Introduction
The narrow escape problem was initially introduced in biology, in the context of the diffusion of molecules in a cell [27 ] , in particular to model the diffusion of neurotransmitters in a synapse. This physical phenomenon can be modeled by a reflected Brownian motion constrained to a domain, with a small absorbing part on its boundary, modeling the exit of the molecule, or its reaction with a receptor. Under the assumption that the absorbing part is small, the goal is to compute the mean exit time of the molecule and, when there are several absorbing parts, the probability of exiting through each of them. This problem has attracted interest both in the physics community [57 , 24 , 49 ] [5] and in the mathematical community [39 , 38 , 30 ] [1, 15]. In this work, we extend the contributions of [38 ] [13, 14] to study different shapes of the absorbing part, within a similar framework: the quasi-stationary approach. This introduction is organized as follows. First, we define the basic setting of the narrow escape problem and the quasi-stationary distribution in § 1.1 ; then we describe the various hole shapes studied in this work and their connection to the literature in § 1.2 ; finally, the contributions and the outline of the paper are given in § 1.3 .
Introduction to the narrow escape problem and the quasi-stationary distribution
In this section, we briefly introduce the mathematical fundamentals of the narrow escape problem and justify the quasi-stationary approach we use in the rest of the paper. Let Ω ⊂ R 3 \Omega\subset\mathbb{R}^{3} Ω ⊂ R 3 ∂ Ω \partial\Omega ∂ Ω ( X t ) t ≥ 0 (X_t)_{t\geq0} ( X t ) t ≥ 0 Ω \Omega Ω ∂ Ω \partial\Omega ∂ Ω Γ D ε ⊂ ∂ Ω \Gamma_D^\varepsilon\subset\partial\Omega Γ D ε ⊂ ∂ Ω ε > 0 \varepsilon> 0 ε > 0 Γ D ε \Gamma_D^\varepsilon Γ D ε
τ ε = inf { t > 0 ∣ X t ∈ Γ D ε ‾ } , (1) \tau_\varepsilon= \inf\{t > 0 \mid X_t \in\overline{\Gamma_D^\varepsilon}\}, \tag*{(1)} τ ε = inf { t > 0 ∣ X t ∈ Γ D ε } , (1) and the exit position
X τ ε ∈ Γ D ε ‾ . (2) X_{\tau_\varepsilon} \in\overline{\Gamma_D^\varepsilon}. \tag*{(2)} X τ ε ∈ Γ D ε . (2) In general, both of these random variables depend on the initial condition X 0 X_0 X 0 ( X t ) t ≥ 0 (X_t)_{t\geq0} ( X t ) t ≥ 0 ν ε \nu_{\varepsilon} ν ε ν ε \nu_{\varepsilon} ν ε Ω \Omega Ω ν ε \nu_{\varepsilon} ν ε X 0 ∈ Ω X_0 \in\Omega X 0 ∈ Ω
Law ( X t ∣ t < τ ε ) ⟶ t → + ∞ ν ε . \operatorname{Law}(X_t \mid t < \tau_{\varepsilon}) \underset{t \to+\infty}{\longrightarrow} \nu_{\varepsilon}. Law ( X t ∣ t < τ ε ) t → + ∞ ⟶ ν ε . Hence, at large times, the law of the process ( X t ) t ≥ 0 (X_t)_{t \ge0} ( X t ) t ≥ 0 Figure 1 for an illustration). This relaxation towards ν ε \nu_{\varepsilon} ν ε ε → 0 \varepsilon\to0 ε → 0 τ ε \tau_{\varepsilon} τ ε ν ε \nu_{\varepsilon} ν ε τ ε \tau_{\varepsilon} τ ε X τ ε X_{\tau_{\varepsilon}} X τ ε ν ε \nu_{\varepsilon} ν ε § 2.3 ). Consequently, the goal of this paper is to study the quasi-stationary distribution ν ε \nu_{\varepsilon} ν ε Γ D ε \Gamma_{\mathrm{D}}^{\varepsilon} Γ D ε τ ε \tau_{\varepsilon} τ ε X τ ε X_{\tau_{\varepsilon}} X τ ε ε → 0 \varepsilon\to0 ε → 0
Figure 1. Histogram of the exit time of 100 000 particles in a unit ball of dimension 3 with 2 absorbing spherical caps of angle 0.2 radians on its boundary. Notice the exponential tail of the distribution, drawn in black, typical of the exit time under the quasi-stationary distribution. The vertical red line represents the time from which one can heuristically consider the particles to be distributed according to the quasi-stationary distribution. In this case, less than 20% of the particles have escaped before this time. This proportion shrinks to 0 in the limit ε → 0 \varepsilon\to0 ε → 0
Hole shapes and bibliography review
Let us now describe the different hole shapes studied in this work, which model various physical phenomena; see the global illustration in Figure 2 .
Figure 2. Illustration of the different hole shapes considered in this work. The gray plane represents the boundary of the domain ∂ Ω \partial\Omega ∂ Ω Γ D ε \Gamma_{\mathrm{D}}^{\varepsilon} Γ D ε ε \varepsilon ε
The narrow escape problem is the standard setting introduced by [27 ] , where the holes are geodesic disks on the boundary of the domain that shrink to a point as ε → 0 \varepsilon\to0 ε → 0 Figure 2 (B). Some notable contributions in the literature on this version of the problem are [64 , 56 , 57 , 45 ] [5, 1, 15] which all focus on the mean exit time or on the exit time from a fixed starting point. The quasi-stationary distribution approach has been used in [14] to study the exit time and exit position in dimension 2. Notice two other contributions [38 ] [13], which use the quasi-stationary distribution but for slightly modified holes that are half-spheres inside the domain; see Figure 2 (A). One goal of this paper is to understand the difference between the two settings.
The narrow slit problem is a variant of the narrow escape problem where one considers a hole that shrinks to a curve of finite length on the boundary of the domain instead of a point. In practice, this is built from a prolate spheroid that shrinks to a curve segment (see Figure 2 (C)). Notice that this problem is fundamentally different, as the hole shrinks to an object of dimension 1. Heuristically, one could expect the mean exit time to diverge logarithmically, as in the two-dimensional narrow escape problem, pointing to the importance of the codimension of the hole instead of the dimension of the domain.
The geometry behind the narrow slit problem is not new and has been studied in a wide range of applications, from electromagnetism [9] and mechanics [10] to hydrodynamics [35 ] and acoustics [43 ] . Yet, to the best of our knowledge, this regime has not been studied in the context of the narrow escape problem. However, using the results on ellipsoidal holes from [58 , 45 ] , we can gain insight into the behavior of the narrow slit problem. Formally, taking the ellipticity of the ellipsoid to scale as ε − 1 \varepsilon^{-1} ε − 1 [45 Theorem 1.3] leads to a logarithmic divergence of the mean exit time, which confirms the heuristic prediction above. Finally, note that this problem can be tackled with capacity methods to obtain the scaling in ε \varepsilon ε
The narrow transition problem is the variant illustrated in purple in Figure 2 (D), where one considers a tube of fixed length with a shrinking cross-section. The exit hole is at the end of the tube. The narrow escape community has considered this problem since [28 ] ; see a list of recent contributions on this topic in [39 , 31 , 30 , 47 ] [6]. Let us also point out the strong connection between this problem and Helmholtz resonators in acoustics/electrostatics [32 , 29 , 33 ] [4].
Another interpretation of this problem arises from statistical physics, in the context of transitions from one state to another (see [65 ] for a discussion of several problems involving Brownian motions trapped in channels, and [22 ] for entropy-driven phenomena). Indeed, if at the end of the tube there is another domain Ω ′ \Omega' Ω ′ Ω \Omega Ω Ω ′ \Omega' Ω ′ [36 , 25 , 62 ] . Usually, the metastability is induced by an energetic barrier between Ω \Omega Ω Ω ′ \Omega' Ω ′ Ω \Omega Ω [37 Remark 1] ). In this work, where Ω \Omega Ω Ω ′ \Omega' Ω ′
The analogy between the narrow transition problem and the energetic barrier framework can be pushed further. The transition time for the latter is given by the Eyring–Kramers law [36 , 25 , 37 ] ,
E [ τ ] ∼ 2 π ∣ λ 1 ( z ) ∣ ∣ det ∇ 2 V ( z ) ∣ det ∇ 2 V ( x ) exp ( V ( z ) − V ( x ) h ) , \mathbb{E}[\tau] \sim\frac{2\pi}{\lvert\lambda_1(z)\rvert}\sqrt{\frac{\lvert\det\nabla^2 V(z)\rvert}{\det\nabla^2 V(x)}}\exp\left(\frac{V(z)-V(x)}{h}\right), E [ τ ] ∼ ∣ λ 1 ( z )∣ 2 π det ∇ 2 V ( x ) ∣ det ∇ 2 V ( z )∣ exp ( h V ( z ) − V ( x ) ) , where h h h V V V λ 1 \lambda_1 λ 1 V V V z z z x x x V V V ∣ Ω ∣ ε 2 L \frac{\lvert\Omega\rvert}{\varepsilon^2 L} ε 2 L ∣ Ω ∣ L L L ε 2 \varepsilon^2 ε 2 ∣ λ 1 ∣ − 1 \lvert\lambda_1\rvert^{-1} ∣ λ 1 ∣ − 1 L 2 L^2 L 2
E [ τ ] ∼ L ∣ Ω ∣ ε 2 . \mathbb{E}[\tau] \sim L\frac{\lvert\Omega\rvert}{\varepsilon^2}. E [ τ ] ∼ L ε 2 ∣ Ω ∣ . In § 1.3 , we see that these heuristics lead to the correct scaling behavior for the mean exit time in the narrow transition problem, making a link between metastability arising from an energetic constraint and from an entropic constraint.
Contribution and outline of the paper
The contribution of this work is to derive the exit time distribution and the exit hole distribution for the narrow escape problem (in two variants, A and B), the slit problem (C), and the transition problem (D), within a single quasi-stationary distribution framework. Let us emphasize that the novelty of this work is to demonstrate that the framework introduced in [38 ] can be applied to a wide range of hole shapes, and to derive asymptotically the exit time distribution and the exit hole distribution from the quasi-stationary distribution, in the limit ε → 0 \varepsilon\to0 ε → 0 § 2.3 . The results of this paper are gathered in Table 1 , which summarizes the different hole shapes and the asymptotics of their mean exit time and the exit hole distribution in a reference setting (one hole for the mean exit time, two holes for the exit hole) in the limit ε → 0 \varepsilon\to0 ε → 0
Table 1. Summary of the different hole shapes considered in this work and their properties.
For the sake of comparison, we also include a column stating how the surface area of each hole scales with ε \varepsilon ε Table 1 are consistent with the heuristics on the mean exit time developed in § 1.2 . Furthermore, we make explicit the difference between the spherical holes on the boundary studied in [14] and the spherical holes studied in [38 ] [13], which are half-spheres inside the domain. The scaling of the mean exit time differs by a factor π / 2 \pi/2 π /2
The notation A, B, C and D refers to the illustrations in Figure 2 . See (Type A), (Type B), (Type C) and (Type D) in § 2.2 for the precise definitions of the holes. The mean exit time is E ν ε [ τ ε ] \mathbb{E}_{\nu_\varepsilon}[\tau_\varepsilon] E ν ε [ τ ε ] ν ε \nu_\varepsilon ν ε ε \varepsilon ε 2 ε 2\varepsilon 2 ε ℓ \ell ℓ L L L ε → 0 \varepsilon\to0 ε → 0
The results presented in Table 1 are believed to be derivable in higher dimensions, at the cost of lengthy computations. This is one goal of the numerical experiments in § 6 , which aim to illustrate the results of this work and to explore the higher-dimensional case.
The paper is organized as follows. In § 2 , we introduce the main notation and explain the quasi-stationary distribution framework. Then, in § 3 , we present our results on the spherical holes A and B. In § 4 , we construct a quasi-mode for the narrow slit problem and derive the asymptotics of the exit time and exit hole distributions. The same is done for the narrow transition problem in § 5 . Finally, we present numerical experiments in § 6 to illustrate the results of this work and explore the higher-dimensional case. Additionally, in Section A, we discuss the Poincaré–Wirtinger inequality uniform in ε \varepsilon ε Theorem 2.2 in Section B. Section C gathers the fundamental properties of the Green function considered in this work, and Section D is a technical section on the computation of several integrals.
Notation and detailed method
In this section we introduce the mathematical notation, first in general in § 2.1 , then for the different hole shapes in § 2.2 . Finally, in § 2.3 , we explain how to use the quasi-stationary distribution to derive the exit time and exit hole distributions, using a quasi-mode.
Notation
2.1. Notation. In this section, we define the notation and the mathematical concepts used throughout this work. We first introduce geometric considerations, before introducing the operator associated with the process ( X t ) t ≥ 0 (X_t)_{t\geq0} ( X t ) t ≥ 0 (3) . Note that in this paper, four geometric settings are considered. For simplicity, we use the same notation in all four settings, with the understanding that the underlying geometric setting is clear from the context.
We recall that Ω \Omega Ω R 3 \mathbb{R}^3 R 3 ∂ Ω \partial\Omega ∂ Ω ( X t ) t ≥ 0 (X_t)_{t\geq0} ( X t ) t ≥ 0 Ω ‾ \overline{\Omega} Ω
d X t = 2 d B t − n ( X t ) d L t , (3) \mathrm{d}X_t = \sqrt{2}\,\mathrm{d}B_t - n(X_t)\,\mathrm{d}L_t, \tag*{(3)} d X t = 2 d B t − n ( X t ) d L t , (3) where ( B t ) t ≥ 0 (B_t)_{t\geq0} ( B t ) t ≥ 0 R 3 \mathbb{R}^3 R 3 ( L t ) t ≥ 0 (L_t)_{t\geq0} ( L t ) t ≥ 0 ( X t ) t ≥ 0 (X_t)_{t\geq0} ( X t ) t ≥ 0 ∂ Ω \partial\Omega ∂ Ω [60 , 63 ] ), and n n n ∂ Ω \partial\Omega ∂ Ω Ω \Omega Ω
For N ≥ 1 N\geq1 N ≥ 1 ε > 0 \varepsilon>0 ε > 0 N N N Γ ε k ⊂ Ω ‾ \Gamma_\varepsilon^k\subset\overline{\Omega} Γ ε k ⊂ Ω k ∈ { 1 , … , N } k\in\{1,\ldots,N\} k ∈ { 1 , … , N } r ε ( k ) > 0 r_\varepsilon^{(k)}>0 r ε ( k ) > 0 w ε ( k ) w_\varepsilon^{(k)} w ε ( k ) Γ D ε = ⋃ k = 1 N Γ ε k \Gamma_\mathrm{D}^\varepsilon=\bigcup_{k=1}^N\Gamma_\varepsilon^k Γ D ε = ⋃ k = 1 N Γ ε k ε → 0 \varepsilon\to0 ε → 0 r ε ( k ) → 0 r_\varepsilon^{(k)}\to0 r ε ( k ) → 0 ∣ Γ ε k ∣ → 0 |\Gamma_\varepsilon^k|\to0 ∣ Γ ε k ∣ → 0 ε \varepsilon ε
The generator of ( X t ) t ≥ 0 (X_t)_{t\geq0} ( X t ) t ≥ 0 The generator of the process ( X t ) t ≥ 0 (X_t)_{t\geq0} ( X t ) t ≥ 0 (3) , killed on Γ D ε \Gamma_\mathrm{D}^\varepsilon Γ D ε Δ \Delta Δ Ω ε \Omega_\varepsilon Ω ε Γ D ε \Gamma_\mathrm{D}^\varepsilon Γ D ε Γ N ε \Gamma_\mathrm{N}^\varepsilon Γ N ε L ε = − Δ \mathcal{L}_\varepsilon=-\Delta L ε = − Δ
D ( L ε ) = { u ∈ H 1 ( Ω ε ) | Δ u ∈ L 2 ( Ω ε ) , u = 0 on Γ D ε , ∂ n u = 0 on Γ N ε } . (4) \mathcal{D}(\mathcal{L}_\varepsilon)=\left\{u\in H^1(\Omega_\varepsilon)\mathrel{}\middle|\mathrel{} \Delta u\in L^2(\Omega_\varepsilon),\ u=0\ \text{on }\Gamma_\mathrm{D}^\varepsilon,\ \partial_nu=0\ \text{on }\Gamma_\mathrm{N}^\varepsilon\right\}. \tag*{(4)} D ( L ε ) = { u ∈ H 1 ( Ω ε ) Δ u ∈ L 2 ( Ω ε ) , u = 0 on Γ D ε , ∂ n u = 0 on Γ N ε } . (4) where the normal derivative ∂ n u \partial_nu ∂ n u H − 1 / 2 ( ∂ Ω ε ) H^{-1/2}(\partial\Omega_\varepsilon) H − 1/2 ( ∂ Ω ε ) L ε \mathcal{L}_\varepsilon L ε
q ε ( u , v ) = ∫ Ω ε ∇ u ⋅ ∇ v , u , v ∈ H Γ D ε 1 ( Ω ε ) , (5) q_\varepsilon(u,v)=\int_{\Omega_\varepsilon}\nabla u\cdot\nabla v,\qquad u,v\in H^1_{\Gamma_\mathrm{D}^\varepsilon}(\Omega_\varepsilon), \tag*{(5)} q ε ( u , v ) = ∫ Ω ε ∇ u ⋅ ∇ v , u , v ∈ H Γ D ε 1 ( Ω ε ) , (5) with form domain
H Γ D ε 1 ( Ω ε ) = { u ∈ H 1 ( Ω ε ) | u = 0 on Γ D ε } . H^1_{\Gamma_\mathrm{D}^\varepsilon}(\Omega_\varepsilon)=\left\{u\in H^1(\Omega_\varepsilon)\mathrel{}\middle|\mathrel{}u=0\ \text{on }\Gamma_\mathrm{D}^\varepsilon\right\}. H Γ D ε 1 ( Ω ε ) = { u ∈ H 1 ( Ω ε ) u = 0 on Γ D ε } . It is classical that L ε \mathcal{L}_\varepsilon L ε [52 Theorem 1] and the Rellich–Kondrachov theorem), so it has a discrete spectrum. Furthermore, its principal eigenvalue is positive and associated with a positive and continuous eigenfunction [23 Theorem 8.38] . The density of the quasi-stationary distribution ν ε \nu_\varepsilon ν ε § 1.1 , is this principal eigenfunction [13, Theorem 1.1]. Explicitly, denoting this density by ν ε \nu_\varepsilon ν ε
{ − Δ ν ε = λ ε ν ε in Ω ε , ∂ n ν ε = 0 on Γ N ε , ν ε = 0 on Γ D ε , ∫ Ω ε ν ε = 1. (6) \begin{cases}
-\Delta\nu_\varepsilon=\lambda_\varepsilon\nu_\varepsilon& \text{in }\Omega_\varepsilon,\\
\partial_n\nu_\varepsilon=0 & \text{on }\Gamma_\mathrm{N}^\varepsilon,\\
\nu_\varepsilon=0 & \text{on }\Gamma_\mathrm{D}^\varepsilon,\\
\int_{\Omega_\varepsilon}\nu_\varepsilon=1. \tag*{(6)}
\end{cases} ⎩ ⎨ ⎧ − Δ ν ε = λ ε ν ε ∂ n ν ε = 0 ν ε = 0 ∫ Ω ε ν ε = 1. in Ω ε , on Γ N ε , on Γ D ε , (6) Notation for the different hole shapes
With this notation, the hole shapes considered in this work (see Figure 2 and § 1.2 ) are defined as follows. For each of the constructions considered in (Type A) and (Type C), removing the holes from (or, in (Type D), attaching the tubes to) Ω \Omega Ω Ω ε \Omega_\varepsilon Ω ε
∂ Ω ε = Γ D ε ∪ Γ N ε \partial\Omega_\varepsilon= \Gamma^\varepsilon_D \cup\Gamma^\varepsilon_N ∂ Ω ε = Γ D ε ∪ Γ N ε where Γ N ε \Gamma^\varepsilon_N Γ N ε ∂ Ω \partial\Omega ∂ Ω k ∈ { 1 , … , N } k \in\{1,\ldots,N\} k ∈ { 1 , … , N } ε > 0 \varepsilon> 0 ε > 0 r ε ( k ) > 0 r_\varepsilon^{(k)} > 0 r ε ( k ) > 0
Spherical holes. The spherical holes are defined as
(Type A)
Γ k ε = ∂ B ( x ( k ) , r ε ( k ) ) ∩ Ω , \Gamma^\varepsilon_k = \partial B\left(x^{(k)},r_\varepsilon^{(k)}\right) \cap\Omega, Γ k ε = ∂ B ( x ( k ) , r ε ( k ) ) ∩ Ω , where x ( k ) x^{(k)} x ( k ) ∂ Ω \partial\Omega ∂ Ω x ( k ) x^{(k)} x ( k ) η > 0 \eta> 0 η > 0 B ( x ( k ) , η ) B(x^{(k)},\eta) B ( x ( k ) , η ) d ∂ Ω ( ⋅ , ⋅ ) d_{\partial\Omega}(\cdot,\cdot) d ∂ Ω ( ⋅ , ⋅ ) ∂ Ω \partial\Omega ∂ Ω
(Type B)
Γ k ε = { x ∈ ∂ Ω ∣ d ∂ Ω ( x , x ( k ) ) < r ε ( k ) } . \Gamma^\varepsilon_k = \{x \in\partial\Omega\mid d_{\partial\Omega}(x,x^{(k)}) < r_\varepsilon^{(k)}\}. Γ k ε = { x ∈ ∂ Ω ∣ d ∂ Ω ( x , x ( k ) ) < r ε ( k ) } . In this work, we choose the geodesic distance in (Type B) for the sake of simplicity. Using the underlying Euclidean distance instead is possible with the same approach, and changes only the error terms in the results.
