + L s ( F ∩ Ω c , F c ∩ Ω ) + L s ( F ∩ Ω , F c ∩ Ω c ) . \operatorname{Per}_s(F,\Omega)=\mathcal{L}_s(F\cap\Omega,F^c\cap\Omega)+\mathcal{L}_s(F\cap\Omega^c,F^c\cap\Omega)+\mathcal{L}_s(F\cap\Omega,F^c\cap\Omega^c). Per s ( F , Ω ) = L s ( F ∩ Ω , F c ∩ Ω ) + L s ( F ∩ Ω c , F c ∩ Ω ) + L s ( F ∩ Ω , F c ∩ Ω c ) . Observe that the result will follow if we show that, for any sets A A A B B B
L s ( F ∩ A , F c ∩ B ) ≤ ( 1 + K ε ) L s ( Φ ε ( F ) ∩ Φ ε ( A ) , Φ ε ( F c ) ∩ Φ ε ( B ) ) , (100) \mathcal{L}_s(F\cap A,F^c\cap B)\leq(1+K\varepsilon)\mathcal{L}_s(\Phi_\varepsilon(F)\cap\Phi_\varepsilon(A),\Phi_\varepsilon(F^c)\cap\Phi_\varepsilon(B)), \tag*{(100)} L s ( F ∩ A , F c ∩ B ) ≤ ( 1 + K ε ) L s ( Φ ε ( F ) ∩ Φ ε ( A ) , Φ ε ( F c ) ∩ Φ ε ( B )) , (100) for some K > 0 K>0 K > 0 n n n s s s
To prove this, To prove this, we use (D.21) and the change of variables ( w , z ) : = ( Φ ε ( x ) , Φ ε ( y ) ) (w,z):=(\Phi_\varepsilon(x),\Phi_\varepsilon(y)) ( w , z ) := ( Φ ε ( x ) , Φ ε ( y ))
L s ( F ∩ A , F c ∩ B ) = ∫ F ∩ A ∫ F c ∩ B d x d y ∣ x − y ∣ n + s ≤ ( 1 + ε ) n + s ∫ F ∩ A ∫ F c ∩ B d x d y ∣ Φ ε ( x ) − Φ ε ( y ) ∣ n + s ≤ ( 1 + ε ) n + s + 2 ∫ Φ ε ( F ∩ A ) ∫ Φ ε ( F c ∩ B ) d w d z ∣ w − z ∣ n + s ≤ ( 1 + K ε ) ∫ Φ ε ( F ∩ A ) ∫ Φ ε ( F c ∩ B ) d w d z ∣ w − z ∣ n + s . \begin{aligned}
\mathcal{L}_s(F\cap A,F^c\cap B)
&=\int_{F\cap A}\int_{F^c\cap B}\frac{dx\,dy}{\lvert x-y\rvert^{n+s}}\\
&\leq(1+\varepsilon)^{n+s}\int_{F\cap A}\int_{F^c\cap B}\frac{dx\,dy}{\lvert\Phi_\varepsilon(x)-\Phi_\varepsilon(y)\rvert^{n+s}}\\
&\leq(1+\varepsilon)^{n+s+2}\int_{\Phi_\varepsilon(F\cap A)}\int_{\Phi_\varepsilon(F^c\cap B)}\frac{dw\,dz}{\lvert w-z\rvert^{n+s}}\\
&\leq(1+K\varepsilon)\int_{\Phi_\varepsilon(F\cap A)}\int_{\Phi_\varepsilon(F^c\cap B)}\frac{dw\,dz}{\lvert w-z\rvert^{n+s}}.