Slit holes. For the slit problem, let H k ⊂ ∂ Ω \mathcal{H}_k \subset\partial\Omega H k ⊂ ∂ Ω ( H j ) j = 1 N (\mathcal{H}_j)_{j=1}^{N} ( H j ) j = 1 N H k \mathcal{H}_k H k γ k : [ − ℓ k , ℓ k ] → H k \gamma_k : [-\ell_k,\ell_k] \to\mathcal{H}_k γ k : [ − ℓ k , ℓ k ] → H k ∣ γ k ′ ∣ = 1 |\gamma'_k| = 1 ∣ γ k ′ ∣ = 1 H k \mathcal{H}_k H k 2 ℓ k 2\ell_k 2 ℓ k
To make the shape of the holes explicit, a local basis is needed to describe the geometry of the slit: the Frenet coordinates. For x x x H k \mathcal{H}_k H k x x x H k \mathcal{H}_k H k π k ( x ) ∈ H k \pi_k(x) \in\mathcal{H}_k π k ( x ) ∈ H k s x ∈ [ − ℓ k , ℓ k ] s_x \in[-\ell_k,\ell_k] s x ∈ [ − ℓ k , ℓ k ] γ k ( s x ) = π k ( x ) \gamma_k(s_x) = \pi_k(x) γ k ( s x ) = π k ( x ) π k ( x ) \pi_k(x) π k ( x ) x = γ k ( ℓ k ) + t τ k ( ℓ k ) x = \gamma_k(\ell_k) + t\tau_k(\ell_k) x = γ k ( ℓ k ) + t τ k ( ℓ k ) t > 0 t > 0 t > 0 τ k ( ⋅ ) \tau_k(\cdot) τ k ( ⋅ ) H k \mathcal{H}_k H k γ k ( ⋅ ) \gamma_k(\cdot) γ k ( ⋅ ) x x x H k \mathcal{H}_k H k
x = γ k ( s x ) + t x ( k ) τ k ( s x ) + ρ x ( k ) , (7) x = \gamma_k(s_x) + t_x^{(k)}\tau_k(s_x) + \rho_x^{(k)}, \tag*{(7)} x = γ k ( s x ) + t x ( k ) τ k ( s x ) + ρ x ( k ) , (7) with t x ( k ) ∈ R t_x^{(k)} \in\mathbb{R} t x ( k ) ∈ R ρ x ( k ) ⊥ τ k ( s x ) \rho_x^{(k)} \perp\tau_k(s_x) ρ x ( k ) ⊥ τ k ( s x ) ε \varepsilon ε
(Type C)
Γ k ε = { x ∈ Ω | ( s x + t x ( k ) ) 2 ( w ε ( k ) ) 2 + ℓ k 2 + ∣ ρ x ( k ) ∣ 2 ( w ε ( k ) ) 2 = 1 } . \Gamma^\varepsilon_k = \left\{x \in\Omega\,\middle|\, \frac{(s_x+t_x^{(k)})^2}{(w_\varepsilon^{(k)})^2+\ell_k^2}+\frac{|\rho_x^{(k)}|^2}{(w_\varepsilon^{(k)})^2}=1\right\}. Γ k ε = { x ∈ Ω ( w ε ( k ) ) 2 + ℓ k 2 ( s x + t x ( k ) ) 2 + ( w ε ( k ) ) 2 ∣ ρ x ( k ) ∣ 2 = 1 } . By design, the sets Γ k ε \Gamma^\varepsilon_k Γ k ε H k \mathcal{H}_k H k (7) that shrink to H k \mathcal{H}_k H k ε → 0 \varepsilon\to0 ε → 0 w ε ( k ) w_\varepsilon^{(k)} w ε ( k ) Γ k ε ⊂ Ω ‾ \Gamma^\varepsilon_k \subset\overline{\Omega} Γ k ε ⊂ Ω
Finally, we define the major semi-axis, which is useful in the analysis of the slit problem, as
a ε ( k ) = ℓ k 2 + ( w ε ( k ) ) 2 , so that ( a ε ( k ) − ℓ k ) ( a ε ( k ) + ℓ k ) = ( w ε ( k ) ) 2 . (8) a_\varepsilon^{(k)} = \sqrt{\ell_k^2+\left(w_\varepsilon^{(k)}\right)^2},
\qquad\text{so that}\qquad
\left(a_\varepsilon^{(k)}-\ell_k\right)\left(a_\varepsilon^{(k)}+\ell_k\right)=\left(w_\varepsilon^{(k)}\right)^2. \tag*{(8)} a ε ( k ) = ℓ k 2 + ( w ε ( k ) ) 2 , so that ( a ε ( k ) − ℓ k ) ( a ε ( k ) + ℓ k ) = ( w ε ( k ) ) 2 . (8) Tubular holes. To construct the tubular holes, pick a point x ( k ) ∈ ∂ Ω x^{(k)} \in\partial\Omega x ( k ) ∈ ∂ Ω D k ⊂ T x ( k ) ( ∂ Ω ) \mathcal{D}_k \subset T_{x^{(k)}}(\partial\Omega) D k ⊂ T x ( k ) ( ∂ Ω ) T x ( k ) ( ∂ Ω ) T_{x^{(k)}}(\partial\Omega) T x ( k ) ( ∂ Ω ) ∂ Ω \partial\Omega ∂ Ω x ( k ) x^{(k)} x ( k )
D k , ε = r ε ( k ) D k , so that ∣ D k , ε ∣ = ( r ε ( k ) ) 2 ∣ D k ∣ . (9) \mathcal{D}_{k,\varepsilon}=r_\varepsilon^{(k)}\mathcal{D}_k,
\qquad\text{so that}\qquad
|\mathcal{D}_{k,\varepsilon}|=\left(r_\varepsilon^{(k)}\right)^2|\mathcal{D}_k|. \tag*{(9)} D k , ε = r ε ( k ) D k , so that ∣ D k , ε ∣ = ( r ε ( k ) ) 2 ∣ D k ∣. (9) where r ε ( k ) r_\varepsilon^{(k)} r ε ( k ) k k k r ε = ∑ j = 1 N r ε ( j ) r_\varepsilon=\sum_{j=1}^{N}r_\varepsilon^{(j)} r ε = ∑ j = 1 N r ε ( j )
Denote by n k n_k n k ∂ Ω \partial\Omega ∂ Ω x ( k ) x^{(k)} x ( k ) x ( k ) x^{(k)} x ( k ) x = x ( k ) + ρ + s n k x=x^{(k)}+\rho+sn_k x = x ( k ) + ρ + s n k ρ ∈ T x ( k ) ( ∂ Ω ) \rho\in T_{x^{(k)}}(\partial\Omega) ρ ∈ T x ( k ) ( ∂ Ω ) s ∈ R s\in\mathbb{R} s ∈ R ∂ Ω \partial\Omega ∂ Ω r 0 > 0 r_0>0 r 0 > 0 h k : B ( 0 , r 0 ) → R h_k:B(0,r_0)\to\mathbb{R} h k : B ( 0 , r 0 ) → R h k ( 0 ) = 0 h_k(0)=0 h k ( 0 ) = 0 ∇ h k ( 0 ) = 0 \nabla h_k(0)=0 ∇ h k ( 0 ) = 0 ∂ Ω \partial\Omega ∂ Ω s = h k ( ρ ) s=h_k(\rho) s = h k ( ρ ) Ω \Omega Ω { s < h k ( ρ ) } \{s<h_k(\rho)\} { s < h k ( ρ )} ∣ ρ ∣ ≤ C r ε ( k ) |\rho|\leq Cr_{\varepsilon}^{(k)} ∣ ρ ∣ ≤ C r ε ( k )
h k = O ( ( r ε ( k ) ) 2 ) , ∇ h k = O ( r ε ( k ) ) . (10) h_k=O\left((r_{\varepsilon}^{(k)})^2\right),\qquad\nabla h_k=O\left(r_{\varepsilon}^{(k)}\right). \tag*{(10)} h k = O ( ( r ε ( k ) ) 2 ) , ∇ h k = O ( r ε ( k ) ) . (10) For ε \varepsilon ε D ‾ k , ε ⊂ B ( 0 , r 0 ) \overline{D}_{k,\varepsilon}\subset B(0,r_0) D k , ε ⊂ B ( 0 , r 0 )
T k , ε = { x ( k ) + ρ + s n k : ρ ∈ D k , ε , h k ( ρ ) < s < L k } , (11) \mathcal{T}_{k,\varepsilon}=\left\{x^{(k)}+\rho+sn_k:\rho\in D_{k,\varepsilon},\ h_k(\rho)<s<L_k\right\}, \tag*{(11)} T k , ε = { x ( k ) + ρ + s n k : ρ ∈ D k , ε , h k ( ρ ) < s < L k } , (11) with L k > 0 L_k>0 L k > 0
(Type D) Γ k ε = { x ( k ) + ρ + L k n k : ρ ∈ D k , ε } \Gamma_k^\varepsilon=\left\{x^{(k)}+\rho+L_kn_k:\rho\in D_{k,\varepsilon}\right\} Γ k ε = { x ( k ) + ρ + L k n k : ρ ∈ D k , ε }
We shall always assume that the tubes never intersect each other or the domain itself.
We denote Σ k , ε = Ω ‾ ∩ T ‾ k , ε \Sigma_{k,\varepsilon}=\overline{\Omega}\cap\overline{\mathcal{T}}_{k,\varepsilon} Σ k , ε = Ω ∩ T k , ε k ∈ { 1 , … , N } k\in\{1,\ldots,N\} k ∈ { 1 , … , N } k k k ε \varepsilon ε
Σ k , ε = { x ( k ) + ρ + h k ( ρ ) n k ∣ ρ ∈ D k , ε } , (12) \Sigma_{k,\varepsilon}=\left\{x^{(k)}+\rho+h_k(\rho)n_k\mid\rho\in D_{k,\varepsilon}\right\}, \tag*{(12)} Σ k , ε = { x ( k ) + ρ + h k ( ρ ) n k ∣ ρ ∈ D k , ε } , (12) where h k h_k h k (10) . To be explicit in this case, the resulting domain Ω ε \Omega_\varepsilon Ω ε T k , ε \mathcal{T}_{k,\varepsilon} T k , ε Ω \Omega Ω
Ω ε = Ω ∪ ⋃ k = 1 N ( T k , ε ∪ Σ k , ε ) ∖ ∂ Σ k , ε . (13) \Omega_\varepsilon=\Omega\cup\bigcup_{k=1}^{N}\left(\mathcal{T}_{k,\varepsilon}\cup\Sigma_{k,\varepsilon}\right)\setminus\partial\Sigma_{k,\varepsilon}. \tag*{(13)} Ω ε = Ω ∪ k = 1 ⋃ N ( T k , ε ∪ Σ k , ε ) ∖ ∂ Σ k , ε . (13) This problem is fundamentally different from the others, as Γ k ε ⊄ Ω ‾ \Gamma_k^\varepsilon\not\subset\overline{\Omega} Γ k ε ⊂ Ω ε \varepsilon ε [48 Theorem 1.2] do not apply directly in order to obtain the Poincaré–Wirtinger inequality uniformly in ε \varepsilon ε Lemma A.1 using a different approach.
Finally, we introduce, for all k ∈ { 1 , … , N } k\in\{1,\ldots,N\} k ∈ { 1 , … , N }
s ε ( k ) = sup ρ ∈ D k , ε ∣ h k ( ρ ) ∣ , (14) s_\varepsilon^{(k)}=\sup_{\rho\in D_{k,\varepsilon}}|h_k(\rho)|, \tag*{(14)} s ε ( k ) = ρ ∈ D k , ε sup ∣ h k ( ρ ) ∣ , (14) so that the length of the tube is at most L k + s ε ( k ) L_k+s_\varepsilon^{(k)} L k + s ε ( k ) ε \varepsilon ε Figure 3 .
Figure 3. Illustration of the different notations used for the holes (Type C) and (Type D).
From the quasi-stationary distribution to the narrow escape problem
In this section, we state Theorem 2.2 that we will apply in different settings throughout the paper, which will give us the results claimed in Table 1 .
As introduced in § 1.1 , the quasi-stationary distribution ν ε \nu_\varepsilon ν ε X 0 ∼ ν ε X_0 \sim\nu_\varepsilon X 0 ∼ ν ε τ ε \tau_\varepsilon τ ε X τ ε X_{\tau_\varepsilon} X τ ε
τ ε ∼ Exp ( λ ε ) , and ∀ k ∈ { 1 , … , N } , P ν ε ( X τ ε ∈ Γ ‾ k ε ) = − λ ε − 1 ∫ Γ k ε ∂ n ν ε d σ . (15) \tau_\varepsilon\sim\operatorname{Exp}(\lambda_\varepsilon), \qquad\text{and} \qquad\forall k \in\{1,\ldots,N\}, \quad\mathbb{P}_{\nu_\varepsilon}\left(X_{\tau_\varepsilon} \in\overline{\Gamma}_k^\varepsilon\right)=-\lambda_\varepsilon^{-1}\int_{\Gamma_k^\varepsilon}\partial_n\nu_\varepsilon\,\mathrm{d}\sigma. \tag*{(15)} τ ε ∼ Exp ( λ ε ) , and ∀ k ∈ { 1 , … , N } , P ν ε ( X τ ε ∈ Γ k ε ) = − λ ε − 1 ∫ Γ k ε ∂ n ν ε d σ . (15) However, in a general setting, it is impossible to obtain an explicit form for ν ε \nu_\varepsilon ν ε φ ε \varphi_\varepsilon φ ε (6) and leverage the results from it. This is the technique applied in [13]. In this work, the quasi-mode is built as a sum of contributions of each hole, in the form
φ ε = 1 − ∑ k = 1 N ψ ε ( k ) . (16) \varphi_\varepsilon=1-\sum_{k=1}^{N}\psi_\varepsilon^{(k)}. \tag*{(16)} φ ε = 1 − k = 1 ∑ N ψ ε ( k ) . (16) where ψ ε ( k ) \psi_\varepsilon^{(k)} ψ ε ( k ) 1 1 1 k k k
Hypothesis 2.1. The principal eigenvalue λ ε \lambda_\varepsilon λ ε L ε \mathcal{L}_\varepsilon L ε 0 0 0 ε → 0 \varepsilon\to0 ε → 0
(Eig)
λ ε → ε → 0 0. \lambda_\varepsilon\xrightarrow[\varepsilon\to0]{} 0. λ ε ε → 0 0. Furthermore, for each k ∈ { 1 , … , N } k\in\{1,\ldots,N\} k ∈ { 1 , … , N } K ε ( k ) K_\varepsilon^{(k)} K ε ( k ) K ε ( k ) → 0 K_\varepsilon^{(k)}\to0 K ε ( k ) → 0 ε → 0 \varepsilon\to0 ε → 0 ψ ε ( k ) ∈ H 1 ( Ω ε ) ∩ C ( Γ ‾ D ) \psi_\varepsilon^{(k)}\in H^1(\Omega_\varepsilon)\cap C(\overline{\Gamma}_D) ψ ε ( k ) ∈ H 1 ( Ω ε ) ∩ C ( Γ D )
(PDE)
{ Δ ψ ε ( k ) = C K ε ( k ) in Ω ε , ∂ n ψ ε ( k ) = 0 on Γ N ε , ∥ ψ ε ( k ) − 1 Γ ‾ k ε ∥ L ∞ ( Γ D ε ) = O ( E 1 ( ( K ε ( j ) ) j = 1 N ) ) , \begin{cases}
\Delta\psi_\varepsilon^{(k)}=CK_\varepsilon^{(k)} & \text{in }\Omega_\varepsilon,\\
\partial_n\psi_\varepsilon^{(k)}=0 & \text{on }\Gamma_N^\varepsilon,\\
\left\|\psi_\varepsilon^{(k)}-1_{\overline{\Gamma}_k^\varepsilon}\right\|_{L^\infty(\Gamma_D^\varepsilon)}
=O\left(\mathcal{E}_1\left((K_\varepsilon^{(j)})_{j=1}^{N}\right)\right),
\end{cases} ⎩ ⎨ ⎧ Δ ψ ε ( k ) = C K ε ( k ) ∂ n ψ ε ( k ) = 0 ψ ε ( k ) − 1 Γ k ε L ∞ ( Γ D ε ) = O ( E 1 ( ( K ε ( j ) ) j = 1 N ) ) , in Ω ε , on Γ N ε , with C > 0 C>0 C > 0 ε \varepsilon ε
K ‾ ε = ∑ k = 1 N K ε ( k ) \overline{K}_\varepsilon=\sum_{k=1}^{N}K_\varepsilon^{(k)} K ε = k = 1 ∑ N K ε ( k ) and E 1 \mathcal{E}_1 E 1 ( K ε ( j ) ) j = 1 N (K_\varepsilon^{(j)})_{j=1}^{N} ( K ε ( j ) ) j = 1 N E 1 ( ( K ε ( j ) ) j = 1 N ) → 0 \mathcal{E}_1((K_\varepsilon^{(j)})_{j=1}^{N})\to0 E 1 (( K ε ( j ) ) j = 1 N ) → 0 ε → 0 \varepsilon\to0 ε → 0
Furthermore, for all k ∈ { 1 , … , N } k\in\{1,\ldots,N\} k ∈ { 1 , … , N } ψ ε ( k ) \psi_\varepsilon^{(k)} ψ ε ( k )
(Norm)
∥ ψ ε ( k ) ∥ L 2 ( Ω ε ) = O ( E 2 ( ( K ε ( j ) ) j = 1 N ) ) , \left\|\psi_\varepsilon^{(k)}\right\|_{L^2(\Omega_\varepsilon)}
=O\left(\mathcal{E}_2\left((K_\varepsilon^{(j)})_{j=1}^{N}\right)\right), ψ ε ( k ) L 2 ( Ω ε ) = O ( E 2 ( ( K ε ( j ) ) j = 1 N ) ) , with E 2 \mathcal{E}_2 E 2 ( K ε ( j ) ) j = 1 N (K_\varepsilon^{(j)})_{j=1}^{N} ( K ε ( j ) ) j = 1 N E 2 ( ( K ε ( j ) ) j = 1 N ) → 0 \mathcal{E}_2((K_\varepsilon^{(j)})_{j=1}^{N})\to0 E 2 (( K ε ( j ) ) j = 1 N ) → 0 ε → 0 \varepsilon\to0 ε → 0
The two functions E 1 \mathcal{E}_1 E 1 E 2 \mathcal{E}_2 E 2 Theorem 2.2 . They depend on the whole family ( K ε ( j ) ) j = 1 N (K_\varepsilon^{(j)})_{j=1}^{N} ( K ε ( j ) ) j = 1 N K ‾ ε \overline{K}_\varepsilon K ε § 5 ). The Dirichlet condition of (PDE) states that ψ ε ( k ) \psi_\varepsilon^{(k)} ψ ε ( k ) 1 1 1 Γ k ε \Gamma_k^\varepsilon Γ k ε
Summing (PDE) and (Norm) over k k k ∑ k = 1 N 1 Γ ‾ k ε = 1 \sum_{k=1}^{N}1_{\overline{\Gamma}_k^\varepsilon}=1 ∑ k = 1 N 1 Γ k ε = 1 Γ ‾ D \overline{\Gamma}_D Γ D φ ε \varphi_\varepsilon φ ε (16) satisfies
\begin{cases}
-\Delta\varphi_\varepsilon=C\overline{K}_\varepsilon& \text{in }\Omega_\varepsilon,\\
\partial_n\varphi_\varepsilon=0 & \text{on }\Gamma_N^\varepsilon,\\
\left\|\varphi_\varepsilon\right\|_{L^\infty(\Gamma_D^\varepsilon)}
=O\left(\mathcal{E}_1\left((K_\varepsilon^{(j)})_{j=1}^{N}\right)\right),
\end{cases}
\qquad\text{and}\qquad
\left\|\varphi_\varepsilon-1\right\|_{L^2(\Omega_\varepsilon)}
=O\left(\mathcal{E}_2\left((K_\varepsilon^{(j)})_{j=1}^{N}\right)\right). \tag*{(17)}\end{cases}
Qualitatively, these hypotheses and this method in general show that φ ε ≃ ∣ Ω ε ∣ ν ε \varphi_\varepsilon\simeq|\Omega_\varepsilon|\nu_\varepsilon φ ε ≃ ∣ Ω ε ∣ ν ε (6) and (17) . From the form of the quasi-mode φ ε \varphi_\varepsilon φ ε (16) , one expects that the quasi-stationary distribution ν ε \nu_\varepsilon ν ε Ω ε \Omega_\varepsilon Ω ε ε \varepsilon ε
We can now state the key result of this work.
Theorem 2.2. Consider ( X t ) t ≥ 0 (X_t)_{t\geq0} ( X t ) t ≥ 0 (3) . If Hypothesis 2.1 holds, then the exit time τ ε \tau_\varepsilon τ ε ν ε \nu_\varepsilon ν ε λ ε \lambda_\varepsilon λ ε
E ν ε [ τ ε ] = 1 λ ε = 1 C K ε ( 1 + O ( E 1 ( ( K ε ( j ) ) j = 1 N ) + E 2 ( ( K ε ( j ) ) j = 1 N ) ) ) . \mathbb{E}_{\nu_\varepsilon}[\tau_\varepsilon]=\frac{1}{\lambda_\varepsilon}=\frac{1}{C K_\varepsilon}\left(1+O\left(\mathcal{E}_1\left((K_\varepsilon^{(j)})_{j=1}^{N}\right)+\mathcal{E}_2\left((K_\varepsilon^{(j)})_{j=1}^{N}\right)\right)\right). E ν ε [ τ ε ] = λ ε 1 = C K ε 1 ( 1 + O ( E 1 ( ( K ε ( j ) ) j = 1 N ) + E 2 ( ( K ε ( j ) ) j = 1 N ) ) ) . The exit position is independent of the exit time, and is distributed as, for k ∈ { 1 , … , N } k\in\{1,\ldots,N\} k ∈ { 1 , … , N }
P ν ε ( κ ε = k ) = K ε ( k ) K ε + O ( E 1 ( ( K ε ( j ) ) j = 1 N ) + E 2 ( ( K ε ( j ) ) j = 1 N ) ) . \mathbb{P}_{\nu_\varepsilon}(\kappa_\varepsilon=k)=\frac{K_\varepsilon^{(k)}}{K_\varepsilon}+O\left(\mathcal{E}_1\left((K_\varepsilon^{(j)})_{j=1}^{N}\right)+\mathcal{E}_2\left((K_\varepsilon^{(j)})_{j=1}^{N}\right)\right). P ν ε ( κ ε = k ) = K ε K ε ( k ) + O ( E 1 ( ( K ε ( j ) ) j = 1 N ) + E 2 ( ( K ε ( j ) ) j = 1 N ) ) . where K ε ( k ) K_\varepsilon^{(k)} K ε ( k ) κ ε \kappa_\varepsilon κ ε κ ε = k \kappa_\varepsilon=k κ ε = k X τ ε ∈ Γ ε k ‾ X_{\tau_\varepsilon}\in\overline{\Gamma_\varepsilon^{k}} X τ ε ∈ Γ ε k
The proof of this theorem can be performed in a similar manner as in [13, Theorems 1.4 and 1.6]; for the sake of completeness, it is given in Section B. This proof also relies on the Poincaré–Wirtinger inequality uniform in ε \varepsilon ε ε → 0 \varepsilon\to0 ε → 0
Spherical hole shapes
In this section, we apply Theorem 2.2 to two different settings: (Type A) and (Type B), by constructing a quasi-mode that verifies Hypothesis 2.1, to obtain the results claimed in Table 1 . For both of them, the main ingredient is the Green function G G G Ω \Omega Ω G G G § 3.1 , and then (Type B) in § 3.2 .
Spherical holes inside the domain with a center on the boundary
In this subsection, we recall what was done in [13] for the case of holes defined in (Type A) for the sake of the comparison. The authors of [13] constructed a quasi-mode φ ε \varphi_\varepsilon φ ε (16) with
ψ ε ( k ) ( x ) = 2 π r ε ( k ) G ( x , x ( k ) ) , k ∈ { 1 , … , N } , \psi_\varepsilon^{(k)}(x)=2\pi r_\varepsilon^{(k)}G(x,x^{(k)}),\qquad k\in\{1,\ldots,N\}, ψ ε ( k ) ( x ) = 2 π r ε ( k ) G ( x , x ( k ) ) , k ∈ { 1 , … , N } , which verify Hypothesis 2.1 in this setting with K ε ( k ) = r ε ( k ) K_\varepsilon^{(k)}=r_\varepsilon^{(k)} K ε ( k ) = r ε ( k ) K ε ‾ = r ε ‾ = ∑ k = 1 N r ε ( k ) \overline{K_\varepsilon}=\overline{r_\varepsilon}=\sum_{k=1}^{N}r_\varepsilon^{(k)} K ε = r ε = ∑ k = 1 N r ε ( k ) C = 2 π ∣ ∂ Ω ∣ C=\frac{2\pi}{|\partial\Omega|} C = ∣ ∂ Ω∣ 2 π E 1 ( ( K ε ( j ) ) j = 1 N ) = K ε ‾ ∣ log ( K ε ‾ ) ∣ \mathcal{E}_1((K_\varepsilon^{(j)})_{j=1}^{N})=\overline{K_\varepsilon}|\log(\overline{K_\varepsilon})| E 1 (( K ε ( j ) ) j = 1 N ) = K ε ∣ log ( K ε ) ∣ E 2 ( ( K ε ( j ) ) j = 1 N ) = K ε ‾ \mathcal{E}_2((K_\varepsilon^{(j)})_{j=1}^{N})=\overline{K_\varepsilon} E 2 (( K ε ( j ) ) j = 1 N ) = K ε Γ ε D ‾ \overline{\Gamma_\varepsilon^{D}} Γ ε D G ( ⋅ , x ( k ) ) G(\cdot,x^{(k)}) G ( ⋅ , x ( k ) ) Ω ‾ ∖ { x ( k ) } \overline{\Omega}\setminus\{x^{(k)}\} Ω ∖ { x ( k ) } x ( k ) ∉ Γ ε D ‾ x^{(k)}\notin\overline{\Gamma_\varepsilon^{D}} x ( k ) ∈ / Γ ε D
Proposition 3.1. Consider the reflected Brownian motion ( X t ) t ≥ 0 (X_t)_{t\geq0} ( X t ) t ≥ 0 Ω ε \Omega_\varepsilon Ω ε
E ν ε [ τ ε ] = ∣ Ω ∣ 2 π 1 r ε ‾ ( 1 + O ( E ( r ε ‾ ) ) ) , (18) \mathbb{E}_{\nu_\varepsilon}[\tau_\varepsilon]=\frac{|\Omega|}{2\pi}\frac{1}{\overline{r_\varepsilon}}\left(1+O(\mathcal{E}(\overline{r_\varepsilon}))\right), \tag*{(18)} E ν ε [ τ ε ] = 2 π ∣Ω∣ r ε 1 ( 1 + O ( E ( r ε )) ) , (18) with E : x ↦ x ∣ log ( x ) ∣ \mathcal{E}:x\mapsto x|\log(x)| E : x ↦ x ∣ log ( x ) ∣ k ∈ { 1 , … , N } k\in\{1,\ldots,N\} k ∈ { 1 , … , N }
P ν ε ( κ ε = k ) = r ε ( k ) r ε ‾ + O ( E ( r ε ‾ ) ) . (19) \mathbb{P}_{\nu_\varepsilon}(\kappa_\varepsilon=k)=\frac{r_\varepsilon^{(k)}}{\overline{r_\varepsilon}}+O(\mathcal{E}(\overline{r_\varepsilon})). \tag*{(19)} P ν ε ( κ ε = k ) = r ε r ε ( k ) + O ( E ( r ε )) . (19) Spherical holes on the boundary of the domain
In this subsection, we consider holes of type (Type B). Recall that d ∂ Ω d_{\partial\Omega} d ∂ Ω ∂ Ω \partial\Omega ∂ Ω Ω ε = Ω \Omega_\varepsilon=\Omega Ω ε = Ω ε > 0 \varepsilon>0 ε > 0
Let us start checking Hypothesis 2.1 by giving a simple proof that the eigenvalue λ ε \lambda_\varepsilon λ ε
Lemma 3.2. Consider the operator L ε \mathcal{L}_\varepsilon L ε Ω ε \Omega_\varepsilon Ω ε λ ε \lambda_\varepsilon λ ε ε → 0 \varepsilon\to0 ε → 0
Proof. As λ ε \lambda_\varepsilon λ ε L ε \mathcal{L}_\varepsilon L ε
λ ε = min u ∈ H Γ D ε 1 ( Ω ε ) ∖ { 0 } ∫ Ω ε ∣ ∇ u ∣ 2 ∫ Ω ε ∣ u ∣ 2 , \lambda_\varepsilon=\min_{u\in H^1_{\Gamma_D^\varepsilon}(\Omega_\varepsilon)\setminus\{0\}}\frac{\int_{\Omega_\varepsilon}|\nabla u|^2}{\int_{\Omega_\varepsilon}|u|^2}, λ ε = u ∈ H Γ D ε 1 ( Ω ε ) ∖ { 0 } min ∫ Ω ε ∣ u ∣ 2 ∫ Ω ε ∣∇ u ∣ 2 , where H Γ D ε 1 ( Ω ε ) H^1_{\Gamma_D^\varepsilon}(\Omega_\varepsilon) H Γ D ε 1 ( Ω ε ) q ε q_\varepsilon q ε L ε \mathcal{L}_\varepsilon L ε (5) . This shows that λ ε \lambda_\varepsilon λ ε k ∈ { 1 , … , N } k\in\{1,\ldots,N\} k ∈ { 1 , … , N } ∂ Ω \partial\Omega ∂ Ω x x x x ( k ) x^{(k)} x ( k )
∣ x − x ( k ) ∣ ≤ d ∂ Ω ( x , x ( k ) ) . |x-x^{(k)}|\le d_{\partial\Omega}(x,x^{(k)}). ∣ x − x ( k ) ∣ ≤ d ∂ Ω ( x , x ( k ) ) . Therefore, one has that Γ k ε ⊂ B ( x ( k ) , r ε ( k ) ) ∩ ∂ Ω \Gamma_k^\varepsilon\subset B\left(x^{(k)},r_\varepsilon^{(k)}\right)\cap\partial\Omega Γ k ε ⊂ B ( x ( k ) , r ε ( k ) ) ∩ ∂ Ω H 1 ( Ω ε ) H^1(\Omega_\varepsilon) H 1 ( Ω ε ) H B ε 1 ( Ω ε ) H^1_{\mathcal{B}_\varepsilon}(\Omega_\varepsilon) H B ε 1 ( Ω ε )
B ε = ⋃ k = 1 N B ( x ( k ) , r ε ( k ) ) ∩ Ω ε . \mathcal{B}_\varepsilon=\bigcup_{k=1}^{N}B\left(x^{(k)},r_\varepsilon^{(k)}\right)\cap\Omega_\varepsilon. B ε = k = 1 ⋃ N B ( x ( k ) , r ε ( k ) ) ∩ Ω ε . It is clear, by extending the functions by 0 0 0 H B ε 1 ( Ω ε ) ⊂ H Γ D ε 1 ( Ω ε ) H^1_{\mathcal{B}_\varepsilon}(\Omega_\varepsilon)\subset H^1_{\Gamma_D^\varepsilon}(\Omega_\varepsilon) H B ε 1 ( Ω ε ) ⊂ H Γ D ε 1 ( Ω ε )
λ ε ≤ min u ∈ H B ε 1 ( Ω ε ) ∖ { 0 } ∫ Ω ε ∣ ∇ u ∣ 2 ∫ Ω ε ∣ u ∣ 2 = min u ∈ H B ε 1 ( Ω ε ) ∖ { 0 } ∫ Ω ε ∖ B ε ∣ ∇ u ∣ 2 ∫ Ω ε ∖ B ε ∣ u ∣ 2 . (20) \lambda_\varepsilon\le\min_{u\in H^1_{\mathcal{B}_\varepsilon}(\Omega_\varepsilon)\setminus\{0\}}\frac{\int_{\Omega_\varepsilon}|\nabla u|^2}{\int_{\Omega_\varepsilon}|u|^2}
=\min_{u\in H^1_{\mathcal{B}_\varepsilon}(\Omega_\varepsilon)\setminus\{0\}}\frac{\int_{\Omega_\varepsilon\setminus\mathcal{B}_\varepsilon}|\nabla u|^2}{\int_{\Omega_\varepsilon\setminus\mathcal{B}_\varepsilon}|u|^2}. \tag*{(20)} λ ε ≤ u ∈ H B ε 1 ( Ω ε ) ∖ { 0 } min ∫ Ω ε ∣ u ∣ 2 ∫ Ω ε ∣∇ u ∣ 2 = u ∈ H B ε 1 ( Ω ε ) ∖ { 0 } min ∫ Ω ε ∖ B ε ∣ u ∣ 2 ∫ Ω ε ∖ B ε ∣∇ u ∣ 2 . (20) Notice that the right-hand side of (20) is exactly the Rayleigh quotient of the eigenvalue associated with the problem with (Type A) holes, described in § 3.1 . In Proposition 3.1, we showed that the eigenvalue of the (Type A) tends to 0 0 0 ε → 0 \varepsilon\to0 ε → 0 λ ε \lambda_\varepsilon λ ε
We now move on to building the quasi-mode φ ε \varphi_\varepsilon φ ε Hypothesis 2.1 in this setting, using again the Green function G G G Ω \Omega Ω k ∈ { 1 , … , N } k\in\{1,\ldots,N\} k ∈ { 1 , … , N }
ρ ε ( k ) : y ∈ Γ k ε ↦ ( ( r ε ( k ) ) 2 − d ∂ Ω ( y , x ( k ) ) 2 ) − 1 2 , \rho_\varepsilon^{(k)}:y\in\Gamma_k^\varepsilon\mapsto\left(\left(r_\varepsilon^{(k)}\right)^2-d_{\partial\Omega}(y,x^{(k)})^2\right)^{-\frac{1}{2}}, ρ ε ( k ) : y ∈ Γ k ε ↦ ( ( r ε ( k ) ) 2 − d ∂ Ω ( y , x ( k ) ) 2 ) − 2 1 , and
ψ ε ( k ) ( x ) = 2 π ∫ Γ k ε ρ ε ( k ) ( y ) G ( x , y ) d σ ( y ) , k ∈ { 1 , … , N } . (21) \psi_\varepsilon^{(k)}(x)=\frac{2}{\pi}\int_{\Gamma_k^\varepsilon}\rho_\varepsilon^{(k)}(y)G(x,y)\,d\sigma(y),\qquad k\in\{1,\ldots,N\}. \tag*{(21)} ψ ε ( k ) ( x ) = π 2 ∫ Γ k ε ρ ε ( k ) ( y ) G ( x , y ) d σ ( y ) , k ∈ { 1 , … , N } . (21) We can now show that Hypothesis 2.1 holds for ψ ε ( k ) \psi_\varepsilon^{(k)} ψ ε ( k )
Proposition 3.3. The functions ψ ε ( k ) \psi_\varepsilon^{(k)} ψ ε ( k ) (21) verify Hypothesis 2.1 with
K ε ( k ) = r ε ( k ) , K ‾ ε = r ‾ ε , C = 4 ∣ Ω ∣ , E 1 ( ( K ε ( j ) ) j = 1 N ) = K ‾ ε ∣ log ( K ‾ ε ) ∣ , E 2 ( ( K ε ( j ) ) j = 1 N ) = K ‾ ε . K_\varepsilon^{(k)}=r_\varepsilon^{(k)},\qquad\overline{K}_\varepsilon=\overline{r}_\varepsilon,\qquad C=\frac{4}{|\Omega|},\qquad\mathcal{E}_1\left(\left(K_\varepsilon^{(j)}\right)_{j=1}^{N}\right)=\overline{K}_\varepsilon|\log(\overline{K}_\varepsilon)|,\qquad\mathcal{E}_2\left(\left(K_\varepsilon^{(j)}\right)_{j=1}^{N}\right)=\overline{K}_\varepsilon. K ε ( k ) = r ε ( k ) , K ε = r ε , C = ∣Ω∣ 4 , E 1 ( ( K ε ( j ) ) j = 1 N ) = K ε ∣ log ( K ε ) ∣ , E 2 ( ( K ε ( j ) ) j = 1 N ) = K ε . Furthermore, the mean exit time and exit position of the reflected Brownian motion ( X t ) t ≥ 0 (X_t)_{t\ge0} ( X t ) t ≥ 0 Ω ε \Omega_\varepsilon Ω ε
E ν ε [ τ ε ] = ∣ Ω ∣ r ‾ ε 1 1 + O ( E ( r ‾ ε ) ) , (22) \mathbb{E}_{\nu_\varepsilon}[\tau_\varepsilon]=\frac{|\Omega|}{\overline{r}_\varepsilon}\frac{1}{1+O\left(\mathcal{E}(\overline{r}_\varepsilon)\right)}, \tag*{(22)} E ν ε [ τ ε ] = r ε ∣Ω∣ 1 + O ( E ( r ε ) ) 1 , (22) with E : x ↦ x ∣ log ( x ) ∣ \mathcal{E}:x\mapsto x|\log(x)| E : x ↦ x ∣ log ( x ) ∣ k ∈ { 1 , … , N } k\in\{1,\ldots,N\} k ∈ { 1 , … , N }
P ν ε ( κ ε = k ) = r ε ( k ) r ‾ ε + O ( E ( r ‾ ε ) ) . (23) \mathbb{P}_{\nu_\varepsilon}(\kappa_\varepsilon=k)=\frac{r_\varepsilon^{(k)}}{\overline{r}_\varepsilon}+O\left(\mathcal{E}(\overline{r}_\varepsilon)\right). \tag*{(23)} P ν ε ( κ ε = k ) = r ε r ε ( k ) + O ( E ( r ε ) ) . (23) Proof. In the light of Lemma 3.2, we only need to show that the functions ψ ε ( k ) \psi_{\varepsilon}^{(k)} ψ ε ( k ) Hypothesis 2.1 to show the results on the mean exit time and exit position. Fix k ∈ { 1 , … , N } k \in\{1,\ldots,N\} k ∈ { 1 , … , N }
PDE. The proof relies on the results from Lemma C.4 . Indeed, by the (Integrability) estimate of Lemma D.1, ρ ε ( k ) ∈ L 4 ( Γ ‾ ε ( k ) ) \rho_{\varepsilon}^{(k)} \in L^4(\overline{\Gamma}_{\varepsilon}^{(k)}) ρ ε ( k ) ∈ L 4 ( Γ ε ( k ) ) ρ ε ( k ) ∉ L 2 ( Γ ‾ ε ( k ) ) \rho_{\varepsilon}^{(k)} \notin L^2(\overline{\Gamma}_{\varepsilon}^{(k)}) ρ ε ( k ) ∈ / L 2 ( Γ ε ( k ) ) Lemma C.4 with p = 4 / 3 p=4/3 p = 4/3 ψ ε ( k ) \psi_{\varepsilon}^{(k)} ψ ε ( k ) H 1 ( Ω ) H^1(\Omega) H 1 ( Ω )
{ Δ ψ ε ( k ) = 2 π ∣ Ω ∣ ∫ Γ ε ( k ) ρ ε ( k ) ( y ) d σ ( y ) in Ω , ∂ n ψ ε ( k ) ( x ) = 2 π ρ ε ( k ) ( x ) 1 Γ ε ( k ) ( x ) for x ∈ ∂ Ω . (24) \begin{cases}
\Delta\psi_{\varepsilon}^{(k)}=\dfrac{2}{\pi|\Omega|}\int_{\Gamma_{\varepsilon}^{(k)}}\rho_{\varepsilon}^{(k)}(y)\,\mathrm{d}\sigma(y) & \text{in }\Omega,\\
\partial_n\psi_{\varepsilon}^{(k)}(x)=\dfrac{2}{\pi}\rho_{\varepsilon}^{(k)}(x)\mathbf{1}_{\Gamma_{\varepsilon}^{(k)}}(x) & \text{for }x\in\partial\Omega. \tag*{(24)}
\end{cases} ⎩ ⎨ ⎧ Δ ψ ε ( k ) = π ∣Ω∣ 2 ∫ Γ ε ( k ) ρ ε ( k ) ( y ) d σ ( y ) ∂ n ψ ε ( k ) ( x ) = π 2 ρ ε ( k ) ( x ) 1 Γ ε ( k ) ( x ) in Ω , for x ∈ ∂ Ω. (24) It is clear from (24) that ∂ n ψ ε ( k ) = 0 \partial_n\psi_{\varepsilon}^{(k)}=0 ∂ n ψ ε ( k ) = 0 Γ N ε \Gamma_{\mathcal{N}}^\varepsilon Γ N ε Γ N ε ∩ Γ D ε = ∅ \Gamma_{\mathcal{N}}^\varepsilon\cap\Gamma_{\mathcal{D}}^\varepsilon=\varnothing Γ N ε ∩ Γ D ε = ∅
∫ Γ ε ( k ) ρ ε ( k ) d σ ( y ) = 2 π r ε ( k ) , so that Δ ψ ε ( k ) = 4 ∣ Ω ∣ r ε ( k ) . \int_{\Gamma_{\varepsilon}^{(k)}}\rho_{\varepsilon}^{(k)}\,\mathrm{d}\sigma(y)=2\pi r_{\varepsilon}^{(k)},\qquad\text{so that}\qquad\Delta\psi_{\varepsilon}^{(k)}=\frac{4}{|\Omega|}r_{\varepsilon}^{(k)}. ∫ Γ ε ( k ) ρ ε ( k ) d σ ( y ) = 2 π r ε ( k ) , so that Δ ψ ε ( k ) = ∣Ω∣ 4 r ε ( k ) . Continuity on Γ ‾ D ε \overline{\Gamma}_{\mathcal{D}}^\varepsilon Γ D ε By Lemma D.2, ψ ε ( k ) \psi_{\varepsilon}^{(k)} ψ ε ( k ) Γ ‾ D ε \overline{\Gamma}_{\mathcal{D}}^\varepsilon Γ D ε ψ ε ( k ) ∈ H 1 ( Ω ) ∩ C 0 ( Γ ‾ D ε ) \psi_{\varepsilon}^{(k)}\in H^1(\Omega)\cap C^0(\overline{\Gamma}_{\mathcal{D}}^\varepsilon) ψ ε ( k ) ∈ H 1 ( Ω ) ∩ C 0 ( Γ D ε ) Hypothesis 2.1 .
Dirichlet estimate. Take x ∈ Γ ε ( k ) x\in\Gamma_{\varepsilon}^{(k)} x ∈ Γ ε ( k ) Lemma C.3 for r ε ( k ) r_{\varepsilon}^{(k)} r ε ( k ) ψ ε ( k ) \psi_{\varepsilon}^{(k)} ψ ε ( k )
ψ ε ( k ) ( x ) = 2 π ∫ Γ ε ( k ) ρ ε ( k ) 1 2 π ∣ x − y ∣ d σ ( y ) + 2 π ∫ Γ ε ( k ) ρ ε ( k ) g ( x , y ) d σ ( y ) , (25) \psi_{\varepsilon}^{(k)}(x)=\frac{2}{\pi}\int_{\Gamma_{\varepsilon}^{(k)}}\rho_{\varepsilon}^{(k)}\frac{1}{2\pi|x-y|}\,\mathrm{d}\sigma(y)+\frac{2}{\pi}\int_{\Gamma_{\varepsilon}^{(k)}}\rho_{\varepsilon}^{(k)}g(x,y)\,\mathrm{d}\sigma(y), \tag*{(25)} ψ ε ( k ) ( x ) = π 2 ∫ Γ ε ( k ) ρ ε ( k ) 2 π ∣ x − y ∣ 1 d σ ( y ) + π 2 ∫ Γ ε ( k ) ρ ε ( k ) g ( x , y ) d σ ( y ) , (25) where g ( x , y ) g(x,y) g ( x , y ) O ( ∣ log ∣ x − y ∣ ∣ ) O(|\log|x-y||) O ( ∣ log ∣ x − y ∣∣ ) ∣ x − y ∣ → 0 |x-y|\to0 ∣ x − y ∣ → 0 (25) have been computed in Lemma D.1. Indeed, by the (Newtonian potential) estimate from Lemma D.1,
2 π ∫ Γ ε ( k ) ρ ε ( k ) 1 2 π ∣ x − y ∣ d σ ( y ) = 2 π π 2 + O ( r ε ( k ) ) . (26) \frac{2}{\pi}\int_{\Gamma_{\varepsilon}^{(k)}}\rho_{\varepsilon}^{(k)}\frac{1}{2\pi|x-y|}\,\mathrm{d}\sigma(y)=\frac{2}{\pi}\frac{\pi}{2}+O(r_{\varepsilon}^{(k)}). \tag*{(26)} π 2 ∫ Γ ε ( k ) ρ ε ( k ) 2 π ∣ x − y ∣ 1 d σ ( y ) = π 2 2 π + O ( r ε ( k ) ) . (26) By the (Logarithmic potential) estimate from Lemma D.1,
∣ ∫ Γ ε ( k ) ρ ε ( k ) g ( x , y ) d σ ( y ) ∣ ≤ C ∫ Γ ε ( k ) ρ ε ( k ) ∣ log ∣ x − y ∣ ∣ d σ ( y ) = O ( r ε ( k ) ∣ log ( r ε ( k ) ) ∣ ) . (27) \left|\int_{\Gamma_{\varepsilon}^{(k)}}\rho_{\varepsilon}^{(k)}g(x,y)\,\mathrm{d}\sigma(y)\right|\le C\int_{\Gamma_{\varepsilon}^{(k)}}\rho_{\varepsilon}^{(k)}|\log|x-y||\,\mathrm{d}\sigma(y)=O(r_{\varepsilon}^{(k)}|\log(r_{\varepsilon}^{(k)})|). \tag*{(27)} ∫ Γ ε ( k ) ρ ε ( k ) g ( x , y ) d σ ( y ) ≤ C ∫ Γ ε ( k ) ρ ε ( k ) ∣ log ∣ x − y ∣∣ d σ ( y ) = O ( r ε ( k ) ∣ log ( r ε ( k ) ) ∣ ) . (27) Combining (25) , (26) and (27) leads to
ψ ε ( k ) ( x ) = 1 + O ( r ε ( k ) ) + O ( r ε ( k ) ∣ log ( r ε ( k ) ) ∣ ) = 1 + O ( r ε ( k ) ∣ log ( r ε ( k ) ) ∣ ) , for all x ∈ Γ ε ( k ) . \psi_{\varepsilon}^{(k)}(x)=1+O(r_{\varepsilon}^{(k)})+O(r_{\varepsilon}^{(k)}|\log(r_{\varepsilon}^{(k)})|)=1+O(r_{\varepsilon}^{(k)}|\log(r_{\varepsilon}^{(k)})|),\qquad\text{for all }x\in\Gamma_{\varepsilon}^{(k)}. ψ ε ( k ) ( x ) = 1 + O ( r ε ( k ) ) + O ( r ε ( k ) ∣ log ( r ε ( k ) ) ∣ ) = 1 + O ( r ε ( k ) ∣ log ( r ε ( k ) ) ∣ ) , for all x ∈ Γ ε ( k ) . Now take x ∈ Γ ε ( j ) x\in\Gamma_{\varepsilon}^{(j)} x ∈ Γ ε ( j ) j ≠ k j\ne k j = k G ( x , ⋅ ) G(x,\cdot) G ( x , ⋅ ) Γ ε ( k ) \Gamma_{\varepsilon}^{(k)} Γ ε ( k ) x x x ε \varepsilon ε ψ ε ( k ) ( x ) = O ( r ε ( k ) ) \psi_{\varepsilon}^{(k)}(x)=O(r_{\varepsilon}^{(k)}) ψ ε ( k ) ( x ) = O ( r ε ( k ) ) x ↦ x ∣ log ( x ) ∣ x\mapsto x|\log(x)| x ↦ x ∣ log ( x ) ∣ ( 0 , e − 1 ) (0,\mathrm{e}^{-1}) ( 0 , e − 1 ) r ε ( k ) ≤ r ‾ ε r_{\varepsilon}^{(k)}\le\overline{r}_{\varepsilon} r ε ( k ) ≤ r ε
∥ ψ ε ( k ) − 1 Γ ε ( k ) ∥ L ∞ ( Γ D ε ) = O ( r ‾ ε ∣ log ( r ‾ ε ) ∣ ) , \left\|\psi_{\varepsilon}^{(k)}-\mathbf{1}_{\Gamma_{\varepsilon}^{(k)}}\right\|_{L^\infty(\Gamma_{\mathcal{D}}^\varepsilon)}=O(\overline{r}_{\varepsilon}|\log(\overline{r}_{\varepsilon})|), ψ ε ( k ) − 1 Γ ε ( k ) L ∞ ( Γ D ε ) = O ( r ε ∣ log ( r ε ) ∣ ) , which is the Dirichlet condition in (PDE) with E 1 ( ( K ε ( j ) ) j = 1 N ) = r ‾ ε ∣ log ( r ‾ ε ) ∣ \mathcal{E}_1((K_{\varepsilon}^{(j)})_{j=1}^N)=\overline{r}_{\varepsilon}|\log(\overline{r}_{\varepsilon})| E 1 (( K ε ( j ) ) j = 1 N ) = r ε ∣ log ( r ε ) ∣
Normalization. Using the Cauchy–Schwarz inequality between ρ ε ( k ) \sqrt{\rho_{\varepsilon}^{(k)}} ρ ε ( k ) ρ ε ( k ) G ( x , ⋅ ) \sqrt{\rho_{\varepsilon}^{(k)}}G(x,\cdot) ρ ε ( k ) G ( x , ⋅ ) x ∈ Ω x\in\Omega x ∈ Ω
( ∫ Γ ε ( k ) ρ ε ( k ) ( y ) G ( x , y ) d σ ( y ) ) 2 ≤ ( ∫ Γ ε ( k ) ρ ε ( k ) ( y ) d σ ( y ) ) ( ∫ Γ ε ( k ) ρ ε ( k ) ( y ) G ( x , y ) 2 d σ ( y ) ) . (28) \left(\int_{\Gamma_{\varepsilon}^{(k)}}\rho_{\varepsilon}^{(k)}(y)G(x,y)\,\mathrm{d}\sigma(y)\right)^2\le\left(\int_{\Gamma_{\varepsilon}^{(k)}}\rho_{\varepsilon}^{(k)}(y)\,\mathrm{d}\sigma(y)\right)\left(\int_{\Gamma_{\varepsilon}^{(k)}}\rho_{\varepsilon}^{(k)}(y)G(x,y)^2\,\mathrm{d}\sigma(y)\right). \tag*{(28)} ( ∫ Γ ε ( k ) ρ ε ( k ) ( y ) G ( x , y ) d σ ( y ) ) 2 ≤ ( ∫ Γ ε ( k ) ρ ε ( k ) ( y ) d σ ( y ) ) ( ∫ Γ ε ( k ) ρ ε ( k ) ( y ) G ( x , y ) 2 d σ ( y ) ) . (28) It is clear by (mass) from Lemma D.1 that the first integral in the right-hand side is O ( r ε ( k ) ) O\left(r_\varepsilon^{(k)}\right) O ( r ε ( k ) ) ε → 0 \varepsilon\to0 ε → 0 x ∈ Ω x \in\Omega x ∈ Ω
∫ Ω ∫ Γ ε ( k ) ρ ε ( k ) ( y ) G ( x , y ) 2 d σ ( y ) d x = ∫ Γ ε ( k ) ρ ε ( k ) ( y ) ( ∫ Ω G ( x , y ) 2 d x ) d σ ( y ) . (29) \int_{\Omega}\int_{\Gamma_\varepsilon^{(k)}} \rho_\varepsilon^{(k)}(y)G(x,y)^2\,\mathrm{d}\sigma(y)\,\mathrm{d}x
=
\int_{\Gamma_\varepsilon^{(k)}} \rho_\varepsilon^{(k)}(y)\left(\int_{\Omega}G(x,y)^2\,\mathrm{d}x\right)\mathrm{d}\sigma(y). \tag*{(29)} ∫ Ω ∫ Γ ε ( k ) ρ ε ( k ) ( y ) G ( x , y ) 2 d σ ( y ) d x = ∫ Γ ε ( k ) ρ ε ( k ) ( y ) ( ∫ Ω G ( x , y ) 2 d x ) d σ ( y ) . (29) Using (28) , (29) , Lemma C.5 and (mass) from Lemma D.1, one has that there exists a constant C > 0 C>0 C > 0
∥ ψ ε ( k ) ∥ L 2 ( Ω ) 2 = 4 π 2 ∫ Ω ( ∫ Γ ε ( k ) ρ ε ( k ) ( y ) G ( x , y ) d σ ( y ) ) 2 d x ≤ C ( r ε ( k ) ) 2 , (30) \left\|\psi_\varepsilon^{(k)}\right\|_{L^2(\Omega)}^2
=
\frac{4}{\pi^2}\int_{\Omega}\left(\int_{\Gamma_\varepsilon^{(k)}}\rho_\varepsilon^{(k)}(y)G(x,y)\,\mathrm{d}\sigma(y)\right)^2\mathrm{d}x
\leq C\left(r_\varepsilon^{(k)}\right)^2, \tag*{(30)} ψ ε ( k ) L 2 ( Ω ) 2 = π 2 4 ∫ Ω ( ∫ Γ ε ( k ) ρ ε ( k ) ( y ) G ( x , y ) d σ ( y ) ) 2 d x ≤ C ( r ε ( k ) ) 2 , (30) so that ∥ ψ ε ( k ) ∥ L 2 ( Ω ) = O ( r ‾ ε ) \left\|\psi_\varepsilon^{(k)}\right\|_{L^2(\Omega)}=O\left(\overline{r}_\varepsilon\right) ψ ε ( k ) L 2 ( Ω ) = O ( r ε ) E 2 ( ( K ε ( j ) ) j = 1 N ) = r ‾ ε E_2\left(\left(K_\varepsilon^{(j)}\right)_{j=1}^N\right)=\overline{r}_\varepsilon E 2 ( ( K ε ( j ) ) j = 1 N ) = r ε
Slit-like holes
In this section, we consider the case of slit-like holes defined in Type C, whose half-width w ε ( k ) w_\varepsilon^{(k)} w ε ( k )
In particular, the foci of these ellipsoids are located at the tips s x + t x ( k ) = ± ℓ k s_x+t_x^{(k)}=\pm\ell_k s x + t x ( k ) = ± ℓ k Ω ε \Omega_\varepsilon Ω ε
Ω ε = Ω ∖ ⋃ k = 1 N { x ∈ Ω ‾ | ( s x + t x ( k ) ) 2 ( a ε ( k ) ) 2 + ∣ ρ x ( k ) ∣ 2 ( w ε ( k ) ) 2 ≤ 1 } . \Omega_\varepsilon=\Omega\setminus\bigcup_{k=1}^{N}\left\{x\in\overline{\Omega}\,\middle|\,\frac{\left(s_x+t_x^{(k)}\right)^2}{\left(a_\varepsilon^{(k)}\right)^2}+\frac{\left|\rho_x^{(k)}\right|^2}{\left(w_\varepsilon^{(k)}\right)^2}\leq1\right\}. Ω ε = Ω ∖ k = 1 ⋃ N ⎩ ⎨ ⎧ x ∈ Ω ( a ε ( k ) ) 2 ( s x + t x ( k ) ) 2 + ( w ε ( k ) ) 2 ρ x ( k ) 2 ≤ 1 ⎭ ⎬ ⎫ . We are now in a position to introduce a quasi-mode φ ε \varphi_\varepsilon φ ε Hypothesis 2.1 in this setting. For k ∈ { 1 , … , N } k\in\{1,\ldots,N\} k ∈ { 1 , … , N }
ψ ε ( k ) ( x ) = π ∣ log w ε ( k ) ∣ ∫ H k G ( x , y ) d y , x ∈ Ω , (31) \psi_\varepsilon^{(k)}(x)=\frac{\pi}{\left|\log w_\varepsilon^{(k)}\right|}\int_{\mathcal{H}_k}G(x,y)\,\mathrm{d}y,\qquad x\in\Omega, \tag*{(31)} ψ ε ( k ) ( x ) = log w ε ( k ) π ∫ H k G ( x , y ) d y , x ∈ Ω , (31) where G G G § 3 , the Green function of the Laplacian on Ω \Omega Ω Definition C.1 . The construction of ψ ε ( k ) \psi_\varepsilon^{(k)} ψ ε ( k ) § 3.2 : a single-layer potential built from G G G H k \mathcal{H}_k H k d y \mathrm{d}y d y Γ ε ( k ) \Gamma_\varepsilon^{(k)} Γ ε ( k ) § 3.2 . This drastically changes the behavior of the singularity of the quasi-mode close to the holes.