\end{aligned} L s ( F ∩ A , F c ∩ B ) = ∫ F ∩ A ∫ F c ∩ B ∣ x − y ∣ n + s d x d y ≤ ( 1 + ε ) n + s ∫ F ∩ A ∫ F c ∩ B ∣ Φ ε ( x ) − Φ ε ( y ) ∣ n + s d x d y ≤ ( 1 + ε ) n + s + 2 ∫ Φ ε ( F ∩ A ) ∫ Φ ε ( F c ∩ B ) ∣ w − z ∣ n + s d w d z ≤ ( 1 + K ε ) ∫ Φ ε ( F ∩ A ) ∫ Φ ε ( F c ∩ B ) ∣ w − z ∣ n + s d w d z . for some K > 0 K>0 K > 0 n n n s s s
Moreover, notice that, for any sets C C C D D D
Φ ε ( C ∩ D ) = Φ ε ( C ) ∩ Φ ε ( D ) . \Phi_\varepsilon(C\cap D)=\Phi_\varepsilon(C)\cap\Phi_\varepsilon(D). Φ ε ( C ∩ D ) = Φ ε ( C ) ∩ Φ ε ( D ) . Therefore, we infer that
L s ( F ∩ A , F c ∩ B ) ≤ ( 1 + K ε ) ∫ Φ ε ( F ) ∩ Φ ε ( A ) ∫ Φ ε ( F c ) ∩ Φ ε ( B ) d w d z ∣ w − z ∣ n + s ≤ ( 1 + K ε ) L s ( Φ ε ( F ) ∩ Φ ε ( A ) , Φ ε ( F c ) ∩ Φ ε ( B ) ) , \begin{aligned}
\mathcal{L}_s(F\cap A,F^c\cap B)
&\leq(1+K\varepsilon)\int_{\Phi_\varepsilon(F)\cap\Phi_\varepsilon(A)}\int_{\Phi_\varepsilon(F^c)\cap\Phi_\varepsilon(B)}\frac{dw\,dz}{\lvert w-z\rvert^{n+s}}\\
&\leq(1+K\varepsilon)\mathcal{L}_s(\Phi_\varepsilon(F)\cap\Phi_\varepsilon(A),\Phi_\varepsilon(F^c)\cap\Phi_\varepsilon(B)),
\end{aligned} L s ( F ∩ A , F c ∩ B ) ≤ ( 1 + K ε ) ∫ Φ ε ( F ) ∩ Φ ε ( A ) ∫ Φ ε ( F c ) ∩ Φ ε ( B ) ∣ w − z ∣ n + s d w d z ≤ ( 1 + K ε ) L s ( Φ ε ( F ) ∩ Φ ε ( A ) , Φ ε ( F c ) ∩ Φ ε ( B )) , establishing (D.22) and concluding the proof. □
As a corollary, we infer that Q Q Q
Example D.6. Let ε > 0 \varepsilon>0 ε > 0 E ε ⊆ R 2 E_\varepsilon\subseteq\mathbb{R}^2 E ε ⊆ R 2 φ ε : R → R \varphi_\varepsilon:\mathbb{R}\to\mathbb{R} φ ε : R → R
φ ε ( x ) : = ∣ x ∣ tan ( ε ) . \varphi_\varepsilon(x):=\lvert x\rvert\tan(\varepsilon). φ ε ( x ) := ∣ x ∣ tan ( ε ) . Moreover, we define the bi-Lipschitz transformation Φ ε : R n → R n \Phi_\varepsilon:\mathbb{R}^n\to\mathbb{R}^n Φ ε : R n → R n
Φ ε ( x 1 , x 2 ) : = ( x 1 x 2 − φ ε ( x 1 ) ) \Phi_\varepsilon(x_1,x_2):=
\begin{pmatrix}
x_1\\
x_2-\varphi_\varepsilon(x_1)
\end{pmatrix} Φ ε ( x 1 , x 2 ) := ( x 1 x 2 − φ ε ( x 1 ) ) Observe that Φ ε ( E ε ) = H : = { x 2 < 0 } \Phi_\varepsilon(E_\varepsilon)=H:=\{x_2<0\} Φ ε ( E ε ) = H := { x 2 < 0 } ∣ det J Φ ε ∣ = 1 \lvert\det J\Phi_\varepsilon\rvert=1 ∣ det J Φ ε ∣ = 1
∣ Φ ε ( x 1 , x 2 ) − Φ ε ( y 1 , y 2 ) ∣ ≤ ( 1 + ε ) ∣ ( x 1 , x 2 ) − ( y 1 , y 2 ) ∣ , \lvert\Phi_\varepsilon(x_1,x_2)-\Phi_\varepsilon(y_1,y_2)\rvert\leq(1+\varepsilon)\lvert(x_1,x_2)-(y_1,y_2)\rvert, ∣ Φ ε ( x 1 , x 2 ) − Φ ε ( y 1 , y 2 )∣ ≤ ( 1 + ε ) ∣( x 1 , x 2 ) − ( y 1 , y 2 )∣ , provided that ε \varepsilon ε