Let us first check that the functions ψ ε ( k ) \psi_\varepsilon^{(k)} ψ ε ( k ) Hypothesis 2.1 .
Lemma 4.1. The functions ψ ε ( k ) \psi_\varepsilon^{(k)} ψ ε ( k ) (31) verify (PDE) and (Norm) of Hypothesis 2.1 with
K ε ( k ) = ℓ k ∣ log w ε ( k ) ∣ , C = 2 π ∣ Ω ∣ , E 1 ( ( K ε ( j ) ) j = 1 N ) = E 2 ( ( K ε ( j ) ) j = 1 N ) = K ‾ ε . K_\varepsilon^{(k)}=\frac{\ell_k}{\left|\log w_\varepsilon^{(k)}\right|},\qquad C=\frac{2\pi}{|\Omega|},\qquad E_1\left(\left(K_\varepsilon^{(j)}\right)_{j=1}^{N}\right)=E_2\left(\left(K_\varepsilon^{(j)}\right)_{j=1}^{N}\right)=\overline{K}_\varepsilon. K ε ( k ) = log w ε ( k ) ℓ k , C = ∣Ω∣ 2 π , E 1 ( ( K ε ( j ) ) j = 1 N ) = E 2 ( ( K ε ( j ) ) j = 1 N ) = K ε . Proof. Fix k ∈ { 1 , … , N } k\in\{1,\ldots,N\} k ∈ { 1 , … , N }
PDE. Fix ε > 0 \varepsilon>0 ε > 0 Ω ε \Omega_\varepsilon Ω ε Ω ‾ ε \overline{\Omega}_\varepsilon Ω ε (7) , every point of the half-spheroid Γ ε ( k ) \Gamma_\varepsilon^{(k)} Γ ε ( k ) H k \mathcal{H}_k H k w ε ( k ) w_\varepsilon^{(k)} w ε ( k ) H k \mathcal{H}_k H k
δ ε = dist ( Ω ‾ ε , H k ) > 0. \delta_\varepsilon=\operatorname{dist}\left(\overline{\Omega}_\varepsilon,\mathcal{H}_k\right)>0. δ ε = dist ( Ω ε , H k ) > 0. By Lemma C.2 (extended by continuity to Ω ‾ ε \overline{\Omega}_\varepsilon Ω ε ∣ G ( x , y ) ∣ ≤ K δ ε − 1 |G(x,y)|\leq K\delta_\varepsilon^{-1} ∣ G ( x , y ) ∣ ≤ K δ ε − 1 ∣ ∇ x G ( x , y ) ∣ ≤ K δ ε − 2 |\nabla_xG(x,y)|\leq K\delta_\varepsilon^{-2} ∣ ∇ x G ( x , y ) ∣ ≤ K δ ε − 2 x ∈ Ω ‾ ε x\in\overline{\Omega}_\varepsilon x ∈ Ω ε y ∈ H k y\in\mathcal{H}_k y ∈ H k ψ ε ( k ) \psi_\varepsilon^{(k)} ψ ε ( k ) Ω ‾ ε \overline{\Omega}_\varepsilon Ω ε (31) , ψ ε ( k ) ∈ W 1 , ∞ ( Ω ε ) ⊂ H 1 ( Ω ε ) \psi_\varepsilon^{(k)}\in W^{1,\infty}(\Omega_\varepsilon)\subset H^1(\Omega_\varepsilon) ψ ε ( k ) ∈ W 1 , ∞ ( Ω ε ) ⊂ H 1 ( Ω ε ) y ∈ H k y\in\mathcal{H}_k y ∈ H k G ( ⋅ , y ) G(\cdot,y) G ( ⋅ , y ) Ω ‾ ε \overline{\Omega}_\varepsilon Ω ε Δ x G ( ⋅ , y ) = 1 / ∣ Ω ∣ \Delta_xG(\cdot,y)=1/|\Omega| Δ x G ( ⋅ , y ) = 1/∣Ω∣ Ω ε \Omega_\varepsilon Ω ε ∂ n G ( ⋅ , y ) = 0 \partial_nG(\cdot,y)=0 ∂ n G ( ⋅ , y ) = 0 Γ N ε \Gamma_{\mathcal{N}}^\varepsilon Γ N ε Definition C.1 . Integrating these relations in y ∈ H k y\in\mathcal{H}_k y ∈ H k ψ ε ( k ) ∈ H 1 ( Ω ε ) ∩ C 0 ( Γ ‾ D ε ) \psi_\varepsilon^{(k)}\in H^1(\Omega_\varepsilon)\cap C^0(\overline{\Gamma}_{\mathcal{D}}^\varepsilon) ψ ε ( k ) ∈ H 1 ( Ω ε ) ∩ C 0 ( Γ D ε )
Δ ψ ε ( k ) = π ∣ log w ε ( k ) ∣ ∫ H k 1 ∣ Ω ∣ d y = π ∣ log w ε ( k ) ∣ 2 ℓ k ∣ Ω ∣ = C K ε ( k ) in Ω ε , ∂ n ψ ε ( k ) = 0 on Γ N ε , \Delta\psi_\varepsilon^{(k)}
=
\frac{\pi}{\left|\log w_\varepsilon^{(k)}\right|}\int_{\mathcal{H}_k}\frac{1}{|\Omega|}\,\mathrm{d}y
=
\frac{\pi}{\left|\log w_\varepsilon^{(k)}\right|}\frac{2\ell_k}{|\Omega|}
=
CK_\varepsilon^{(k)}
\quad\text{in }\Omega_\varepsilon,\qquad
\partial_n\psi_\varepsilon^{(k)}=0\quad\text{on }\Gamma_{\mathcal{N}}^\varepsilon, Δ ψ ε ( k ) = log w ε ( k ) π ∫ H k ∣Ω∣ 1 d y = log w ε ( k ) π ∣Ω∣ 2 ℓ k = C K ε ( k ) in Ω ε , ∂ n ψ ε ( k ) = 0 on Γ N ε , where we used ∫ H k d y = 2 ℓ k \int_{\mathcal{H}_k}dy=2\ell_k ∫ H k d y = 2 ℓ k Γ N ε ⊂ ∂ Ω ∖ H k \Gamma^\varepsilon_N\subset\partial\Omega\setminus\mathcal{H}_k Γ N ε ⊂ ∂ Ω ∖ H k
Dirichlet estimate. The proof of this estimate is postponed to Lemma D.3 in the appendix below, where we show that
∫ H k G ( x , y ) d y = 1 π ∣ log w ε ( k ) ∣ + O ( 1 ) , x ∈ Γ k ε (32) \int_{\mathcal{H}_k}G(x,y)\,dy=\frac{1}{\pi}\lvert\log w_\varepsilon^{(k)}\rvert+O(1),\qquad x\in\Gamma_k^\varepsilon \tag*{(32)} ∫ H k G ( x , y ) d y = π 1 ∣ log w ε ( k ) ∣ + O ( 1 ) , x ∈ Γ k ε (32) so that, by the very choice of the prefactor π / ∣ log w ε ( k ) ∣ \pi/\lvert\log w_\varepsilon^{(k)}\rvert π / ∣ log w ε ( k ) ∣ (31) ,
ψ ε ( k ) ( x ) = 1 + O ( 1 ∣ log w ε ( k ) ∣ ) , x ∈ Γ k ε . (33) \psi_\varepsilon^{(k)}(x)=1+O\left(\frac{1}{\lvert\log w_\varepsilon^{(k)}\rvert}\right),\qquad x\in\Gamma_k^\varepsilon. \tag*{(33)} ψ ε ( k ) ( x ) = 1 + O ( ∣ log w ε ( k ) ∣ 1 ) , x ∈ Γ k ε . (33) As the curves H j \mathcal{H}_j H j ε → 0 \varepsilon\to0 ε → 0 j ≠ k j\ne k j = k (32) is bounded uniformly for x ∈ Γ j ε x\in\Gamma_j^\varepsilon x ∈ Γ j ε
ψ ε ( k ) ( x ) = O ( 1 ∣ log w ε ( k ) ∣ ) , x ∈ Γ D ε ∖ Γ k ε . (34) \psi_\varepsilon^{(k)}(x)=O\left(\frac{1}{\lvert\log w_\varepsilon^{(k)}\rvert}\right),\qquad x\in\Gamma_D^\varepsilon\setminus\Gamma_k^\varepsilon. \tag*{(34)} ψ ε ( k ) ( x ) = O ( ∣ log w ε ( k ) ∣ 1 ) , x ∈ Γ D ε ∖ Γ k ε . (34) The Dirichlet estimate follows from (33) and (34) .
Normalization. Notice from Lemma C.5 that
sup y ∈ ∂ Ω ∫ Ω G ( x , y ) 2 d x ≤ M < + ∞ , \sup_{y\in\partial\Omega}\int_\Omega G(x,y)^2\,dx\le M<+\infty, y ∈ ∂ Ω sup ∫ Ω G ( x , y ) 2 d x ≤ M < + ∞ , where M M M ε \varepsilon ε L 2 L^2 L 2 ψ ε ( k ) \psi_\varepsilon^{(k)} ψ ε ( k )
∥ ψ ε ( k ) ∥ L 2 ( Ω ε ) ≤ π ∣ log w ε ( k ) ∣ ( ∫ Ω ε ( ∫ H k G ( x , y ) d y ) 2 d x ) 1 2 . (35) \left\|\psi_\varepsilon^{(k)}\right\|_{L^2(\Omega_\varepsilon)}
\le\frac{\pi}{\lvert\log w_\varepsilon^{(k)}\rvert}
\left(\int_{\Omega_\varepsilon}\left(\int_{\mathcal{H}_k}G(x,y)\,dy\right)^2dx\right)^{\frac12}. \tag*{(35)} ψ ε ( k ) L 2 ( Ω ε ) ≤ ∣ log w ε ( k ) ∣ π ( ∫ Ω ε ( ∫ H k G ( x , y ) d y ) 2 d x ) 2 1 . (35) Using the Cauchy–Schwarz inequality, the inclusion Ω ε ⊂ Ω \Omega_\varepsilon\subset\Omega Ω ε ⊂ Ω (35) by constants independent of ε \varepsilon ε
∥ ψ ε ( k ) ∥ L 2 ( Ω ε ) = O ( 1 ∣ log w ε ( k ) ∣ ) = O ( K ‾ ε ) , \left\|\psi_\varepsilon^{(k)}\right\|_{L^2(\Omega_\varepsilon)}
=O\left(\frac{1}{\lvert\log w_\varepsilon^{(k)}\rvert}\right)
=O(\overline{K}_\varepsilon), ψ ε ( k ) L 2 ( Ω ε ) = O ( ∣ log w ε ( k ) ∣ 1 ) = O ( K ε ) , which is (Norm) with E 2 ( ( K ε ( j ) ) j = 1 N ) = K ‾ ε \mathcal{E}_2((K_\varepsilon^{(j)})_{j=1}^{N})=\overline{K}_\varepsilon E 2 (( K ε ( j ) ) j = 1 N ) = K ε y ↦ 1 y\mapsto1 y ↦ 1 y ↦ G ( x , y ) y\mapsto G(x,y) y ↦ G ( x , y ) H k \mathcal{H}_k H k x ∈ Ω ε x\in\Omega_\varepsilon x ∈ Ω ε
( ∫ H k G ( x , y ) d y ) 2 ≤ ∣ H k ∣ ∫ H k G ( x , y ) 2 d y = 2 ℓ k ∫ H k G ( x , y ) 2 d y . \left(\int_{\mathcal{H}_k}G(x,y)\,dy\right)^2
\le\lvert\mathcal{H}_k\rvert\int_{\mathcal{H}_k}G(x,y)^2\,dy
=2\ell_k\int_{\mathcal{H}_k}G(x,y)^2\,dy. ( ∫ H k G ( x , y ) d y ) 2 ≤ ∣ H k ∣ ∫ H k G ( x , y ) 2 d y = 2 ℓ k ∫ H k G ( x , y ) 2 d y . Integrating over x ∈ Ω ε ⊂ Ω x\in\Omega_\varepsilon\subset\Omega x ∈ Ω ε ⊂ Ω Lemma C.5 ,
∫ Ω ε ( ∫ H k G ( x , y ) d y ) 2 d x ≤ 2 ℓ k ∫ H k ( ∫ Ω G ( x , y ) 2 d x ) d y ≤ 2 ℓ k ∫ H k M d y = 4 M ℓ k 2 , \int_{\Omega_\varepsilon}\left(\int_{\mathcal{H}_k}G(x,y)\,dy\right)^2dx
\le2\ell_k\int_{\mathcal{H}_k}\left(\int_\Omega G(x,y)^2\,dx\right)dy
\le2\ell_k\int_{\mathcal{H}_k}M\,dy=4M\ell_k^2, ∫ Ω ε ( ∫ H k G ( x , y ) d y ) 2 d x ≤ 2 ℓ k ∫ H k ( ∫ Ω G ( x , y ) 2 d x ) d y ≤ 2 ℓ k ∫ H k M d y = 4 M ℓ k 2 , so that ∥ u k ∥ L 2 ( Ω ) = O ( 1 ) \|u_k\|_{L^2(\Omega)}=O(1) ∥ u k ∥ L 2 ( Ω ) = O ( 1 ) ε \varepsilon ε u k u_k u k ε \varepsilon ε
∥ ψ ε ( k ) ∥ L 2 ( Ω ε ) ≤ π ∣ log w ε ( k ) ∣ ∥ u k ∥ L 2 ( Ω ) = O ( 1 ∣ log w ε ( k ) ∣ ) = O ( K ‾ ε ) , \left\|\psi_\varepsilon^{(k)}\right\|_{L^2(\Omega_\varepsilon)}
\le\frac{\pi}{\lvert\log w_\varepsilon^{(k)}\rvert}\|u_k\|_{L^2(\Omega)}
=O\left(\frac{1}{\lvert\log w_\varepsilon^{(k)}\rvert}\right)
=O(\overline{K}_\varepsilon), ψ ε ( k ) L 2 ( Ω ε ) ≤ ∣ log w ε ( k ) ∣ π ∥ u k ∥ L 2 ( Ω ) = O ( ∣ log w ε ( k ) ∣ 1 ) = O ( K ε ) , which is (Norm) with E 2 ( ( K ε ( j ) ) j = 1 N ) = K ‾ ε \mathcal{E}_2((K_\varepsilon^{(j)})_{j=1}^{N})=\overline{K}_\varepsilon E 2 (( K ε ( j ) ) j = 1 N ) = K ε
It remains to check that the eigenvalue λ ε \lambda_\varepsilon λ ε 0 0 0
Proposition 4.2. Consider the reflected Brownian motion ( X t ) t ≥ 0 (X_t)_{t\geq0} ( X t ) t ≥ 0 Ω ε \Omega_\varepsilon Ω ε ε → 0 \varepsilon\to0 ε → 0
E ν ε [ τ ε ] = ∣ Ω ∣ 2 π 1 K ‾ ε ( 1 + O ( K ‾ ε ) ) . (36) \mathbb{E}_{\nu_\varepsilon}[\tau_\varepsilon] = \frac{|\Omega|}{2\pi}\frac{1}{\overline{K}_\varepsilon}\left(1+O(\overline{K}_\varepsilon)\right). \tag*{(36)} E ν ε [ τ ε ] = 2 π ∣Ω∣ K ε 1 ( 1 + O ( K ε ) ) . (36) Then the exit position is independent of the exit time, and is distributed as, for k ∈ { 1 , … , N } k \in\{1,\ldots,N\} k ∈ { 1 , … , N }
P ν ε ( κ ε = k ) = K ε ( k ) K ε + O ( K ‾ ε ) . (37) \mathbb{P}_{\nu_\varepsilon}(\kappa_\varepsilon=k)=\frac{K_\varepsilon^{(k)}}{K_\varepsilon}+O(\overline{K}_\varepsilon). \tag*{(37)} P ν ε ( κ ε = k ) = K ε K ε ( k ) + O ( K ε ) . (37) Proof. In view of Lemma 4.1 , it remains to prove that λ ε → 0 \lambda_\varepsilon\to0 λ ε → 0 ε → 0 \varepsilon\to0 ε → 0 Theorem 2.2 . As in Lemma 3.2, we rely on the Rayleigh quotient
λ ε = min u ∈ H Γ D ε 1 ( Ω ε ) ∖ { 0 } ∫ Ω ε ∣ ∇ u ∣ 2 ∫ Ω ε ∣ u ∣ 2 . (38) \lambda_\varepsilon=\min_{u\in H^1_{\Gamma_D^\varepsilon}(\Omega_\varepsilon)\setminus\{0\}}\frac{\int_{\Omega_\varepsilon}|\nabla u|^2}{\int_{\Omega_\varepsilon}|u|^2}. \tag*{(38)} λ ε = u ∈ H Γ D ε 1 ( Ω ε ) ∖ { 0 } min ∫ Ω ε ∣ u ∣ 2 ∫ Ω ε ∣∇ u ∣ 2 . (38) As φ ε \varphi_\varepsilon φ ε (16) using the ( ψ ε ( k ) ) k = 1 N (\psi_\varepsilon^{(k)})_{k=1}^N ( ψ ε ( k ) ) k = 1 N (31) , belongs to H 1 ( Ω ε ) H^1(\Omega_\varepsilon) H 1 ( Ω ε ) Γ D ε \Gamma_D^\varepsilon Γ D ε [38 Proof of item (ii) of Theorem 1.2] , and consider, for α > 0 \alpha>0 α > 0
φ ε α ( x ) = 1 − α ∑ k = 1 N ψ ε ( k ) ( x ) . \varphi_\varepsilon^\alpha(x)=1-\alpha\sum_{k=1}^N\psi_\varepsilon^{(k)}(x). φ ε α ( x ) = 1 − α k = 1 ∑ N ψ ε ( k ) ( x ) . By the Dirichlet estimate of Lemma 4.1 , ∑ k = 1 N ψ ε ( k ) = 1 + O ( K ‾ ε ) \sum_{k=1}^N\psi_\varepsilon^{(k)}=1+O(\overline{K}_\varepsilon) ∑ k = 1 N ψ ε ( k ) = 1 + O ( K ε ) Γ D ε \Gamma_D^\varepsilon Γ D ε
φ ε α = ( 1 − α ) + α O ( K ‾ ε ) , uniformly on Γ D ε . \varphi_\varepsilon^\alpha=(1-\alpha)+\alpha O(\overline{K}_\varepsilon),\qquad\text{uniformly on }\Gamma_D^\varepsilon. φ ε α = ( 1 − α ) + α O ( K ε ) , uniformly on Γ D ε . Hence, for a fixed α ∗ > 1 \alpha^*>1 α ∗ > 1 α ∗ = 2 \alpha^*=2 α ∗ = 2 ε \varepsilon ε φ ε α ∗ \varphi_\varepsilon^{\alpha^*} φ ε α ∗ Γ D ε \Gamma_D^\varepsilon Γ D ε φ ε α ∗ = ( φ ε α ∗ ) + \varphi_\varepsilon^{\alpha^*}=(\varphi_\varepsilon^{\alpha^*})_+ φ ε α ∗ = ( φ ε α ∗ ) + φ ε α ∗ \varphi_\varepsilon^{\alpha^*} φ ε α ∗ Γ D ε \Gamma_D^\varepsilon Γ D ε H Γ D ε 1 ( Ω ε ) H^1_{\Gamma_D^\varepsilon}(\Omega_\varepsilon) H Γ D ε 1 ( Ω ε )
By the same computation as in the normalization part of the proof of Lemma 4.1 (with the extra factor α ∗ \alpha^* α ∗
∥ φ ε α ∗ ∥ L 2 ( Ω ε ) 2 = ∣ Ω ε ∣ + O ( K ‾ ε ) . (39) \|\varphi_\varepsilon^{\alpha^*}\|_{L^2(\Omega_\varepsilon)}^2=|\Omega_\varepsilon|+O(\overline{K}_\varepsilon). \tag*{(39)} ∥ φ ε α ∗ ∥ L 2 ( Ω ε ) 2 = ∣ Ω ε ∣ + O ( K ε ) . (39) For the gradient, using ∇ φ ε α ∗ = − α ∗ ∑ k = 1 N ∇ ψ ε ( k ) \nabla\varphi_\varepsilon^{\alpha^*}=-\alpha^*\sum_{k=1}^N\nabla\psi_\varepsilon^{(k)} ∇ φ ε α ∗ = − α ∗ ∑ k = 1 N ∇ ψ ε ( k ) Lemma C.2 to bound ∣ ∇ x G ( x , y ) ∣ ≤ K ∣ x − y ∣ − 2 |\nabla_xG(x,y)|\leq K|x-y|^{-2} ∣ ∇ x G ( x , y ) ∣ ≤ K ∣ x − y ∣ − 2
∫ Ω ε ∣ ∇ φ ε α ∗ ∣ 2 ≤ ∫ Ω ε ∣ ∇ φ ε α ∗ ∣ 2 ≤ N ( α ∗ ) 2 ∑ k = 1 N π 2 ∣ log w ε ( k ) ∣ 2 ∫ Ω ε ( ∫ H k K ∣ x − y ∣ − 2 d y ) 2 d x . \int_{\Omega_\varepsilon}|\nabla\varphi_\varepsilon^{\alpha^*}|^2\leq\int_{\Omega_\varepsilon}|\nabla\varphi_\varepsilon^{\alpha^*}|^2\leq N(\alpha^*)^2\sum_{k=1}^N\frac{\pi^2}{|\log w_\varepsilon^{(k)}|^2}\int_{\Omega_\varepsilon}\left(\int_{\mathcal{H}_k}K|x-y|^{-2}\,dy\right)^2dx. ∫ Ω ε ∣∇ φ ε α ∗ ∣ 2 ≤ ∫ Ω ε ∣∇ φ ε α ∗ ∣ 2 ≤ N ( α ∗ ) 2 k = 1 ∑ N ∣ log w ε ( k ) ∣ 2 π 2 ∫ Ω ε ( ∫ H k K ∣ x − y ∣ − 2 d y ) 2 d x . Fix k k k H k \mathcal{H}_k H k ∫ H k ∣ x − y ∣ − 2 d y ≤ C d ( x , H k ) − 1 \int_{\mathcal{H}_k}|x-y|^{-2}\,dy\leq C d(x,\mathcal{H}_k)^{-1} ∫ H k ∣ x − y ∣ − 2 d y ≤ C d ( x , H k ) − 1 H k \mathcal{H}_k H k a ε ( k ) − ℓ k = ( w ε ( k ) ) 2 / ( a ε ( k ) + ℓ k ) a_\varepsilon^{(k)}-\ell_k=(w_\varepsilon^{(k)})^2/(a_\varepsilon^{(k)}+\ell_k) a ε ( k ) − ℓ k = ( w ε ( k ) ) 2 / ( a ε ( k ) + ℓ k ) (8) , so that d ( x , H k ) ≥ c ( w ε ( k ) ) 2 d(x,\mathcal{H}_k)\geq c(w_\varepsilon^{(k)})^2 d ( x , H k ) ≥ c ( w ε ( k ) ) 2 x ∈ Ω ε x\in\Omega_\varepsilon x ∈ Ω ε ∣ { x ∈ R 3 ∣ d ( x , H k ) < t } ∣ ≤ C t 2 |\{x\in\mathbb{R}^3\mid d(x,\mathcal{H}_k)<t\}|\leq Ct^2 ∣ { x ∈ R 3 ∣ d ( x , H k ) < t } ∣ ≤ C t 2
∫ Ω ε ( ∫ H k K ∣ x − y ∣ − 2 d y ) 2 d x ≤ C ∫ Ω ε d ( x , H k ) − 2 d x ≤ C ∫ c ( w ε ( k ) ) 2 r 0 d t t + C = O ( ∣ log w ε ( k ) ∣ ) , \int_{\Omega_\varepsilon}\left(\int_{\mathcal{H}_k}K|x-y|^{-2}\,dy\right)^2dx\leq C\int_{\Omega_\varepsilon}d(x,\mathcal{H}_k)^{-2}dx\leq C\int_{c(w_\varepsilon^{(k)})^2}^{r_0}\frac{dt}{t}+C=O\left(|\log w_\varepsilon^{(k)}|\right), ∫ Ω ε ( ∫ H k K ∣ x − y ∣ − 2 d y ) 2 d x ≤ C ∫ Ω ε d ( x , H k ) − 2 d x ≤ C ∫ c ( w ε ( k ) ) 2 r 0 t d t + C = O ( ∣ log w ε ( k ) ∣ ) , for some fixed $r_0>0. Plugging this bound into the previous display,
∫ Ω ε ∣ ∇ φ ε α ∗ ∣ 2 = O ( ∑ k = 1 N ∣ log w ε ( k ) ∣ ∣ log w ε ( k ) ∣ 2 ) = O ( K ‾ ε ) . (40) \int_{\Omega_\varepsilon}|\nabla\varphi_\varepsilon^{\alpha^*}|^2=O\left(\sum_{k=1}^N\frac{|\log w_\varepsilon^{(k)}|}{|\log w_\varepsilon^{(k)}|^2}\right)=O(\overline{K}_\varepsilon). \tag*{(40)} ∫ Ω ε ∣∇ φ ε α ∗ ∣ 2 = O ( k = 1 ∑ N ∣ log w ε ( k ) ∣ 2 ∣ log w ε ( k ) ∣ ) = O ( K ε ) . (40) As φ ε α ∗ \varphi_\varepsilon^{\alpha^*} φ ε α ∗ H Γ D ε 1 ( Ω ε ) H^1_{\Gamma_D^\varepsilon}(\Omega_\varepsilon) H Γ D ε 1 ( Ω ε ) (39) and (40) into (38) shows that
λ ε ≤ O ( K ‾ ε ) 1 + O ( K ‾ ε ) = O ( K ‾ ε ) → ε → 0 0. \lambda_\varepsilon\leq\frac{O(\overline{K}_\varepsilon)}{1+O(\overline{K}_\varepsilon)}=O(\overline{K}_\varepsilon)\xrightarrow{\varepsilon\to0}0. λ ε ≤ 1 + O ( K ε ) O ( K ε ) = O ( K ε ) ε → 0 0. Having checked both hypotheses (Eig) and (PDE)–(Norm) of Hypothesis 2.1, Theorem 2.2 together with Lemma 4.1 directly give (36) and (37) . □
Tubular holes
In this section we consider the narrow escape problem for the hole shape defined in (Type D). Unlike in the previous sections, the functions ψ ε ( k ) \psi_{\varepsilon}^{(k)} ψ ε ( k ) ψ ε ( k ) \psi_{\varepsilon}^{(k)} ψ ε ( k ) § 5.1 as the solution of a Neumann problem where a flux Φ ε ( k ) \Phi_{\varepsilon}^{(k)} Φ ε ( k ) Ω ε \Omega_{\varepsilon} Ω ε k k k Γ ε k \Gamma_{\varepsilon}^{k} Γ ε k Proposition 5.1 by computing the energy matrix of the family ( ψ ε ( k ) ) k = 1 N (\psi_{\varepsilon}^{(k)})_{k=1}^{N} ( ψ ε ( k ) ) k = 1 N § 5.2 , we read the means of ψ ε ( k ) \psi_{\varepsilon}^{(k)} ψ ε ( k ) Proposition 5.2 ). We conclude with Proposition 5.3 in § 5.3 .
Definition of the quasi-mode
Throughout this section, with a slight abuse of notation, we still denote q ε ( u , v ) = ∫ Ω ε ∇ u ⋅ ∇ v q_{\varepsilon}(u,v)=\int_{\Omega_{\varepsilon}}\nabla u\cdot\nabla v q ε ( u , v ) = ∫ Ω ε ∇ u ⋅ ∇ v (5) , now for u , v ∈ H 1 ( Ω ε ) u,v\in H^{1}(\Omega_{\varepsilon}) u , v ∈ H 1 ( Ω ε )
For k ∈ { 1 , … , N } k\in\{1,\ldots,N\} k ∈ { 1 , … , N } Φ ∈ R \Phi\in\mathbb{R} Φ ∈ R Φ \Phi Φ Ω ε \Omega_{\varepsilon} Ω ε k k k
{ Δ ψ Φ ( k ) = Φ ∣ Ω ε ∣ in Ω ε , ∂ n ψ Φ ( k ) = 0 on ∂ Ω ε ∖ Γ ε k , ∂ n ψ Φ ( k ) = Φ ∣ Γ ε k ∣ on Γ ε k . (41) \begin{cases}
\Delta\psi_{\Phi}^{(k)}=\dfrac{\Phi}{|\Omega_{\varepsilon}|} & \text{in }\Omega_{\varepsilon},\\
\partial_{n}\psi_{\Phi}^{(k)}=0 & \text{on }\partial\Omega_{\varepsilon}\setminus\Gamma_{\varepsilon}^{k},\\
\partial_{n}\psi_{\Phi}^{(k)}=\dfrac{\Phi}{|\Gamma_{\varepsilon}^{k}|} & \text{on }\Gamma_{\varepsilon}^{k}. \tag*{(41)}
\end{cases} ⎩ ⎨ ⎧ Δ ψ Φ ( k ) = ∣ Ω ε ∣ Φ ∂ n ψ Φ ( k ) = 0 ∂ n ψ Φ ( k ) = ∣ Γ ε k ∣ Φ in Ω ε , on ∂ Ω ε ∖ Γ ε k , on Γ ε k . (41) A weak solution of (41) is a function ψ Φ ( k ) ∈ H 1 ( Ω ε ) \psi_{\Phi}^{(k)}\in H^{1}(\Omega_{\varepsilon}) ψ Φ ( k ) ∈ H 1 ( Ω ε )
∀ v ∈ H 1 ( Ω ε ) , q ε ( ψ Φ ( k ) , v ) = Φ l k ( v ) = − Φ ∣ Ω ε ∣ ∫ Ω ε v + Φ ∣ Γ ε k ∣ ∫ Γ ε k v , (42) \forall v\in H^{1}(\Omega_{\varepsilon}),\qquad q_{\varepsilon}(\psi_{\Phi}^{(k)},v)=\Phi l_{k}(v)=-\frac{\Phi}{|\Omega_{\varepsilon}|}\int_{\Omega_{\varepsilon}}v+\frac{\Phi}{|\Gamma_{\varepsilon}^{k}|}\int_{\Gamma_{\varepsilon}^{k}}v, \tag*{(42)} ∀ v ∈ H 1 ( Ω ε ) , q ε ( ψ Φ ( k ) , v ) = Φ l k ( v ) = − ∣ Ω ε ∣ Φ ∫ Ω ε v + ∣ Γ ε k ∣ Φ ∫ Γ ε k v , (42) where l k l_{k} l k H 1 ( Ω ε ) H^{1}(\Omega_{\varepsilon}) H 1 ( Ω ε ) (42) . As the compatibility condition is verified in (41) , it has a unique zero-mean weak solution (see [51 Theorem 8.8 and Remark 8.11] ). By linearity, ψ Φ ( k ) = Φ ψ 1 ( k ) \psi_{\Phi}^{(k)}=\Phi\psi_{1}^{(k)} ψ Φ ( k ) = Φ ψ 1 ( k )
Take Φ = 1 \Phi=1 Φ = 1 v = ψ 1 ( k ) v=\psi_{1}^{(k)} v = ψ 1 ( k ) (42) , as ψ 1 ( k ) \psi_{1}^{(k)} ψ 1 ( k ) Ω ε \Omega_{\varepsilon} Ω ε
q ε ( ψ 1 ( k ) , ψ 1 ( k ) ) = 1 ∣ Γ ε k ∣ ∫ Γ ε k ψ 1 ( k ) , q_{\varepsilon}(\psi_{1}^{(k)},\psi_{1}^{(k)})=\frac{1}{|\Gamma_{\varepsilon}^{k}|}\int_{\Gamma_{\varepsilon}^{k}}\psi_{1}^{(k)}, q ε ( ψ 1 ( k ) , ψ 1 ( k ) ) = ∣ Γ ε k ∣ 1 ∫ Γ ε k ψ 1 ( k ) , hence its mean on Γ ε k \Gamma_{\varepsilon}^{k} Γ ε k ψ ε ( k ) \psi_{\varepsilon}^{(k)} ψ ε ( k ) Γ ε k \Gamma_{\varepsilon}^{k} Γ ε k 1 1 1 Φ ε ( k ) \Phi_{\varepsilon}^{(k)} Φ ε ( k )
Φ ε ( k ) = ∥ ∇ ψ 1 ( k ) ∥ L 2 ( Ω ε ) − 2 , so that 1 ∣ Γ ε k ∣ ∫ Γ ε k ψ Φ ε ( k ) ( k ) = 1. (43) \Phi_{\varepsilon}^{(k)}=\left\|\nabla\psi_{1}^{(k)}\right\|_{L^{2}(\Omega_{\varepsilon})}^{-2},
\qquad\text{so that}\qquad
\frac{1}{|\Gamma_{\varepsilon}^{k}|}\int_{\Gamma_{\varepsilon}^{k}}\psi_{\Phi_{\varepsilon}^{(k)}}^{(k)}=1. \tag*{(43)} Φ ε ( k ) = ∇ ψ 1 ( k ) L 2 ( Ω ε ) − 2 , so that ∣ Γ ε k ∣ 1 ∫ Γ ε k ψ Φ ε ( k ) ( k ) = 1. (43) We finally define the quasi-mode ψ ε ( k ) \psi_{\varepsilon}^{(k)} ψ ε ( k )
ψ ε ( k ) = ψ Φ ε ( k ) ( k ) , (44) \psi_{\varepsilon}^{(k)}=\psi_{\Phi_{\varepsilon}^{(k)}}^{(k)}, \tag*{(44)} ψ ε ( k ) = ψ Φ ε ( k ) ( k ) , (44) i.e., the unique zero-mean weak solution of (41) with Φ = Φ ε ( k ) \Phi=\Phi_{\varepsilon}^{(k)} Φ = Φ ε ( k )
q ε ( ψ ε ( k ) , ψ ε ( k ) ) = Φ ε ( k ) . (45) q_{\varepsilon}(\psi_{\varepsilon}^{(k)},\psi_{\varepsilon}^{(k)})=\Phi_{\varepsilon}^{(k)}. \tag*{(45)} q ε ( ψ ε ( k ) , ψ ε ( k ) ) = Φ ε ( k ) . (45) Moreover, taking v = ψ ε ( j ) v=\psi_{\varepsilon}^{(j)} v = ψ ε ( j ) (42) for some j ∈ { 1 , … , N } j\in\{1,\ldots,N\} j ∈ { 1 , … , N } ψ ε ( j ) \psi_{\varepsilon}^{(j)} ψ ε ( j ) Ω ε \Omega_{\varepsilon} Ω ε
q ε ( ψ ε ( k ) , ψ ε ( j ) ) = Φ ε ( k ) 1 ∣ Γ ε k ∣ ∫ Γ ε k ψ ε ( j ) . (46) q_{\varepsilon}(\psi_{\varepsilon}^{(k)},\psi_{\varepsilon}^{(j)})=\Phi_{\varepsilon}^{(k)}\frac{1}{|\Gamma_{\varepsilon}^{k}|}\int_{\Gamma_{\varepsilon}^{k}}\psi_{\varepsilon}^{(j)}. \tag*{(46)} q ε ( ψ ε ( k ) , ψ ε ( j ) ) = Φ ε ( k ) ∣ Γ ε k ∣ 1 ∫ Γ ε k ψ ε ( j ) . (46) Hence the symmetric energy matrix Q ε = ( q ε ( ψ ε ( i ) , ψ ε ( j ) ) ) 1 ≤ i , j ≤ N Q_{\varepsilon}=\left(q_{\varepsilon}(\psi_{\varepsilon}^{(i)},\psi_{\varepsilon}^{(j)})\right)_{1\leq i,j\leq N} Q ε = ( q ε ( ψ ε ( i ) , ψ ε ( j ) ) ) 1 ≤ i , j ≤ N Φ ε ( k ) \Phi_{\varepsilon}^{(k)} Φ ε ( k ) ψ ε ( k ) \psi_{\varepsilon}^{(k)} ψ ε ( k ) Φ ε ( k ) \Phi_{\varepsilon}^{(k)} Φ ε ( k )
Proposition 5.1. There are C , ε 0 > 0 C,\varepsilon_0>0 C , ε 0 > 0 ε < ε 0 \varepsilon<\varepsilon_0 ε < ε 0 j ≠ k ∈ { 1 , … , N } j\ne k\in\{1,\ldots,N\} j = k ∈ { 1 , … , N }
Φ ε ( k ) = ( r ε ( k ) ) 2 ∣ D k ∣ L k ( 1 + O ( ( r ε ( k ) ) 2 ) ) , ∣ q ε ( ψ ε ( j ) , ψ ε ( k ) ) ∣ ≤ C ( r ε ( j ) r ε ( k ) ) 3 / 2 . (47) \Phi_\varepsilon^{(k)}=(r_\varepsilon^{(k)})^2\frac{|D_{k}|}{L_k}\left(1+O((r_\varepsilon^{(k)})^2)\right),\qquad\left|q_\varepsilon(\psi_\varepsilon^{(j)},\psi_\varepsilon^{(k)})\right|\le C(r_\varepsilon^{(j)}r_\varepsilon^{(k)})^{3/2}. \tag*{(47)} Φ ε ( k ) = ( r ε ( k ) ) 2 L k ∣ D k ∣ ( 1 + O (( r ε ( k ) ) 2 ) ) , q ε ( ψ ε ( j ) , ψ ε ( k ) ) ≤ C ( r ε ( j ) r ε ( k ) ) 3/2 . (47) Proof. In this proof, C C C ε \varepsilon ε i ∈ { 1 , … , N } i\in\{1,\ldots,N\} i ∈ { 1 , … , N } (14) and (10) , ∣ h i ∣ ≤ s ε ( i ) = O ( ( r ε ( i ) ) 2 ) |h_i|\le s_\varepsilon^{(i)}=O((r_\varepsilon^{(i)})^2) ∣ h i ∣ ≤ s ε ( i ) = O (( r ε ( i ) ) 2 ) D i , ε D_{i,\varepsilon} D i , ε
∣ T i , ε ∣ = ∫ D i , ε ( L i − h i ( ρ ) ) d ρ = L i ∣ D i , ε ∣ ( 1 + O ( ( r ε ( i ) ) 2 ) ) . (48) |\mathcal{T}_{i,\varepsilon}|=\int_{D_{i,\varepsilon}}(L_i-h_i(\rho))\,\mathrm{d}\rho=L_i|D_{i,\varepsilon}|\left(1+O((r_\varepsilon^{(i)})^2)\right). \tag*{(48)} ∣ T i , ε ∣ = ∫ D i , ε ( L i − h i ( ρ )) d ρ = L i ∣ D i , ε ∣ ( 1 + O (( r ε ( i ) ) 2 ) ) . (48) Fix k ∈ { 1 , … , N } k\in\{1,\ldots,N\} k ∈ { 1 , … , N } Φ ε ( k ) \Phi_\varepsilon^{(k)} Φ ε ( k ) Q ε Q^\varepsilon Q ε
Upper bound. Let v ∈ H 1 ( Ω ε ) v\in H^1(\Omega_\varepsilon) v ∈ H 1 ( Ω ε ) l k ( v ) ≠ 0 l_k(v)\ne0 l k ( v ) = 0 v v v (42) with Φ = Φ ε ( k ) \Phi=\Phi_\varepsilon^{(k)} Φ = Φ ε ( k ) (45) give Φ ε ( k ) ∣ l k ( v ) ∣ = ∣ q ε ( ψ ε ( k ) , v ) ∣ ≤ Φ ε ( k ) q ε ( v , v ) \Phi_\varepsilon^{(k)}|l_k(v)|=|q_\varepsilon(\psi_\varepsilon^{(k)},v)|\le\sqrt{\Phi_\varepsilon^{(k)}}\sqrt{q_\varepsilon(v,v)} Φ ε ( k ) ∣ l k ( v ) ∣ = ∣ q ε ( ψ ε ( k ) , v ) ∣ ≤ Φ ε ( k ) q ε ( v , v )
Φ ε ( k ) ≤ q ε ( v , v ) l k ( v ) 2 . (49) \Phi_\varepsilon^{(k)}\le\frac{q_\varepsilon(v,v)}{l_k(v)^2}. \tag*{(49)} Φ ε ( k ) ≤ l k ( v ) 2 q ε ( v , v ) . (49) Let us now construct a v v v v = 0 v=0 v = 0 T k , ε \mathcal{T}_{k,\varepsilon} T k , ε ( ρ , s ) (\rho,s) ( ρ , s ) (11) ,
v ( ρ , s ) = s − h k ( ρ ) L k in T k , ε . v(\rho,s)=\frac{s-h_k(\rho)}{L_k}\qquad\text{in }\mathcal{T}_{k,\varepsilon}. v ( ρ , s ) = L k s − h k ( ρ ) in T k , ε . As v v v Σ k , ε \Sigma_{k,\varepsilon} Σ k , ε v ∈ H 1 ( Ω ε ) v\in H^1(\Omega_\varepsilon) v ∈ H 1 ( Ω ε ) Γ k ε = { s = L k } \Gamma_k^\varepsilon=\{s=L_k\} Γ k ε = { s = L k } v = ( L k − h k ( ρ ) ) / L k v=(L_k-h_k(\rho))/L_k v = ( L k − h k ( ρ )) / L k ∣ Γ k ε ∣ = ∣ D k , ε ∣ |\Gamma_k^\varepsilon|=|D_{k,\varepsilon}| ∣ Γ k ε ∣ = ∣ D k , ε ∣ Γ k ε \Gamma_k^\varepsilon Γ k ε ∣ T k , ε ∣ / ( L k ∣ D k , ε ∣ ) = 1 + O ( ( r ε ( k ) ) 2 ) |\mathcal{T}_{k,\varepsilon}|/(L_k|D_{k,\varepsilon}|)=1+O((r_\varepsilon^{(k)})^2) ∣ T k , ε ∣/ ( L k ∣ D k , ε ∣ ) = 1 + O (( r ε ( k ) ) 2 ) (48) . Moreover, as 0 ≤ v ≤ C 0\le v\le C 0 ≤ v ≤ C T k , ε \mathcal{T}_{k,\varepsilon} T k , ε ∣ T k , ε ∣ ≤ C ( r ε ( k ) ) 2 |\mathcal{T}_{k,\varepsilon}|\le C(r_\varepsilon^{(k)})^2 ∣ T k , ε ∣ ≤ C ( r ε ( k ) ) 2 v v v Ω ε \Omega_\varepsilon Ω ε C ( r ε ( k ) ) 2 C(r_\varepsilon^{(k)})^2 C ( r ε ( k ) ) 2 l k ( v ) = 1 + O ( ( r ε ( k ) ) 2 ) l_k(v)=1+O((r_\varepsilon^{(k)})^2) l k ( v ) = 1 + O (( r ε ( k ) ) 2 ) T k , ε \mathcal{T}_{k,\varepsilon} T k , ε ∇ v = ( n k − ∇ h k ) / L k \nabla v=(n_k-\nabla h_k)/L_k ∇ v = ( n k − ∇ h k ) / L k ∇ h k ⊥ n k \nabla h_k\perp n_k ∇ h k ⊥ n k ∣ ∇ v ∣ 2 = ( 1 + O ( ( r ε ( k ) ) 2 ) ) / L k 2 |\nabla v|^2=(1+O((r_\varepsilon^{(k)})^2))/L_k^2 ∣∇ v ∣ 2 = ( 1 + O (( r ε ( k ) ) 2 )) / L k 2 (10) , and q ε ( v , v ) = ∣ T k , ε ∣ ( 1 + O ( ( r ε ( k ) ) 2 ) ) / L k 2 q_\varepsilon(v,v)=|\mathcal{T}_{k,\varepsilon}|(1+O((r_\varepsilon^{(k)})^2))/L_k^2 q ε ( v , v ) = ∣ T k , ε ∣ ( 1 + O (( r ε ( k ) ) 2 )) / L k 2 (49) , and using (48) , we obtain
Φ ε ( k ) ≤ ∣ D k , ε ∣ L k ( 1 + C ( r ε ( k ) ) 2 ) . (50) \Phi_\varepsilon^{(k)}\le\frac{|D_{k,\varepsilon}|}{L_k}\left(1+C(r_\varepsilon^{(k)})^2\right). \tag*{(50)} Φ ε ( k ) ≤ L k ∣ D k , ε ∣ ( 1 + C ( r ε ( k ) ) 2 ) . (50) Lower bound. This part is based on an idea of Prager–Synge (see the original paper [46 ] and [7] for a modern presentation). Assume there exists F ∈ L 2 ( Ω ε ; R 3 ) F\in L^2(\Omega_\varepsilon;\mathbb{R}^3) F ∈ L 2 ( Ω ε ; R 3 )
∫ Ω ε F ⋅ ∇ v = l k ( v ) for every v ∈ H 1 ( Ω ε ) . (51) \int_{\Omega_\varepsilon}F\cdot\nabla v=l_k(v)\qquad\text{for every }v\in H^1(\Omega_\varepsilon). \tag*{(51)} ∫ Ω ε F ⋅ ∇ v = l k ( v ) for every v ∈ H 1 ( Ω ε ) . (51) Taking v = ψ ε ( k ) v=\psi_\varepsilon^{(k)} v = ψ ε ( k ) (51) and comparing it with (42) for Φ = Φ ε ( k ) \Phi=\Phi_\varepsilon^{(k)} Φ = Φ ε ( k )
Φ ε ( k ) ∫ Ω ε F ⋅ ∇ ψ ε ( k ) = Φ ε ( k ) l k ( ψ ε ( k ) ) = q ε ( ψ ε ( k ) , ψ ε ( k ) ) . \Phi_\varepsilon^{(k)}\int_{\Omega_\varepsilon}F\cdot\nabla\psi_\varepsilon^{(k)}=\Phi_\varepsilon^{(k)}l_k(\psi_\varepsilon^{(k)})=q_\varepsilon(\psi_\varepsilon^{(k)},\psi_\varepsilon^{(k)}). Φ ε ( k ) ∫ Ω ε F ⋅ ∇ ψ ε ( k ) = Φ ε ( k ) l k ( ψ ε ( k ) ) = q ε ( ψ ε ( k ) , ψ ε ( k ) ) . Hence, expanding the square and using (45) ,
∫ Ω ε ∣ ∇ ψ ε ( k ) − Φ ε ( k ) F ∣ 2 = q ε ( ψ ε ( k ) , ψ ε ( k ) ) − 2 q ε ( ψ ε ( k ) , ψ ε ( k ) ) + ( Φ ε ( k ) ) 2 ∫ Ω ε ∣ F ∣ 2 = ( Φ ε ( k ) ) 2 ∫ Ω ε ∣ F ∣ 2 − Φ ε ( k ) . (52) \begin{aligned}
\int_{\Omega_\varepsilon}|\nabla\psi_\varepsilon^{(k)}-\Phi_\varepsilon^{(k)}F|^2
&=q_\varepsilon(\psi_\varepsilon^{(k)},\psi_\varepsilon^{(k)})-2q_\varepsilon(\psi_\varepsilon^{(k)},\psi_\varepsilon^{(k)})+(\Phi_\varepsilon^{(k)})^2\int_{\Omega_\varepsilon}|F|^2\\
&=(\Phi_\varepsilon^{(k)})^2\int_{\Omega_\varepsilon}|F|^2-\Phi_\varepsilon^{(k)}. \tag*{(52)}
\end{aligned} ∫ Ω ε ∣∇ ψ ε ( k ) − Φ ε ( k ) F ∣ 2 = q ε ( ψ ε ( k ) , ψ ε ( k ) ) − 2 q ε ( ψ ε ( k ) , ψ ε ( k ) ) + ( Φ ε ( k ) ) 2 ∫ Ω ε ∣ F ∣ 2 = ( Φ ε ( k ) ) 2 ∫ Ω ε ∣ F ∣ 2 − Φ ε ( k ) . (52) Therefore, to show the desired lower bound, it suffices to construct F F F (51) and with ∥ F ∥ L 2 ( Ω ε ) 2 \|F\|_{L^2(\Omega_\varepsilon)}^2 ∥ F ∥ L 2 ( Ω ε ) 2 L k / ∣ D k , ε ∣ + C / ( r ε ( k ) ) 2 L_k/|D_{k,\varepsilon}|+C/(r_\varepsilon^{(k)})^2 L k /∣ D k , ε ∣ + C / ( r ε ( k ) ) 2
In each tube T i , ε \mathcal{T}_{i,\varepsilon} T i , ε ( ρ , s ) (\rho,s) ( ρ , s ) (11) , we define
F k = f k , i n i in T i , ε , with f k , i ( ρ , s ) = δ i k ∣ D k , ε ∣ L i − s ∣ Ω ε ∣ . (53) F_k=f_{k,i}n_i\quad\text{in }\mathcal{T}_{i,\varepsilon},\qquad\text{with}\qquad f_{k,i}(\rho,s)=\frac{\delta_{ik}}{|D_{k,\varepsilon}|}\frac{L_i-s}{|\Omega_\varepsilon|}. \tag*{(53)} F k = f k , i n i in T i , ε , with f k , i ( ρ , s ) = ∣ D k , ε ∣ δ ik ∣ Ω ε ∣ L i − s . (53) In Ω \Omega Ω F k = ∇ f ^ k F_k = \nabla\hat{f}_k F k = ∇ f ^ k f ^ k \hat{f}_k f ^ k f k , i f_{k,i} f k , i
{ Δ f ^ k = 1 ∣ Ω ε ∣ in Ω , ∂ n f ^ k = f k , i ( n i ⋅ n ) on Σ i , ε , i ∈ { 1 , … , N } , ∂ n f ^ k = 0 elsewhere on ∂ Ω . (54) \begin{cases}
\Delta\hat{f}_k = \dfrac{1}{|\Omega_\varepsilon|} & \text{in } \Omega,\\
\partial_n \hat{f}_k = f_{k,i}(n_i \cdot n) & \text{on } \Sigma_{i,\varepsilon},\quad i \in\{1,\ldots,N\},\\
\partial_n \hat{f}_k = 0 & \text{elsewhere on } \partial\Omega. \tag*{(54)}
\end{cases} ⎩ ⎨ ⎧ Δ f ^ k = ∣ Ω ε ∣ 1 ∂ n f ^ k = f k , i ( n i ⋅ n ) ∂ n f ^ k = 0 in Ω , on Σ i , ε , i ∈ { 1 , … , N } , elsewhere on ∂ Ω. (54) From [2, Proposition 2.3], since the sources in (54) are bounded, there exists a unique f ^ k ∈ W 1 , 4 ( Ω ) ⊂ C 0 ( Ω ‾ ) \hat{f}_k \in W^{1,4}(\Omega) \subset C^0(\overline{\Omega}) f ^ k ∈ W 1 , 4 ( Ω ) ⊂ C 0 ( Ω ) (54) if and only if
∑ i = 1 N ∫ Σ i , ε f k , i ( n i ⋅ n ) d σ = ∣ Ω ∣ ∣ Ω ε ∣ . (55) \sum_{i=1}^{N}\int_{\Sigma_{i,\varepsilon}} f_{k,i}(n_i \cdot n)\,\mathrm{d}\sigma= \frac{|\Omega|}{|\Omega_\varepsilon|}. \tag*{(55)} i = 1 ∑ N ∫ Σ i , ε f k , i ( n i ⋅ n ) d σ = ∣ Ω ε ∣ ∣Ω∣ . (55) Let us show that this condition is satisfied. For a fixed i ∈ { 1 , … , N } i \in\{1,\ldots,N\} i ∈ { 1 , … , N } Σ i , ε \Sigma_{i,\varepsilon} Σ i , ε (12) , one has d σ = 1 + ∣ ∇ h i ∣ 2 d ρ \mathrm{d}\sigma= \sqrt{1+|\nabla h_i|^2}\,\mathrm{d}\rho d σ = 1 + ∣∇ h i ∣ 2 d ρ Σ i , ε \Sigma_{i,\varepsilon} Σ i , ε Ω \Omega Ω { s < h i ( ρ ) } \{s<h_i(\rho)\} { s < h i ( ρ )} n = ( n i − ∇ h i ) / 1 + ∣ ∇ h i ∣ 2 n=(n_i-\nabla h_i)/\sqrt{1+|\nabla h_i|^2} n = ( n i − ∇ h i ) / 1 + ∣∇ h i ∣ 2 ∇ h i ⊥ n i \nabla h_i \perp n_i ∇ h i ⊥ n i ( n i ⋅ n ) d σ = d ρ (n_i\cdot n)\,\mathrm{d}\sigma=\mathrm{d}\rho ( n i ⋅ n ) d σ = d ρ
∫ Σ i , ε f k , i ( n i ⋅ n ) d σ = ∫ D i , ε ( δ i k ∣ D k , ε ∣ − L i − h i ( ρ ) ∣ Ω ε ∣ ) d ρ = δ i k − ∣ T i , ε ∣ ∣ Ω ε ∣ . \int_{\Sigma_{i,\varepsilon}} f_{k,i}(n_i\cdot n)\,\mathrm{d}\sigma
=\int_{D_{i,\varepsilon}}\left(\frac{\delta_{ik}}{|D_{k,\varepsilon}|}-\frac{L_i-h_i(\rho)}{|\Omega_\varepsilon|}\right)\mathrm{d}\rho
=\delta_{ik}-\frac{|\mathcal{T}_{i,\varepsilon}|}{|\Omega_\varepsilon|}. ∫ Σ i , ε f k , i ( n i ⋅ n ) d σ = ∫ D i , ε ( ∣ D k , ε ∣ δ ik − ∣ Ω ε ∣ L i − h i ( ρ ) ) d ρ = δ ik − ∣ Ω ε ∣ ∣ T i , ε ∣ . Summing over i i i ∑ i = 1 N ∣ T i , ε ∣ = ∣ Ω ε ∣ − ∣ Ω ∣ \sum_{i=1}^{N}|\mathcal{T}_{i,\varepsilon}|=|\Omega_\varepsilon|-|\Omega| ∑ i = 1 N ∣ T i , ε ∣ = ∣ Ω ε ∣ − ∣Ω∣ (55) .