Now, let F ε ⊆ R 2 F_\varepsilon\subseteq\mathbb{R}^2 F ε ⊆ R 2 E ε △ F ε ⊆ B ~ r E_\varepsilon\mathbin{\triangle}F_\varepsilon\subseteq\widetilde{B}_r E ε △ F ε ⊆ B r B ~ r ⋐ B 1 \widetilde{B}_r\Subset B_1 B r ⋐ B 1
Figure 2. The set E ε E_\varepsilon E ε
Then, employing Lemma D.5 and using the minimality of the half-space H H H
Pers s ( E ε , B r ) ≤ ( 1 + K ε ) Pers s ( H , Φ ε ( B r ) ) ≤ ( 1 + K ε ) Pers s ( Φ ε ( F ε ) , Φ ε ( B r ) ) ≤ ( 1 + K ε ) Pers s ( F ε , B r ) , \operatorname{Pers}_{s}(E_\varepsilon,B_r) \le(1+K\varepsilon)\operatorname{Pers}_{s}(H,\Phi_\varepsilon(B_r)) \\
\le(1+K\varepsilon)\operatorname{Pers}_{s}(\Phi_\varepsilon(F_\varepsilon),\Phi_\varepsilon(B_r)) \le(1+K\varepsilon)\operatorname{Pers}_{s}(F_\varepsilon,B_r), Pers s ( E ε , B r ) ≤ ( 1 + K ε ) Pers s ( H , Φ ε ( B r )) ≤ ( 1 + K ε ) Pers s ( Φ ε ( F ε ) , Φ ε ( B r )) ≤ ( 1 + K ε ) Pers s ( F ε , B r ) , for some constant K > 0 K>0 K > 0 s s s E ε E_\varepsilon E ε ( 1 + K ε ) (1+K\varepsilon) ( 1 + K ε ) B 1 B_1 B 1
Acknowledgments
Part of this project was carried out while SD, EV, and RV was visiting Columbia University. For this reason, SD, EV, and RV would like to thank Columbia University for the hospitality. SD and EV are members of the Australian Mathematical Society (AustMS). SD and RV are supported by the Australian Research Council Future Fellowship FT230100333 “New perspectives on nonlocal equations”. OS is supported by NSF Grant DMS-2349794. EV is supported by the Australian Laureate Fellowship FL190100081 “Minimal surfaces, free boundaries and partial differential equations”. RV is supported by a Scholarship for International Research Fees at the University of Western Australia.
References
Serena Dipierro, Department of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Perth, WA 6009, Australia Email address: serena.dipierro@uwa.edu.au
Ovidiu Savin, Department of Mathematics, Columbia University, 2990 Broadway, New York, NY 10027, United States of America Email address: savin@math.columbia.edu
Enrico Valdinoci, Department of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Perth, WA 6009, Australia Email address: enrico.valdinoci@uwa.edu.au
Riccardo Villa, Department of Mathematics and Statistics, The University of Western Australia, 35 Stirling Highway, Perth, WA 6009, Australia Email address: riccardo.villa@research.uwa.edu.au