Let us now show that F k F_k F k (51) . Let v ∈ H 1 ( Ω ε ) v\in H^1(\Omega_\varepsilon) v ∈ H 1 ( Ω ε ) Ω \Omega Ω T i , ε \mathcal{T}_{i,\varepsilon} T i , ε i ∈ { 1 , … , N } i\in\{1,\ldots,N\} i ∈ { 1 , … , N } F k F_k F k T i , ε \mathcal{T}_{i,\varepsilon} T i , ε ∇ ⋅ F k = ∂ s f k , i = 1 / ∣ Ω ε ∣ \nabla\cdot F_k=\partial_s f_{k,i}=1/|\Omega_\varepsilon| ∇ ⋅ F k = ∂ s f k , i = 1/∣ Ω ε ∣ F k F_k F k T i , ε \mathcal{T}_{i,\varepsilon} T i , ε T i , ε \mathcal{T}_{i,\varepsilon} T i , ε n i n_i n i Γ i ε \Gamma_i^\varepsilon Γ i ε − n -n − n Σ i , ε \Sigma_{i,\varepsilon} Σ i , ε f k , i = δ i k / ∣ D k , ε ∣ f_{k,i}=\delta_{ik}/|D_{k,\varepsilon}| f k , i = δ ik /∣ D k , ε ∣ Γ i ε \Gamma_i^\varepsilon Γ i ε
∫ T i , ε F k ⋅ ∇ v = − 1 ∣ Ω ε ∣ ∫ T i , ε v + δ i k ∣ D k , ε ∣ ∫ Γ i ε v − ∫ Σ i , ε f k , i ( n i ⋅ n ) v d σ . (56) \int_{\mathcal{T}_{i,\varepsilon}} F_k\cdot\nabla v
=-\frac{1}{|\Omega_\varepsilon|}\int_{\mathcal{T}_{i,\varepsilon}}v
+\frac{\delta_{ik}}{|D_{k,\varepsilon}|}\int_{\Gamma_i^\varepsilon}v
-\int_{\Sigma_{i,\varepsilon}}f_{k,i}(n_i\cdot n)v\,\mathrm{d}\sigma. \tag*{(56)} ∫ T i , ε F k ⋅ ∇ v = − ∣ Ω ε ∣ 1 ∫ T i , ε v + ∣ D k , ε ∣ δ ik ∫ Γ i ε v − ∫ Σ i , ε f k , i ( n i ⋅ n ) v d σ . (56) In Ω \Omega Ω v ∣ Ω ∈ H 1 ( Ω ) v|_\Omega\in H^1(\Omega) v ∣ Ω ∈ H 1 ( Ω ) (54) reads
∫ Ω F k ⋅ ∇ v = ∫ Ω ∇ f ^ k ⋅ ∇ v = − 1 ∣ Ω ε ∣ ∫ Ω v + ∑ i = 1 N ∫ Σ i , ε f k , i ( n i ⋅ n ) v d σ . (57) \int_{\Omega}F_k\cdot\nabla v
=\int_{\Omega}\nabla\hat{f}_k\cdot\nabla v
=-\frac{1}{|\Omega_\varepsilon|}\int_{\Omega}v
+\sum_{i=1}^{N}\int_{\Sigma_{i,\varepsilon}}f_{k,i}(n_i\cdot n)v\,\mathrm{d}\sigma. \tag*{(57)} ∫ Ω F k ⋅ ∇ v = ∫ Ω ∇ f ^ k ⋅ ∇ v = − ∣ Ω ε ∣ 1 ∫ Ω v + i = 1 ∑ N ∫ Σ i , ε f k , i ( n i ⋅ n ) v d σ . (57) Summing (57) and (56) over i i i (51) , as ∣ Γ k ε ∣ = ∣ D k , ε ∣ |\Gamma_k^\varepsilon|=|D_{k,\varepsilon}| ∣ Γ k ε ∣ = ∣ D k , ε ∣
It remains to bound ∥ F k ∥ L 2 ( Ω ε ) 2 \|F_k\|_{L^2(\Omega_\varepsilon)}^2 ∥ F k ∥ L 2 ( Ω ε ) 2 i ∈ { 1 , … , N } i\in\{1,\ldots,N\} i ∈ { 1 , … , N } ( ρ , s ) (\rho,s) ( ρ , s ) ρ ∈ D i , ε \rho\in D_{i,\varepsilon} ρ ∈ D i , ε h i ( ρ ) ≤ s ≤ L i h_i(\rho)\leq s\leq L_i h i ( ρ ) ≤ s ≤ L i 0 ≤ L i − s ≤ L i + s ε ( i ) ≤ C 0\leq L_i-s\leq L_i+s_\varepsilon^{(i)}\leq C 0 ≤ L i − s ≤ L i + s ε ( i ) ≤ C ∣ D k , ε ∣ = ( r ε ( k ) ) 2 ∣ D k ∣ |D_{k,\varepsilon}|=(r_\varepsilon^{(k)})^2|D_k| ∣ D k , ε ∣ = ( r ε ( k ) ) 2 ∣ D k ∣
∣ f k , i − δ i k ∣ D k , ε ∣ ∣ ≤ C , and ∣ f k , i ∣ ≤ δ i k ∣ D k , ε ∣ + C ≤ C ( r ε ( i ) ) 2 . (58) \left|f_{k,i}-\frac{\delta_{ik}}{|D_{k,\varepsilon}|}\right|\leq C,\qquad\text{and}\qquad|f_{k,i}|\leq\frac{\delta_{ik}}{|D_{k,\varepsilon}|}+C\leq\frac{C}{(r_\varepsilon^{(i)})^2}. \tag*{(58)} f k , i − ∣ D k , ε ∣ δ ik ≤ C , and ∣ f k , i ∣ ≤ ∣ D k , ε ∣ δ ik + C ≤ ( r ε ( i ) ) 2 C . (58) Integrating ∣ f k , i ∣ 2 |f_{k,i}|^2 ∣ f k , i ∣ 2 T i , ε \mathcal{T}_{i,\varepsilon} T i , ε ∣ T i , ε ∣ ≤ C ( r ε ( i ) ) 2 |\mathcal{T}_{i,\varepsilon}|\leq C(r_\varepsilon^{(i)})^2 ∣ T i , ε ∣ ≤ C ( r ε ( i ) ) 2 (48) , and summing over i i i
∑ i = 1 N ∫ T i , ε ∣ F k ∣ 2 ≤ ∣ T k , ε ∣ ∣ D k , ε ∣ 2 + C ≤ L k ∣ D k , ε ∣ + C . (59) \sum_{i=1}^{N}\int_{\mathcal{T}_{i,\varepsilon}}|F_k|^2
\leq\frac{|\mathcal{T}_{k,\varepsilon}|}{|D_{k,\varepsilon}|^2}+C
\leq\frac{L_k}{|D_{k,\varepsilon}|}+C. \tag*{(59)} i = 1 ∑ N ∫ T i , ε ∣ F k ∣ 2 ≤ ∣ D k , ε ∣ 2 ∣ T k , ε ∣ + C ≤ ∣ D k , ε ∣ L k + C . (59) Now for the bulk part, taking v = f ^ k v=\hat{f}_k v = f ^ k (54) , as f ^ k \hat{f}_k f ^ k Ω \Omega Ω ( n i ⋅ n ) d σ = d ρ (n_i\cdot n)\,\mathrm{d}\sigma=\mathrm{d}\rho ( n i ⋅ n ) d σ = d ρ Σ i , ε \Sigma_{i,\varepsilon} Σ i , ε ∫ D i , ε ∣ f k , i ∣ ≤ δ i k + C ∣ D i , ε ∣ \int_{D_{i,\varepsilon}}|f_{k,i}|\leq\delta_{ik}+C|D_{i,\varepsilon}| ∫ D i , ε ∣ f k , i ∣ ≤ δ ik + C ∣ D i , ε ∣ (58) ,
∫ Ω ∣ F k ∣ 2 = ∑ i = 1 N ∫ Σ i , ε f k , i f ^ k ( n i ⋅ n ) d σ ≤ C ∥ f ^ k ∥ L ∞ ( Ω ) . (60) \int_{\Omega}|F_k|^2=\sum_{i=1}^{N}\int_{\Sigma_{i,\varepsilon}}f_{k,i}\hat{f}_k(n_i\cdot n)\,\mathrm{d}\sigma\leq C\|\hat{f}_k\|_{L^\infty(\Omega)}. \tag*{(60)} ∫ Ω ∣ F k ∣ 2 = i = 1 ∑ N ∫ Σ i , ε f k , i f ^ k ( n i ⋅ n ) d σ ≤ C ∥ f ^ k ∥ L ∞ ( Ω ) . (60) By Lemma C.4 , f ^ k ( x ) = ∫ ∂ Ω G ( x , y ) g ( y ) d σ ( y ) \hat{f}_k(x)=\int_{\partial\Omega}G(x,y)g(y)\,\mathrm{d}\sigma(y) f ^ k ( x ) = ∫ ∂ Ω G ( x , y ) g ( y ) d σ ( y ) x ∈ Ω x\in\Omega x ∈ Ω G G G Definition C.1 and g = ∂ n f ^ k g=\partial_n\hat{f}_k g = ∂ n f ^ k (54) . Hence, by Lemma C.2 and (58) ,
∣ f ^ k ( x ) ∣ ≤ C ∑ i = 1 N ( δ i k ∣ D k , ε ∣ + 1 ) ∫ Σ i , ε d σ ( y ) ∣ x − y ∣ , x ∈ Ω . (61) \lvert\hat{f}_k(x)\rvert\leq C\sum_{i=1}^{N}\left(\frac{\delta_{ik}}{\lvert D_{k,\varepsilon}\rvert}+1\right)\int_{\Sigma_{i,\varepsilon}}\frac{\mathrm{d}\sigma(y)}{\lvert x-y\rvert},\qquad x\in\Omega. \tag*{(61)} ∣ f ^ k ( x )∣ ≤ C i = 1 ∑ N ( ∣ D k , ε ∣ δ ik + 1 ) ∫ Σ i , ε ∣ x − y ∣ d σ ( y ) , x ∈ Ω. (61) To compute the integral in (61) , fix i ∈ { 1 , … , N } i\in\{1,\ldots,N\} i ∈ { 1 , … , N } x ∈ Ω x\in\Omega x ∈ Ω ρ x \rho_x ρ x x − x ( i ) x-x^{(i)} x − x ( i ) T x ( i ) ( ∂ Ω ) T_{x^{(i)}}(\partial\Omega) T x ( i ) ( ∂ Ω ) y = x ( i ) + ρ + h i ( ρ ) n i ∈ Σ i , ε y=x^{(i)}+\rho+h_i(\rho)n_i\in\Sigma_{i,\varepsilon} y = x ( i ) + ρ + h i ( ρ ) n i ∈ Σ i , ε ∣ x − y ∣ ≥ ∣ ρ x − ρ ∣ \lvert x-y\rvert\geq\lvert\rho_x-\rho\rvert ∣ x − y ∣ ≥ ∣ ρ x − ρ ∣ ρ ↦ ∣ ρ x − ρ ∣ − 1 \rho\mapsto\lvert\rho_x-\rho\rvert^{-1} ρ ↦ ∣ ρ x − ρ ∣ − 1 ρ x \rho_x ρ x D i , ε D_{i,\varepsilon} D i , ε B ( ρ x , a i ) B(\rho_x,a_i) B ( ρ x , a i ) π a i 2 = ∣ D i , ε ∣ = ( r ε ( i ) ) 2 ∣ D i ∣ \pi a_i^2=\lvert D_{i,\varepsilon}\rvert=(r_\varepsilon^{(i)})^2\lvert D_i\rvert π a i 2 = ∣ D i , ε ∣ = ( r ε ( i ) ) 2 ∣ D i ∣ d σ ≤ 2 d ρ \mathrm{d}\sigma\leq2\,\mathrm{d}\rho d σ ≤ 2 d ρ ε \varepsilon ε
∫ Σ i , ε d σ ( y ) ∣ x − y ∣ ≤ 2 ∫ D i , ε d ρ ∣ ρ x − ρ ∣ ≤ 2 ∫ B ( ρ x , a i ) d ρ ∣ ρ x − ρ ∣ = 4 π a i = 4 π ∣ D i ∣ r ε ( i ) . \int_{\Sigma_{i,\varepsilon}}\frac{\mathrm{d}\sigma(y)}{\lvert x-y\rvert}\leq2\int_{D_{i,\varepsilon}}\frac{\mathrm{d}\rho}{\lvert\rho_x-\rho\rvert}\leq2\int_{B(\rho_x,a_i)}\frac{\mathrm{d}\rho}{\lvert\rho_x-\rho\rvert}=4\pi a_i=4\sqrt{\pi\lvert D_i\rvert}\,r_\varepsilon^{(i)}. ∫ Σ i , ε ∣ x − y ∣ d σ ( y ) ≤ 2 ∫ D i , ε ∣ ρ x − ρ ∣ d ρ ≤ 2 ∫ B ( ρ x , a i ) ∣ ρ x − ρ ∣ d ρ = 4 π a i = 4 π ∣ D i ∣ r ε ( i ) . Plugging this into (61) gives ∥ f ^ k ∥ L ∞ ( Ω ) ≤ C / r ε ( k ) \lVert\hat{f}_k\rVert_{L^\infty(\Omega)}\leq C/r_\varepsilon^{(k)} ∥ f ^ k ∥ L ∞ ( Ω ) ≤ C / r ε ( k ) (60) becomes
∫ Ω ∣ F k ∣ 2 = ∫ Ω ∣ ∇ f ^ k ∣ 2 ≤ C r ε ( k ) . (62) \int_{\Omega}\lvert F_k\rvert^2=\int_{\Omega}\lvert\nabla\hat{f}_k\rvert^2\leq\frac{C}{r_\varepsilon^{(k)}}. \tag*{(62)} ∫ Ω ∣ F k ∣ 2 = ∫ Ω ∣ ∇ f ^ k ∣ 2 ≤ r ε ( k ) C . (62) Finally, the combination of (59) and (62) gives, as ∣ D k , ε ∣ = ( r ε ( k ) ) 2 ∣ D k ∣ \lvert D_{k,\varepsilon}\rvert=(r_\varepsilon^{(k)})^2\lvert D_k\rvert ∣ D k , ε ∣ = ( r ε ( k ) ) 2 ∣ D k ∣
∥ F k ∥ L 2 ( Ω ε ) 2 ≤ L k ∣ D k , ε ∣ + C r ε ( k ) ≤ L k ∣ D k , ε ∣ ( 1 + C r ε ( k ) ) , so that Φ ε ( k ) ≥ ∣ D k , ε ∣ L k ( 1 − C r ε ( k ) ) . \lVert F_k\rVert_{L^2(\Omega_\varepsilon)}^2\leq\frac{L_k}{\lvert D_{k,\varepsilon}\rvert}+\frac{C}{r_\varepsilon^{(k)}}\leq\frac{L_k}{\lvert D_{k,\varepsilon}\rvert}\left(1+Cr_\varepsilon^{(k)}\right),\qquad\text{so that}\qquad\Phi_\varepsilon^{(k)}\geq\frac{\lvert D_{k,\varepsilon}\rvert}{L_k}\left(1-Cr_\varepsilon^{(k)}\right). ∥ F k ∥ L 2 ( Ω ε ) 2 ≤ ∣ D k , ε ∣ L k + r ε ( k ) C ≤ ∣ D k , ε ∣ L k ( 1 + C r ε ( k ) ) , so that Φ ε ( k ) ≥ L k ∣ D k , ε ∣ ( 1 − C r ε ( k ) ) . Together with (50) , this gives the first estimate of (47) . Moreover, (52) with F = F k F=F_k F = F k (50) and the previous bound on ∥ F k ∥ L 2 ( Ω ε ) \lVert F_k\rVert_{L^2(\Omega_\varepsilon)} ∥ F k ∥ L 2 ( Ω ε )
∥ ∇ ψ ε ( k ) − Φ ε ( k ) F k ∥ L 2 ( Ω ε ) 2 = Φ ε ( k ) ( Φ ε ( k ) ∥ F k ∥ L 2 ( Ω ε ) 2 − 1 ) ≤ Φ ε ( k ) ( ( 1 + C ( r ε ( k ) ) 2 ) ( 1 + C r ε ( k ) ) − 1 ) ≤ C ( r ε ( k ) ) 3 . (63) \left\lVert\nabla\psi_\varepsilon^{(k)}-\Phi_\varepsilon^{(k)}F_k\right\rVert_{L^2(\Omega_\varepsilon)}^2=\Phi_\varepsilon^{(k)}\left(\Phi_\varepsilon^{(k)}\lVert F_k\rVert_{L^2(\Omega_\varepsilon)}^2-1\right)\leq\Phi_\varepsilon^{(k)}\left(\left(1+C(r_\varepsilon^{(k)})^2\right)\left(1+Cr_\varepsilon^{(k)}\right)-1\right)\leq C(r_\varepsilon^{(k)})^3. \tag*{(63)} ∇ ψ ε ( k ) − Φ ε ( k ) F k L 2 ( Ω ε ) 2 = Φ ε ( k ) ( Φ ε ( k ) ∥ F k ∥ L 2 ( Ω ε ) 2 − 1 ) ≤ Φ ε ( k ) ( ( 1 + C ( r ε ( k ) ) 2 ) ( 1 + C r ε ( k ) ) − 1 ) ≤ C ( r ε ( k ) ) 3 . (63) Interactions. Let j ≠ k ∈ { 1 , … , N } j\neq k\in\{1,\ldots,N\} j = k ∈ { 1 , … , N } (51) and (42) , Φ ε ( k ) ∫ Ω ε F k ⋅ ∇ ψ ε ( j ) = Φ ε ( k ) l k ( ψ ε ( j ) ) = q ε ( ψ ε ( j ) , ψ ε ( k ) ) \Phi_\varepsilon^{(k)}\int_{\Omega_\varepsilon}F_k\cdot\nabla\psi_\varepsilon^{(j)}=\Phi_\varepsilon^{(k)}l_k(\psi_\varepsilon^{(j)})=q_\varepsilon(\psi_\varepsilon^{(j)},\psi_\varepsilon^{(k)}) Φ ε ( k ) ∫ Ω ε F k ⋅ ∇ ψ ε ( j ) = Φ ε ( k ) l k ( ψ ε ( j ) ) = q ε ( ψ ε ( j ) , ψ ε ( k ) ) j j j k k k (52) , expanding the product gives
∫ Ω ε ( ∇ ψ ε ( j ) − Φ ε ( j ) F j ) ⋅ ( ∇ ψ ε ( k ) − Φ ε ( k ) F k ) = q ε ( ψ ε ( j ) , ψ ε ( k ) ) − 2 q ε ( ψ ε ( j ) , ψ ε ( k ) ) + Φ ε ( j ) Φ ε ( k ) ∫ Ω ε F j ⋅ F k , \int_{\Omega_\varepsilon}\left(\nabla\psi_\varepsilon^{(j)}-\Phi_\varepsilon^{(j)}F_j\right)\cdot\left(\nabla\psi_\varepsilon^{(k)}-\Phi_\varepsilon^{(k)}F_k\right)=q_\varepsilon(\psi_\varepsilon^{(j)},\psi_\varepsilon^{(k)})-2q_\varepsilon(\psi_\varepsilon^{(j)},\psi_\varepsilon^{(k)})+\Phi_\varepsilon^{(j)}\Phi_\varepsilon^{(k)}\int_{\Omega_\varepsilon}F_j\cdot F_k, ∫ Ω ε ( ∇ ψ ε ( j ) − Φ ε ( j ) F j ) ⋅ ( ∇ ψ ε ( k ) − Φ ε ( k ) F k ) = q ε ( ψ ε ( j ) , ψ ε ( k ) ) − 2 q ε ( ψ ε ( j ) , ψ ε ( k ) ) + Φ ε ( j ) Φ ε ( k ) ∫ Ω ε F j ⋅ F k , that is,
q ε ( ψ ε ( j ) , ψ ε ( k ) ) = Φ ε ( j ) Φ ε ( k ) ∫ Ω ε F j ⋅ F k − ∫ Ω ε ( ∇ ψ ε ( j ) − Φ ε ( j ) F j ) ⋅ ( ∇ ψ ε ( k ) − Φ ε ( k ) F k ) . (64) q_\varepsilon(\psi_\varepsilon^{(j)},\psi_\varepsilon^{(k)})=\Phi_\varepsilon^{(j)}\Phi_\varepsilon^{(k)}\int_{\Omega_\varepsilon}F_j\cdot F_k-\int_{\Omega_\varepsilon}\left(\nabla\psi_\varepsilon^{(j)}-\Phi_\varepsilon^{(j)}F_j\right)\cdot\left(\nabla\psi_\varepsilon^{(k)}-\Phi_\varepsilon^{(k)}F_k\right). \tag*{(64)} q ε ( ψ ε ( j ) , ψ ε ( k ) ) = Φ ε ( j ) Φ ε ( k ) ∫ Ω ε F j ⋅ F k − ∫ Ω ε ( ∇ ψ ε ( j ) − Φ ε ( j ) F j ) ⋅ ( ∇ ψ ε ( k ) − Φ ε ( k ) F k ) . (64) Let us now bound the right-hand side in (64) . By the Cauchy–Schwarz inequality and (63) , the last integral is bounded by C ( r ε ( j ) r ε ( k ) ) 3 / 2 C(r_\varepsilon^{(j)}r_\varepsilon^{(k)})^{3/2} C ( r ε ( j ) r ε ( k ) ) 3/2 T i , ε T_{i,\varepsilon} T i , ε δ i j δ i k = 0 \delta_{ij}\delta_{ik}=0 δ ij δ ik = 0 (58) gives ∣ f j , i f k , i ∣ ≤ C ( r ε ( i ) ) − 2 \lvert f_{j,i}f_{k,i}\rvert\leq C(r_\varepsilon^{(i)})^{-2} ∣ f j , i f k , i ∣ ≤ C ( r ε ( i ) ) − 2 ∣ T i , ε ∣ ≤ C ( r ε ( i ) ) 2 \lvert T_{i,\varepsilon}\rvert\leq C(r_\varepsilon^{(i)})^2 ∣ T i , ε ∣ ≤ C ( r ε ( i ) ) 2
∣ ∫ T i , ε F j ⋅ F k ∣ ≤ C . \left\lvert\int_{T_{i,\varepsilon}}F_j\cdot F_k\right\rvert\leq C. ∫ T i , ε F j ⋅ F k ≤ C . In Ω \Omega Ω (62) give ∣ ∫ Ω ∇ f ^ j ⋅ ∇ f ^ k ∣ ≤ C ( r ε ( j ) r ε ( k ) ) − 1 / 2 \left\lvert\int_{\Omega}\nabla\hat{f}_j\cdot\nabla\hat{f}_k\right\rvert\leq C(r_\varepsilon^{(j)}r_\varepsilon^{(k)})^{-1/2} ∫ Ω ∇ f ^ j ⋅ ∇ f ^ k ≤ C ( r ε ( j ) r ε ( k ) ) − 1/2 Φ ε ( j ) Φ ε ( k ) ≤ C ( r ε ( j ) r ε ( k ) ) 2 \Phi_\varepsilon^{(j)}\Phi_\varepsilon^{(k)}\leq C(r_\varepsilon^{(j)}r_\varepsilon^{(k)})^2 Φ ε ( j ) Φ ε ( k ) ≤ C ( r ε ( j ) r ε ( k ) ) 2 (47) ,
∣ q ε ( ψ ε ( j ) , ψ ε ( k ) ) ∣ ≤ C ( r ε ( j ) r ε ( k ) ) 3 / 2 , \left\lvert q_\varepsilon(\psi_\varepsilon^{(j)},\psi_\varepsilon^{(k)})\right\rvert\leq C(r_\varepsilon^{(j)}r_\varepsilon^{(k)})^{3/2}, q ε ( ψ ε ( j ) , ψ ε ( k ) ) ≤ C ( r ε ( j ) r ε ( k ) ) 3/2 , which is the second estimate of (47) . □
The Dirichlet error
In this subsection, we control the trace of ψ ε ( k ) \psi_{\varepsilon}^{(k)} ψ ε ( k ) Γ j ε \Gamma_{j}^{\varepsilon} Γ j ε j ∈ { 1 , … , N } j \in\{1,\ldots,N\} j ∈ { 1 , … , N }
M j k ε = 1 ∣ Γ j ε ∣ ∫ Γ j ε ψ ε ( k ) (65) M_{jk}^{\varepsilon}=\frac{1}{\lvert\Gamma_{j}^{\varepsilon}\rvert}\int_{\Gamma_{j}^{\varepsilon}}\psi_{\varepsilon}^{(k)} \tag*{(65)} M j k ε = ∣ Γ j ε ∣ 1 ∫ Γ j ε ψ ε ( k ) (65) and its fluctuation around this mean. By (46) , the matrix of the means is M ε = diag ( Φ ε ( 1 ) , … , Φ ε ( N ) ) − 1 Q ε M^{\varepsilon}=\operatorname{diag}(\Phi_{\varepsilon}^{(1)},\ldots,\Phi_{\varepsilon}^{(N)})^{-1}Q^{\varepsilon} M ε = diag ( Φ ε ( 1 ) , … , Φ ε ( N ) ) − 1 Q ε M k k ε = 1 M_{kk}^{\varepsilon}=1 M k k ε = 1 (43) , and, for j ≠ k j\ne k j = k Proposition 5.1 gives
∣ M j k ε ∣ = ∣ Q j k ε ∣ Φ ε ( j ) ≤ C ( ( r ‾ ε ( k ) ) 3 r ‾ ε ( j ) ) 1 / 2 . \lvert M_{jk}^{\varepsilon}\rvert=\frac{\lvert Q_{jk}^{\varepsilon}\rvert}{\Phi_{\varepsilon}^{(j)}}\le C\left(\frac{(\overline{r}_{\varepsilon}^{(k)})^{3}}{\overline{r}_{\varepsilon}^{(j)}}\right)^{1/2}. ∣ M j k ε ∣ = Φ ε ( j ) ∣ Q j k ε ∣ ≤ C ( r ε ( j ) ( r ε ( k ) ) 3 ) 1/2 . Hence, denoting
η ε = ( ( r ‾ ε ) 3 min 1 ≤ j ≤ N r ‾ ε ( j ) ) 1 / 2 ≥ r ‾ ε , (66) \eta_{\varepsilon}=\left(\frac{(\overline{r}_{\varepsilon})^{3}}{\min_{1\le j\le N}\overline{r}_{\varepsilon}^{(j)}}\right)^{1/2}\ge\overline{r}_{\varepsilon}, \tag*{(66)} η ε = ( min 1 ≤ j ≤ N r ε ( j ) ( r ε ) 3 ) 1/2 ≥ r ε , (66) it holds that, for all j , k ∈ { 1 , … , N } j,k\in\{1,\ldots,N\} j , k ∈ { 1 , … , N }
∣ M j k ε − δ j k ∣ ≤ C η ε . (67) \lvert M_{jk}^{\varepsilon}-\delta_{jk}\rvert\le C\eta_{\varepsilon}. \tag*{(67)} ∣ M j k ε − δ j k ∣ ≤ C η ε . (67) For the fluctuations around the mean, we expand ψ ε ( k ) \psi_{\varepsilon}^{(k)} ψ ε ( k ) j j j ψ ε ( k ) \psi_{\varepsilon}^{(k)} ψ ε ( k )
Proposition 5.2. There is C > 0 C>0 C > 0 ε \varepsilon ε j , k ∈ { 1 , … , N } j,k\in\{1,\ldots,N\} j , k ∈ { 1 , … , N } ψ ε ( k ) \psi_{\varepsilon}^{(k)} ψ ε ( k ) Γ ‾ j ε \overline{\Gamma}_{j}^{\varepsilon} Γ j ε
∥ ψ ε ( k ) − M j k ε ∥ L ∞ ( Γ j ε ) ≤ C r ε ( j ) ∥ ∇ ψ ε ( k ) ∥ L 2 ( Ω ε ) ≤ C ( r ‾ ε ) 2 , so that ∥ ψ ε ( k ) − δ j k ∥ L ∞ ( Γ j ε ) ≤ C η ε , (68) \lVert\psi_{\varepsilon}^{(k)}-M_{jk}^{\varepsilon}\rVert_{L^{\infty}(\Gamma_{j}^{\varepsilon})}\le Cr_{\varepsilon}^{(j)}\lVert\nabla\psi_{\varepsilon}^{(k)}\rVert_{L^{2}(\Omega_{\varepsilon})}\le C(\overline{r}_{\varepsilon})^{2},\qquad\text{so that}\qquad\lVert\psi_{\varepsilon}^{(k)}-\delta_{jk}\rVert_{L^{\infty}(\Gamma_{j}^{\varepsilon})}\le C\eta_{\varepsilon}, \tag*{(68)} ∥ ψ ε ( k ) − M j k ε ∥ L ∞ ( Γ j ε ) ≤ C r ε ( j ) ∥ ∇ ψ ε ( k ) ∥ L 2 ( Ω ε ) ≤ C ( r ε ) 2 , so that ∥ ψ ε ( k ) − δ j k ∥ L ∞ ( Γ j ε ) ≤ C η ε , (68) with η ε \eta_{\varepsilon} η ε (66) .
Proof. Fix j , k ∈ { 1 , … , N } j,k\in\{1,\ldots,N\} j , k ∈ { 1 , … , N } ψ = ψ ε ( k ) \psi=\psi_{\varepsilon}^{(k)} ψ = ψ ε ( k ) r = r ε ( j ) r=r_{\varepsilon}^{(j)} r = r ε ( j ) ( ρ , s ) (\rho,s) ( ρ , s ) (11) in the j j j (68) follow from the first one: by (45) and (47) , ∥ ∇ ψ ∥ L 2 ( Ω ε ) = ( Φ ε ( k ) ) 1 / 2 ≤ C r ε ( k ) \lVert\nabla\psi\rVert_{L^{2}(\Omega_{\varepsilon})}=(\Phi_{\varepsilon}^{(k)})^{1/2}\le Cr_{\varepsilon}^{(k)} ∥ ∇ ψ ∥ L 2 ( Ω ε ) = ( Φ ε ( k ) ) 1/2 ≤ C r ε ( k ) r r ε ( k ) ≤ ( r ‾ ε ) 2 rr_{\varepsilon}^{(k)}\le(\overline{r}_{\varepsilon})^{2} r r ε ( k ) ≤ ( r ε ) 2 (67) give ∥ ψ − δ j k ∥ L ∞ ( Γ j ε ) ≤ C ( r ‾ ε ) 2 + ∣ M j k ε − δ j k ∣ ≤ C η ε \lVert\psi-\delta_{jk}\rVert_{L^{\infty}(\Gamma_{j}^{\varepsilon})}\le C(\overline{r}_{\varepsilon})^{2}+\lvert M_{jk}^{\varepsilon}-\delta_{jk}\rvert\le C\eta_{\varepsilon} ∥ ψ − δ j k ∥ L ∞ ( Γ j ε ) ≤ C ( r ε ) 2 + ∣ M j k ε − δ j k ∣ ≤ C η ε ( r ‾ ε ) 2 ≤ r ‾ ε ≤ η ε (\overline{r}_{\varepsilon})^{2}\le\overline{r}_{\varepsilon}\le\eta_{\varepsilon} ( r ε ) 2 ≤ r ε ≤ η ε ε \varepsilon ε (68) .
Transverse modes. Denote by 0 = μ 0 < μ 1 ≤ μ 2 ≤ … 0=\mu_{0}<\mu_{1}\le\mu_{2}\le\ldots 0 = μ 0 < μ 1 ≤ μ 2 ≤ … D j D_{j} D j ( ϕ n ) n ≥ 0 (\phi_{n})_{n\ge0} ( ϕ n ) n ≥ 0 L 2 ( D j ) L^{2}(D_{j}) L 2 ( D j ) ϕ 0 \phi_{0} ϕ 0 ϕ n ε ( ρ ) = r − 1 ϕ n ( ρ / r ) \phi_{n}^{\varepsilon}(\rho)=r^{-1}\phi_{n}(\rho/r) ϕ n ε ( ρ ) = r − 1 ϕ n ( ρ / r ) L 2 ( D j , ε ) L^{2}(D_{j,\varepsilon}) L 2 ( D j , ε ) D j , ε = r D j D_{j,\varepsilon}=rD_{j} D j , ε = r D j μ n / r 2 \mu_{n}/r^{2} μ n / r 2
∀ v ∈ H 1 ( D j , ε ) , ∫ D j , ε ∇ ϕ n ε ⋅ ∇ v = μ n r 2 ∫ D j , ε ϕ n ε v . (69) \forall v\in H^{1}(D_{j,\varepsilon}),\qquad\int_{D_{j,\varepsilon}}\nabla\phi_{n}^{\varepsilon}\cdot\nabla v=\frac{\mu_{n}}{r^{2}}\int_{D_{j,\varepsilon}}\phi_{n}^{\varepsilon}v. \tag*{(69)} ∀ v ∈ H 1 ( D j , ε ) , ∫ D j , ε ∇ ϕ n ε ⋅ ∇ v = r 2 μ n ∫ D j , ε ϕ n ε v . (69) The goal of this step is to decompose ψ \psi ψ ϕ n ε \phi_{n}^{\varepsilon} ϕ n ε s ε ( j ) = O ( r 2 ) s_{\varepsilon}^{(j)}=O(r^{2}) s ε ( j ) = O ( r 2 ) (14) and (10) , for ε \varepsilon ε T j , ε T_{j,\varepsilon} T j , ε s ∈ ( L j / 2 , L j ) s\in(L_{j}/2,L_{j}) s ∈ ( L j /2 , L j ) D j , ε = r D j D_{j,\varepsilon}=rD_{j} D j , ε = r D j n ≥ 1 n\ge1 n ≥ 1
a n ( s ) = ∫ D j , ε ψ ( ρ , s ) ϕ n ε ( ρ ) d ρ , s ∈ [ L j / 2 , L j ] . (70) a_{n}(s)=\int_{D_{j,\varepsilon}}\psi(\rho,s)\phi_{n}^{\varepsilon}(\rho)\,d\rho,\qquad s\in[L_{j}/2,L_{j}]. \tag*{(70)} a n ( s ) = ∫ D j , ε ψ ( ρ , s ) ϕ n ε ( ρ ) d ρ , s ∈ [ L j /2 , L j ] . (70) As ψ ∈ H 1 ( D j , ε × ( L j / 2 , L j ) ) \psi\in H^{1}(D_{j,\varepsilon}\times(L_{j}/2,L_{j})) ψ ∈ H 1 ( D j , ε × ( L j /2 , L j )) ψ \psi ψ ρ ∈ D j , ε \rho\in D_{j,\varepsilon} ρ ∈ D j , ε s ↦ ψ ( ρ , s ) s\mapsto\psi(\rho,s) s ↦ ψ ( ρ , s ) [ L j / 2 , L j ] [L_{j}/2,L_{j}] [ L j /2 , L j ] a n a_{n} a n C 0 ( [ L j / 2 , L j ] ) ∩ H 1 ( L j / 2 , L j ) C^{0}([L_{j}/2,L_{j}])\cap H^{1}(L_{j}/2,L_{j}) C 0 ([ L j /2 , L j ]) ∩ H 1 ( L j /2 , L j ) χ ∈ C ∞ ( [ L j / 2 , L j ] ) \chi\in C^{\infty}([L_j/2,L_j]) χ ∈ C ∞ ([ L j /2 , L j ]) L j / 2 L_j/2 L j /2 w = χ ( s ) φ n ε ( ρ ) w=\chi(s)\varphi_n^\varepsilon(\rho) w = χ ( s ) φ n ε ( ρ ) D j , ε × ( L j / 2 , L j ) D_{j,\varepsilon}\times(L_j/2,L_j) D j , ε × ( L j /2 , L j ) w ∈ H 1 ( Ω ε ) w\in H^1(\Omega_\varepsilon) w ∈ H 1 ( Ω ε ) φ n ε \varphi_n^\varepsilon φ n ε D j , ε D_{j,\varepsilon} D j , ε ∫ Ω ε w = 0 \int_{\Omega_\varepsilon} w=0 ∫ Ω ε w = 0 l k ( w ) = 0 l_k(w)=0 l k ( w ) = 0 (42) , as ∇ w = χ ∇ ρ φ n ε + χ ′ φ n ε n j \nabla w=\chi\nabla_\rho\varphi_n^\varepsilon+\chi'\varphi_n^\varepsilon n_j ∇ w = χ ∇ ρ φ n ε + χ ′ φ n ε n j (69) with v = ψ ( ⋅ , s ) v=\psi(\cdot,s) v = ψ ( ⋅ , s )
0 = q ε ( ψ , w ) = ∫ L j / 2 L j ( χ ′ ( s ) ∫ D j , ε ∂ s ψ ( ⋅ , s ) φ n ε + χ ( s ) ∫ D j , ε ∇ ρ ψ ( ⋅ , s ) ⋅ ∇ ρ φ n ε ) d s = ∫ L j / 2 L j ( χ ′ a n ′ + μ n r 2 χ a n ) d s , (71) \begin{aligned}
0=q_\varepsilon(\psi,w)=\int_{L_j/2}^{L_j}\left(\chi'(s)\int_{D_{j,\varepsilon}}\partial_s\psi(\cdot,s)\varphi_n^\varepsilon+\chi(s)\int_{D_{j,\varepsilon}}\nabla_\rho\psi(\cdot,s)\cdot\nabla_\rho\varphi_n^\varepsilon\right)\,ds \\
&=\int_{L_j/2}^{L_j}\left(\chi'a_n'+\frac{\mu_n}{r^2}\chi a_n\right)\,ds, \tag*{(71)}
\end{aligned} 0 = q ε ( ψ , w ) = ∫ L j /2 L j ( χ ′ ( s ) ∫ D j , ε ∂ s ψ ( ⋅ , s ) φ n ε + χ ( s ) ∫ D j , ε ∇ ρ ψ ( ⋅ , s ) ⋅ ∇ ρ φ n ε ) d s = ∫ L j /2 L j ( χ ′ a n ′ + r 2 μ n χ a n ) d s , (71) where we used the definition of a n a_n a n (70) and the expression of a n ′ a_n' a n ′
First, taking χ ∈ C c ∞ ( ( L j / 2 , L j ) ) \chi\in C^\infty_c((L_j/2,L_j)) χ ∈ C c ∞ (( L j /2 , L j )) (71) shows that
a n ′ ′ = μ n r − 2 a n , (72) a_n''=\mu_n r^{-2}a_n, \tag*{(72)} a n ′′ = μ n r − 2 a n , (72) in the sense of distributions on ( L j / 2 , L j ) (L_j/2,L_j) ( L j /2 , L j ) a n a_n a n a n ∈ C 2 ( [ L j / 2 , L j ] ) a_n\in C^2([L_j/2,L_j]) a n ∈ C 2 ([ L j /2 , L j ]) (72) holds pointwise on [ L j / 2 , L j ] [L_j/2,L_j] [ L j /2 , L j ] χ \chi χ χ ( L j / 2 ) = 0 \chi(L_j/2)=0 χ ( L j /2 ) = 0 χ ( L j ) = 1 \chi(L_j)=1 χ ( L j ) = 1
∫ L j / 2 L j χ ′ a n ′ = χ ( L j ) a n ′ ( L j ) − ∫ L j / 2 L j χ a n ′ ′ . \int_{L_j/2}^{L_j}\chi'a_n'=\chi(L_j)a_n'(L_j)-\int_{L_j/2}^{L_j}\chi a_n''. ∫ L j /2 L j χ ′ a n ′ = χ ( L j ) a n ′ ( L j ) − ∫ L j /2 L j χ a n ′′ . Using (72) and (71) shows that a n ′ ( L j ) = 0 a_n'(L_j)=0 a n ′ ( L j ) = 0 (72) with the boundary condition a n ′ ( L j ) = 0 a_n'(L_j)=0 a n ′ ( L j ) = 0
a n ( s ) = a n ( L j ) cosh ( μ n r ( L j − s ) ) , s ∈ [ L j / 2 , L j ] . (73) a_n(s)=a_n(L_j)\cosh\left(\frac{\sqrt{\mu_n}}{r}(L_j-s)\right),\qquad s\in[L_j/2,L_j]. \tag*{(73)} a n ( s ) = a n ( L j ) cosh ( r μ n ( L j − s ) ) , s ∈ [ L j /2 , L j ] . (73) Energy bound. By (69) , the functions r μ n − 1 / 2 φ n ε r\mu_n^{-1/2}\varphi_n^\varepsilon r μ n − 1/2 φ n ε n ≥ 1 n\geq1 n ≥ 1 ( θ , θ ′ ) ↦ ∫ D j , ε ∇ θ ⋅ ∇ θ ′ (\theta,\theta')\mapsto\int_{D_{j,\varepsilon}}\nabla\theta\cdot\nabla\theta' ( θ , θ ′ ) ↦ ∫ D j , ε ∇ θ ⋅ ∇ θ ′ s ∈ ( L j / 2 , L j ) s\in(L_j/2,L_j) s ∈ ( L j /2 , L j ) (70) and (73) with cosh 2 ξ ≥ ξ 2 \cosh^2\xi\geq\xi^2 cosh 2 ξ ≥ ξ 2
L j 3 24 r 4 ∑ n ≥ 1 μ n 2 a n ( L j ) 2 ≤ ∑ n ≥ 1 μ n r 2 ∫ L j / 2 L j a n ( s ) 2 d s ≤ ∥ ∇ ψ ∥ L 2 ( Ω ε ) 2 . (74) \frac{L_j^3}{24r^4}\sum_{n\geq1}\mu_n^2a_n(L_j)^2\leq\sum_{n\geq1}\frac{\mu_n}{r^2}\int_{L_j/2}^{L_j}a_n(s)^2\,ds\leq\|\nabla\psi\|_{L^2(\Omega_\varepsilon)}^2. \tag*{(74)} 24 r 4 L j 3 n ≥ 1 ∑ μ n 2 a n ( L j ) 2 ≤ n ≥ 1 ∑ r 2 μ n ∫ L j /2 L j a n ( s ) 2 d s ≤ ∥∇ ψ ∥ L 2 ( Ω ε ) 2 . (74) Uniform bound on the hole. It remains to turn the bound (74) on the coefficients of the trace of ψ \psi ψ Γ j ε \Gamma_j^\varepsilon Γ j ε D j D_j D j
f ( y ) = ψ ( r y , L j ) − M j k ε , y ∈ D j , f(y)=\psi(ry,L_j)-M_{jk}^\varepsilon,\qquad y\in D_j, f ( y ) = ψ ( r y , L j ) − M j k ε , y ∈ D j , so that the left-hand side of (68) is ∥ f ∥ L ∞ ( D j ) \|f\|_{L^\infty(D_j)} ∥ f ∥ L ∞ ( D j ) f n = ∫ D j f φ n f_n=\int_{D_j}f\varphi_n f n = ∫ D j f φ n ( φ n ) n ≥ 0 (\varphi_n)_{n\geq0} ( φ n ) n ≥ 0 f f f D j D_j D j f 0 = 0 f_0=0 f 0 = 0 n ≥ 1 n\geq1 n ≥ 1 φ n \varphi_n φ n φ n ε \varphi_n^\varepsilon φ n ε a n a_n a n
f n = ∫ D j ψ ( r y , L j ) φ n ( y ) d y = 1 r 2 ∫ D j , ε ψ ( ρ , L j ) φ n ( ρ / r ) d ρ = 1 r ∫ D j , ε ψ ( ρ , L j ) φ n ε ( ρ ) d ρ = a n ( L j ) r . f_n=\int_{D_j}\psi(ry,L_j)\varphi_n(y)\,dy=\frac{1}{r^2}\int_{D_{j,\varepsilon}}\psi(\rho,L_j)\varphi_n(\rho/r)\,d\rho=\frac{1}{r}\int_{D_{j,\varepsilon}}\psi(\rho,L_j)\varphi_n^\varepsilon(\rho)\,d\rho=\frac{a_n(L_j)}{r}. f n = ∫ D j ψ ( r y , L j ) φ n ( y ) d y = r 2 1 ∫ D j , ε ψ ( ρ , L j ) φ n ( ρ / r ) d ρ = r 1 ∫ D j , ε ψ ( ρ , L j ) φ n ε ( ρ ) d ρ = r a n ( L j ) . Moreover, for n ≥ 1 n\geq1 n ≥ 1 μ n ≥ μ 1 > 0 \mu_n\geq\mu_1>0 μ n ≥ μ 1 > 0 1 + μ n ≤ ( 1 + μ 1 − 1 ) μ n 1+\mu_n\leq(1+\mu_1^{-1})\mu_n 1 + μ n ≤ ( 1 + μ 1 − 1 ) μ n (74) ,
∑ n ≥ 1 ( 1 + μ n ) 2 f n 2 ≤ ( 1 + μ 1 − 1 ) 2 r − 2 ∑ n ≥ 1 μ n 2 a n ( L j ) 2 ≤ 24 ( 1 + μ 1 − 1 ) 2 L j 3 r 2 ∥ ∇ ψ ∥ L 2 ( Ω ε ) 2 . (75) \sum_{n\geq1}(1+\mu_n)^2f_n^2\leq(1+\mu_1^{-1})^2r^{-2}\sum_{n\geq1}\mu_n^2a_n(L_j)^2\leq\frac{24(1+\mu_1^{-1})^2}{L_j^3}r^2\|\nabla\psi\|_{L^2(\Omega_\varepsilon)}^2. \tag*{(75)} n ≥ 1 ∑ ( 1 + μ n ) 2 f n 2 ≤ ( 1 + μ 1 − 1 ) 2 r − 2 n ≥ 1 ∑ μ n 2 a n ( L j ) 2 ≤ L j 3 24 ( 1 + μ 1 − 1 ) 2 r 2 ∥∇ ψ ∥ L 2 ( Ω ε ) 2 . (75) In particular, the series g = ∑ n ≥ 1 ( 1 + μ n ) f n φ n g=\sum_{n\geq1}(1+\mu_n)f_n\varphi_n g = ∑ n ≥ 1 ( 1 + μ n ) f n φ n L 2 ( D j ) L^2(D_j) L 2 ( D j ) ∥ g ∥ L 2 ( D j ) 2 \|g\|_{L^2(D_j)}^2 ∥ g ∥ L 2 ( D j ) 2 (75) .
Let us show that f f f − Δ u + u = g -\Delta u+u=g − Δ u + u = g D j D_j D j ∂ n u = 0 \partial_nu=0 ∂ n u = 0 ∂ D j \partial D_j ∂ D j u ∈ H 1 ( D j ) u\in H^1(D_j) u ∈ H 1 ( D j )
∀ v ∈ H 1 ( D j ) , ∫ D j ∇ u ⋅ ∇ v + ∫ D j u v = ∫ D j g v . \forall v\in H^1(D_j),\qquad\int_{D_j}\nabla u\cdot\nabla v+\int_{D_j}uv=\int_{D_j}gv. ∀ v ∈ H 1 ( D j ) , ∫ D j ∇ u ⋅ ∇ v + ∫ D j uv = ∫ D j g v . Taking v = ϕ n v=\phi_n v = ϕ n ∫ D j ∇ ϕ n ⋅ ∇ u = μ n ∫ D j ϕ n u \int_{D_j}\nabla\phi_n\cdot\nabla u=\mu_n\int_{D_j}\phi_n u ∫ D j ∇ ϕ n ⋅ ∇ u = μ n ∫ D j ϕ n u ( μ n , ϕ n ) (\mu_n,\phi_n) ( μ n , ϕ n ) ( 1 + μ n ) ∫ D j u ϕ n = ∫ D j g ϕ n = ( 1 + μ n ) f n (1+\mu_n)\int_{D_j}u\phi_n=\int_{D_j}g\phi_n=(1+\mu_n)f_n ( 1 + μ n ) ∫ D j u ϕ n = ∫ D j g ϕ n = ( 1 + μ n ) f n n ≥ 0 n\geq0 n ≥ 0 u u u f f f ( ϕ n ) (\phi_n) ( ϕ n ) L 2 ( D j ) L^2(D_j) L 2 ( D j ) u = f u=f u = f
Finally, as D j D_j D j H 2 H^2 H 2 f ∈ H 2 ( D j ) f\in H^2(D_j) f ∈ H 2 ( D j ) ∥ f ∥ H 2 ( D j ) ≤ C ∥ g ∥ L 2 ( D j ) \|f\|_{H^2(D_j)}\leq C\|g\|_{L^2(D_j)} ∥ f ∥ H 2 ( D j ) ≤ C ∥ g ∥ L 2 ( D j ) H 2 ( D j ) ⊂ C ( D j ‾ ) H^2(D_j)\subset C(\overline{D_j}) H 2 ( D j ) ⊂ C ( D j ) f f f D j ‾ \overline{D_j} D j (75) , this gives
∥ f ∥ L ∞ ( D j ) ≤ C ∥ f ∥ H 2 ( D j ) ≤ C ∥ g ∥ L 2 ( D j ) ≤ C r ε ∥ ∇ ψ ∥ L 2 ( Ω ε ) . \|f\|_{L^\infty(D_j)}\leq C\|f\|_{H^2(D_j)}\leq C\|g\|_{L^2(D_j)}\leq Cr_{\varepsilon}\|\nabla\psi\|_{L^2(\Omega_\varepsilon)}. ∥ f ∥ L ∞ ( D j ) ≤ C ∥ f ∥ H 2 ( D j ) ≤ C ∥ g ∥ L 2 ( D j ) ≤ C r ε ∥∇ ψ ∥ L 2 ( Ω ε ) . where C C C D j D_j D j L j L_j L j f f f ψ \psi ψ Γ j ε ‾ \overline{\Gamma_j^\varepsilon} Γ j ε (68) . □
The quasi-mode is admissible
We can now show that Hypothesis 2.1 holds for the functions ψ ε ( k ) \psi_\varepsilon^{(k)} ψ ε ( k ) (44) , and derive the main result of this section.
Proposition 5.3. Assume that the radii satisfy
( r ‾ ε ) 3 = o ( min 1 ≤ j ≤ N r ε ( j ) ) as ε → 0 , (76) (\overline r_\varepsilon)^3=o\left(\min_{1\leq j\leq N}r_\varepsilon^{(j)}\right)\qquad\text{as }\varepsilon\to0, \tag*{(76)} ( r ε ) 3 = o ( 1 ≤ j ≤ N min r ε ( j ) ) as ε → 0 , (76) that is, η ε → 0 \eta_\varepsilon\to0 η ε → 0 η ε \eta_\varepsilon η ε (66) . Then the functions ψ ε ( k ) \psi_\varepsilon^{(k)} ψ ε ( k ) (44) verify Hypothesis 2.1 with
K ε ( k ) = Φ ε ( k ) ∣ Ω ε ∣ = ( r ε ( k ) ) 2 ∣ D k ∣ ∣ Ω ∣ L k ( 1 + O ( r ‾ ε ) ) , C = 1 , (77) K_\varepsilon^{(k)}=\frac{\Phi_\varepsilon^{(k)}}{|\Omega_\varepsilon|}=\frac{(r_\varepsilon^{(k)})^2|D_k|}{|\Omega|L_k}(1+O(\overline r_\varepsilon)),\qquad C=1, \tag*{(77)} K ε ( k ) = ∣ Ω ε ∣ Φ ε ( k ) = ∣Ω∣ L k ( r ε ( k ) ) 2 ∣ D k ∣ ( 1 + O ( r ε )) , C = 1 , (77) and
E 1 ( ( K ε ( j ) ) j = 1 N ) = ( ∑ j = 1 N ( K ε ( j ) ) 1 / 2 ) 3 / 2 min 1 ≤ j ≤ N ( K ε ( j ) ) 1 / 4 , E 2 ( ( K ε ( j ) ) j = 1 N ) = ∑ j = 1 N ( K ε ( j ) ) 1 / 2 . (78) \mathcal{E}_1\left((K_\varepsilon^{(j)})_{j=1}^N\right)=\frac{\left(\sum_{j=1}^N(K_\varepsilon^{(j)})^{1/2}\right)^{3/2}}{\min_{1\leq j\leq N}(K_\varepsilon^{(j)})^{1/4}},\qquad\mathcal{E}_2\left((K_\varepsilon^{(j)})_{j=1}^N\right)=\sum_{j=1}^N(K_\varepsilon^{(j)})^{1/2}. \tag*{(78)} E 1 ( ( K ε ( j ) ) j = 1 N ) = min 1 ≤ j ≤ N ( K ε ( j ) ) 1/4 ( ∑ j = 1 N ( K ε ( j ) ) 1/2 ) 3/2 , E 2 ( ( K ε ( j ) ) j = 1 N ) = j = 1 ∑ N ( K ε ( j ) ) 1/2 . (78) Furthermore, the mean exit time and exit position of the reflected Brownian motion ( X t ) t ≥ 0 (X_t)_{t\geq0} ( X t ) t ≥ 0 Ω ε \Omega_\varepsilon Ω ε
E ν ε [ τ ε ] = ∣ Ω ∣ ( ∑ k = 1 N ( r ε ( k ) ) 2 ∣ D k ∣ L k ) − 1 ( 1 + O ( η ε ) ) . (79) \mathbb{E}_{\nu_\varepsilon}[\tau_\varepsilon]=|\Omega|\left(\sum_{k=1}^N\frac{(r_\varepsilon^{(k)})^2|D_k|}{L_k}\right)^{-1}(1+O(\eta_\varepsilon)). \tag*{(79)} E ν ε [ τ ε ] = ∣Ω∣ ( k = 1 ∑ N L k ( r ε ( k ) ) 2 ∣ D k ∣ ) − 1 ( 1 + O ( η ε )) . (79) Then the exit position is independent of the exit time, and is distributed as, for k ∈ { 1 , … , N } k\in\{1,\ldots,N\} k ∈ { 1 , … , N }
P ν ε ( κ ε = k ) = ( r ε ( k ) ) 2 ∣ D k ∣ / L k ∑ j = 1 N ( r ε ( j ) ) 2 ∣ D j ∣ / L j + O ( η ε ) . (80) \mathbb{P}_{\nu_\varepsilon}(\kappa_\varepsilon=k)=\frac{(r_\varepsilon^{(k)})^2|D_k|/L_k}{\sum_{j=1}^N(r_\varepsilon^{(j)})^2|D_j|/L_j}+O(\eta_\varepsilon). \tag*{(80)} P ν ε ( κ ε = k ) = ∑ j = 1 N ( r ε ( j ) ) 2 ∣ D j ∣/ L j ( r ε ( k ) ) 2 ∣ D k ∣/ L k + O ( η ε ) . (80) Remark 5.4. When the radii are comparable, i.e. min j r ε ( j ) ≥ c r ‾ ε \min_j r_\varepsilon^{(j)}\geq c\overline r_\varepsilon min j r ε ( j ) ≥ c r ε η ε ≤ C r ‾ ε \eta_\varepsilon\leq C\overline r_\varepsilon η ε ≤ C r ε (79) and (80) are O ( r ‾ ε ) O(\overline r_\varepsilon) O ( r ε ) r ε ( j ) = ε α j r_\varepsilon^{(j)}=\varepsilon^{\alpha_j} r ε ( j ) = ε α j (76) holds if and only if max j α j < 3 min j α j \max_j\alpha_j<3\min_j\alpha_j max j α j < 3 min j α j η ε = ε ( 3 min j α j − max j α j ) / 2 \eta_\varepsilon=\varepsilon^{(3\min_j\alpha_j-\max_j\alpha_j)/2} η ε = ε ( 3 m i n j α j − m a x j α j ) /2
Proof. The asymptotics of K ε ( k ) K_\varepsilon^{(k)} K ε ( k ) (77) follow from (47) , as ∣ Ω ε ∣ = ∣ Ω ∣ + O ( ( r ‾ ε ) 2 ) |\Omega_\varepsilon|=|\Omega|+O((\overline r_\varepsilon)^2) ∣ Ω ε ∣ = ∣Ω∣ + O (( r ε ) 2 ) c > 0 c>0 c > 0 c ( r ε ( k ) ) 2 ≤ K ε ( k ) ≤ c − 1 ( r ε ( k ) ) 2 c(r_\varepsilon^{(k)})^2\leq K_\varepsilon^{(k)}\leq c^{-1}(r_\varepsilon^{(k)})^2 c ( r ε ( k ) ) 2 ≤ K ε ( k ) ≤ c − 1 ( r ε ( k ) ) 2 k k k (66) and (78) , E 1 ( ( K ε ( j ) ) j = 1 N ) \mathcal{E}_1((K_\varepsilon^{(j)})_{j=1}^N) E 1 (( K ε ( j ) ) j = 1 N ) η ε \eta_\varepsilon η ε E 2 ( ( K ε ( j ) ) j = 1 N ) \mathcal{E}_2((K_\varepsilon^{(j)})_{j=1}^N) E 2 (( K ε ( j ) ) j = 1 N ) r ‾ ε \overline r_\varepsilon r ε r ‾ ε ≤ η ε \overline r_\varepsilon\leq\eta_\varepsilon r ε ≤ η ε O ( η ε ) O(\eta_\varepsilon) O ( η ε ) (76) . Fix k ∈ { 1 , … , N } k\in\{1,\ldots,N\} k ∈ { 1 , … , N } Hypothesis 2.1 .
PDE. By (41) with Φ = Φ ε ( k ) \Phi=\Phi_\varepsilon^{(k)} Φ = Φ ε ( k ) Δ ψ ε ( k ) = Φ ε ( k ) / ∣ Ω ε ∣ = K ε ( k ) \Delta\psi_\varepsilon^{(k)}=\Phi_\varepsilon^{(k)}/|\Omega_\varepsilon|=K_\varepsilon^{(k)} Δ ψ ε ( k ) = Φ ε ( k ) /∣ Ω ε ∣ = K ε ( k ) Ω ε \Omega_\varepsilon Ω ε ∂ n ψ ε ( k ) = 0 \partial_n\psi_\varepsilon^{(k)}=0 ∂ n ψ ε ( k ) = 0 Γ ε N \Gamma_\varepsilon^N Γ ε N
Dirichlet estimate. By Proposition 5.2, ψ ε ( k ) \psi_\varepsilon^{(k)} ψ ε ( k ) Γ D ε ‾ \overline{\Gamma_D^\varepsilon} Γ D ε ∥ ψ ε ( k ) − 1 Γ D , k ε ∥ L ∞ ( Γ D ε ) ≤ C η ε \|\psi_\varepsilon^{(k)}-\mathbf{1}_{\Gamma_{D,k}^\varepsilon}\|_{L^\infty(\Gamma_D^\varepsilon)}\leq C\eta_\varepsilon ∥ ψ ε ( k ) − 1 Γ D , k ε ∥ L ∞ ( Γ D ε ) ≤ C η ε E 1 \mathcal{E}_1 E 1 (78) . Normalization. As ψ ε ( k ) \psi_\varepsilon^{(k)} ψ ε ( k ) Ω ε \Omega_\varepsilon Ω ε Lemma A.1 , (45) and (47) give
∥ ψ ε ( k ) ∥ L 2 ( Ω ε ) ≤ C ∥ ∇ ψ ε ( k ) ∥ L 2 ( Ω ε ) = C Φ ε ( k ) ≤ C r ε , \|\psi_\varepsilon^{(k)}\|_{L^2(\Omega_\varepsilon)} \le C\|\nabla\psi_\varepsilon^{(k)}\|_{L^2(\Omega_\varepsilon)}=C\sqrt{\Phi_\varepsilon^{(k)}}\le Cr_\varepsilon, ∥ ψ ε ( k ) ∥ L 2 ( Ω ε ) ≤ C ∥∇ ψ ε ( k ) ∥ L 2 ( Ω ε ) = C Φ ε ( k ) ≤ C r ε , which is (Norm) with E 2 E_2 E 2 (78) .
Eigenvalue. Both techniques used in Lemma 3.2 and in the proof of Proposition 4.2 apply to prove (Eig), with no additional complexity.
Conclusion. Having checked both hypotheses (Eig) and (PDE)–(Norm) of Hypothesis 2.1 , Theorem 2.2 holds. □
Numerical simulations
This section has two goals: checking the asymptotics of Table 1 in dimension 3, including the exit hole distribution, and testing their natural extension to dimensions d ≥ 4 d \ge4 d ≥ 4 § 6.1 , the three-dimensional results are presented in § 6.2 , and the higher-dimensional ones in § 6.3 .
Method
Method. The Walk on Stars. Narrow escape simulations are costly, since τ ε \tau_\varepsilon τ ε ε → 0 \varepsilon\to0 ε → 0 (3) must have steps smaller than the holes: it needs a time step h ≪ ε 2 h \ll\varepsilon^2 h ≪ ε 2 E [ τ ε ] / ε 2 \mathbb{E}[\tau_\varepsilon]/\varepsilon^2 E [ τ ε ] / ε 2 [44 , 50 ] away from the boundary, but Euler–Maruyama steps with h = 10 − 11 h=10^{-11} h = 1 0 − 11 d ≥ 4 d \ge4 d ≥ 4 Walk on Stars (WoSt) algorithm [54 ] , which extends WoS to Neumann boundary conditions and belongs to a recent family of grid-free Monte Carlo solvers [53 , 55 , 42 ] . WoSt has neither time step nor mesh: it only queries distances to the boundary and intersections of rays with it, in any dimension.
Let us give the main ideas of the Walk on Stars algorithm, following [54 Section 4] . The mean exit time u ( x ) = E x [ τ ε ] u(x)=\mathbb{E}_x[\tau_\varepsilon] u ( x ) = E x [ τ ε ]
{ − Δ u = 1 in Ω ε , u = 0 on Γ D ε , ∂ n u = 0 on Γ N ε . (81) \begin{cases}
-\Delta u = 1 & \text{in }\Omega_\varepsilon,\\
u = 0 & \text{on }\Gamma_\mathrm{D}^\varepsilon,\\
\partial_n u = 0 & \text{on }\Gamma_\mathrm{N}^\varepsilon. \tag*{(81)}
\end{cases} ⎩ ⎨ ⎧ − Δ u = 1 u = 0 ∂ n u = 0 in Ω ε , on Γ D ε , on Γ N ε . (81) Fix x ∈ Ω ε x\in\Omega_\varepsilon x ∈ Ω ε R > 0 R>0 R > 0 B ( x , R ) B(x,R) B ( x , R ) Γ D ε \Gamma_\mathrm{D}^\varepsilon Γ D ε St ( x , R ) = B ( x , R ) ∩ Ω ε \operatorname{St}(x,R)=B(x,R)\cap\Omega_\varepsilon St ( x , R ) = B ( x , R ) ∩ Ω ε St ( x , R ) \operatorname{St}(x,R) St ( x , R ) x x x R R R
G R ( x , y ) = ∣ y − x ∣ 2 − d − R 2 − d ( d − 2 ) ∣ S d − 1 ∣ G_R(x,y)=\frac{|y-x|^{2-d}-R^{2-d}}{(d-2)|\mathbb{S}^{d-1}|} G R ( x , y ) = ( d − 2 ) ∣ S d − 1 ∣ ∣ y − x ∣ 2 − d − R 2 − d the Dirichlet Green function of B ( x , R ) B(x,R) B ( x , R ) St ( x , R ) \operatorname{St}(x,R) St ( x , R ) (81) give
u ( x ) = ∫ St ( x , R ) G R ( x , y ) d y + ∫ ∂ St ( x , R ) u ( z ) ( − ∂ n z G R ( x , z ) ) d σ ( z ) . (82) u(x)=\int_{\operatorname{St}(x,R)}G_R(x,y)\,\mathrm{d}y+\int_{\partial\operatorname{St}(x,R)}u(z)\left(-\partial_{n_z}G_R(x,z)\right)\,\mathrm{d}\sigma(z). \tag*{(82)} u ( x ) = ∫ St ( x , R ) G R ( x , y ) d y + ∫ ∂ St ( x , R ) u ( z ) ( − ∂ n z G R ( x , z ) ) d σ ( z ) . (82) Following the same approach as [54 ] , we can write both integrals of (82) in polar coordinates around x x x u ( x ) u(x) u ( x )
u ( x ) = E ν [ τ R ( t ( ν ) ) + u ( x + t ( ν ) ν ) ] , ν ∼ U ( S d − 1 ) , (83) u(x)=\mathbb{E}_\nu\left[\tau_R(t(\nu))+u(x+t(\nu)\nu)\right],\qquad\nu\sim\mathcal{U}(\mathbb{S}^{d-1}), \tag*{(83)} u ( x ) = E ν [ τ R ( t ( ν )) + u ( x + t ( ν ) ν ) ] , ν ∼ U ( S d − 1 ) , (83) where
τ R ( t ) = ∣ S d − 1 ∣ ∫ 0 t G R ( x , x + ρ ν ) ρ d − 1 d ρ = 1 d − 2 ( t 2 2 − t d d R d − 2 ) . (84) \tau_R(t)=|\mathbb{S}^{d-1}|\int_0^tG_R(x,x+\rho\nu)\rho^{d-1}\,\mathrm{d}\rho=\frac{1}{d-2}\left(\frac{t^2}{2}-\frac{t^d}{dR^{d-2}}\right). \tag*{(84)} τ R ( t ) = ∣ S d − 1 ∣ ∫ 0 t G R ( x , x + ρ ν ) ρ d − 1 d ρ = d − 2 1 ( 2 t 2 − d R d − 2 t d ) . (84) Formula (83) expresses u ( x ) u(x) u ( x ) u u u x 1 = x + t ( ν ) ν x_1=x+t(\nu)\nu x 1 = x + t ( ν ) ν x 1 x_1 x 1 x 2 x_2 x 2
u ( x 0 ) = E [ ∑ n ≥ 0 τ R n ( t n ) ] . u(x_0)=\mathbb{E}\left[\sum_{n\geq0}\tau_{R_n}(t_n)\right]. u ( x 0 ) = E [ n ≥ 0 ∑ τ R n ( t n ) ] . where the sum runs over the steps of the random walk ( x n ) n ≥ 0 (x_n)_{n\geq0} ( x n ) n ≥ 0 Γ D ε \Gamma_D^\varepsilon Γ D ε u = 0 u=0 u = 0 κ ε \kappa_\varepsilon κ ε k k k Walk on Stars algorithm, detailed in Algorithm 1.
Algorithm 1 Walk on Stars for the mean exit time and the exit hole.
Require: starting point x 0 ∈ Ω ε x_0\in\Omega_\varepsilon x 0 ∈ Ω ε η > 0 \eta>0 η > 0
x ← x 0 , T ← 0 x\leftarrow x_0,\quad T\leftarrow0 x ← x 0 , T ← 0
loop
if d ( x , Γ k ε ) < η d(x,\Gamma_k^\varepsilon)<\eta d ( x , Γ k ε ) < η k ∈ { 1 , … , N } k\in\{1,\ldots,N\} k ∈ { 1 , … , N } then
return ( T , k ) (T,k) ( T , k ) k k k
end if
R ← d ( x , Γ D ε ) R\leftarrow d(x,\Gamma_D^\varepsilon) R ← d ( x , Γ D ε ) St ( x , R ) \operatorname{St}(x,R) St ( x , R )
if x ∈ Γ N ε x\in\Gamma_N^\varepsilon x ∈ Γ N ε then
draw ν \nu ν x x x
else
draw ν \nu ν S d − 1 \mathbb{S}^{d-1} S d − 1
end if
s ← s\leftarrow s ← x x x Γ N ε \Gamma_N^\varepsilon Γ N ε ν \nu ν
t ← min { R , s } t\leftarrow\min\{R,s\} t ← min { R , s }
T ← T + τ R ( t ) T\leftarrow T+\tau_R(t) T ← T + τ R ( t ) (84)
x ← x + t ν x\leftarrow x+t\nu x ← x + t ν x ∈ Γ N ε x\in\Gamma_N^\varepsilon x ∈ Γ N ε t = s t=s t = s
end loop
Explanation of Algorithm 1. The walk is constructed as follows. At the current position x x x R = d ( x , Γ D ε ) R=d(x,\Gamma_D^\varepsilon) R = d ( x , Γ D ε ) B ( x , R ) B(x,R) B ( x , R ) ν \nu ν S d − 1 \mathbb{S}^{d-1} S d − 1 x x x Ω ε \Omega_\varepsilon Ω ε x ∈ Γ N ε x\in\Gamma_N^\varepsilon x ∈ Γ N ε Γ N ε \Gamma_N^\varepsilon Γ N ε u ( x ) / 2 u(x)/2 u ( x ) /2 1 / 2 1/2 1/2 [54 Footnote 1 and Section 4.4] ). Next, we follow the ray from x x x ν \nu ν St ( x , R ) \operatorname{St}(x,R) St ( x , R ) ∂ B ( x , R ) \partial B(x,R) ∂ B ( x , R ) t = R t=R t = R Γ N ε \Gamma_N^\varepsilon Γ N ε t = s < R t=s<R t = s < R T T T τ R ( t ) \tau_R(t) τ R ( t ) G R ( x , ⋅ ) G_R(x,\cdot) G R ( x , ⋅ ) [ x , x + t ν ] [x,x+t\nu] [ x , x + t ν ] t = R t=R t = R τ R ( R ) = R 2 / ( 2 d ) \tau_R(R)=R^2/(2d) τ R ( R ) = R 2 / ( 2 d ) ( X t ) t ≥ 0 (X_t)_{t\geq0} ( X t ) t ≥ 0 B ( x , R ) B(x,R) B ( x , R ) [44 ] ; if t < R t<R t < R Ω ε \Omega_\varepsilon Ω ε τ R ( t ) < R 2 / ( 2 d ) \tau_R(t)<R^2/(2d) τ R ( t ) < R 2 / ( 2 d ) τ R ( t ) \tau_R(t) τ R ( t ) G R ( x , ⋅ ) G_R(x,\cdot) G R ( x , ⋅ ) [54 ] , which reduces the variance. Finally, the walker moves to x + t ν x+t\nu x + t ν
Since t ≤ d ( x , Γ D ε ) t\leq d(x,\Gamma_D^\varepsilon) t ≤ d ( x , Γ D ε ) Γ D ε \Gamma_D^\varepsilon Γ D ε { d ( ⋅ , Γ k ε ) < η } \{d(\cdot,\Gamma_k^\varepsilon)<\eta\} { d ( ⋅ , Γ k ε ) < η } k k k η \eta η η = 10 − 4 ε \eta=10^{-4}\varepsilon η = 1 0 − 4 ε η / ε \eta/\varepsilon η / ε [41 ] , well below the statistical errors (see [8] for the influence of η \eta η
Initial distribution. WoSt simulates no trajectory, hence cannot sample ν ε \nu_\varepsilon ν ε Ω ε \Omega_\varepsilon Ω ε ν ε \nu_\varepsilon ν ε ν ε \nu_\varepsilon ν ε ν ε \nu_\varepsilon ν ε φ ε / ∣ Ω ε ∣ \varphi_\varepsilon/|\Omega_\varepsilon| φ ε /∣ Ω ε ∣ φ ε \varphi_\varepsilon φ ε Ω ε \Omega_\varepsilon Ω ε ψ ε ( k ) \psi_\varepsilon^{(k)} ψ ε ( k ) E \mathbb{E} E P \mathbb{P} P E ν ε \mathbb{E}_{\nu_\varepsilon} E ν ε P ν ε \mathbb{P}_{\nu_\varepsilon} P ν ε
As a check, starting instead from the center of the ball (Type B, d = 3 d = 3 d = 3 E [ τ ε ] \mathbb{E}[\tau_{\varepsilon}] E [ τ ε ] ε ≤ 10 − 2 \varepsilon\le10^{-2} ε ≤ 1 0 − 2
Implementation and statistics. The code is written in Rust [61 ] , generic in the dimension 3 ≤ d ≤ 16 3 \le d \le16 3 ≤ d ≤ 16 [59 ] . The domains are the unit ball and the cube [ − 1 , 1 ] d [-1,1]^d [ − 1 , 1 ] d w = 10 − 8 w = 10^{-8} w = 1 0 − 8 n n n E [ τ ε ] \mathbb{E}[\tau_{\varepsilon}] E [ τ ε ] n − 1 / 2 n^{-1/2} n − 1/2 0.16 % 0.16\% 0.16% n = 4 ⋅ 10 5 n = 4 \cdot10^5 n = 4 ⋅ 1 0 5 5.9 ⋅ 10 7 5.9 \cdot10^7 5.9 ⋅ 1 0 7 Table 2 .
Table 2. Numerical setup. Slits lie along e 2 e_2 e 2 e 2 e_2 e 2 e 3 e_3 e 3 ℓ = 1 / 2 \ell= 1/2 ℓ = 1/2 L = 1 L = 1 L = 1 10 4 10^4 1 0 4 Figure 8 are those of Figure 4 .
Figure 8. Type D: tube of radius ε \varepsilon ε L = 1 L=1 L = 1 R d \mathbb{R}^{d} R d 3 ≤ d ≤ 6 3 \le d \le6 3 ≤ d ≤ 6 E [ τ ε ] \mathbb{E}[\tau_\varepsilon] E [ τ ε ] ε d − 1 \varepsilon^{d-1} ε d − 1 10 d − 3 10^{d-3} 1 0 d − 3
Results in dimension 3.
Mean exit time. Figure 4 shows E [ τ ε ] \mathbb{E}[\tau_{\varepsilon}] E [ τ ε ] Table 1 on the smallest holes. The fitted exponents agree with Table 1 within 0.3%, and the prefactors within 1% for A, B and C, and within 3.3% (1.5 standard errors) for D.
Figure 4. Mean exit time E [ τ ε ] E[\tau_{\varepsilon}] E [ τ ε ] [ − 1 , 1 ] 3 [-1,1]^3 [ − 1 , 1 ] 3 ∣ Ω ∣ = 8 |\Omega| = 8 ∣Ω∣ = 8 L = 1 L = 1 L = 1 ∣ Ω ∣ = 4 π / 3 |\Omega| = 4\pi/3 ∣Ω∣ = 4 π /3 Table 1 .
Exit hole. Figure 5 shows the probability of exiting through the smaller of two holes. By Propositions 3.1, 3.3 and 5.3, it tends to 1/3 for A and B and to 1/5 for D. For slits, in the case of two slits of half-widths w w w 2 w 2w 2 w ℓ / 2 \ell/2 ℓ /2 ℓ \ell ℓ
Figure 5. Probability of exiting through the smaller of two holes in dimension 3: A (half-balls), B (geodesic caps) and D in the unit ball, with sizes ε \varepsilon ε 2 ε 2\varepsilon 2 ε ± e 1 \pm e_1 ± e 1 w w w 2 w 2w 2 w ℓ \ell ℓ ℓ / 2 \ell/2 ℓ /2 ℓ \ell ℓ w w w
Extending the results to higher dimensions
The constructions of § 3 to § 5 have formal analogues in any dimension d ≥ 4 d \ge4 d ≥ 4 Table 3 . Only for (Type A) have these results been proved, in [13, Theorems 1.4 and 1.6]. Figs. 6 to 8 illustrate that the scalings of {xref}Table 3 are verified up to $d = 16$. In {xref}Fig. 8`, the simulations are stopped at d = 6 d = 6 d = 6 Walk on Stars algorithm is not efficient for non-convex geometries. Around the connection with the tube, R R R St ( x , R ) \operatorname{St}(x,R) St ( x , R )
Table 3. Leading-order scaling of the mean exit time in dimension d d d
Results in dimension 3
Figure 6. Types A (left) and B (right) in the cube [ − 1 , 1 ] d [-1,1]^d [ − 1 , 1 ] d 3 ≤ d ≤ 16 3 \le d \le16 3 ≤ d ≤ 16 E [ τ ε ] \mathbb{E}[\tau_\varepsilon] E [ τ ε ] ε d − 2 \varepsilon^{d-2} ε d − 2
Figure 7. Type C. Left: line slits in the cube [ − 1 , 1 ] d [-1,1]^d [ − 1 , 1 ] d 4 ≤ d ≤ 10 4 \le d \le10 4 ≤ d ≤ 10 E [ τ w ] \mathbb{E}[\tau_w] E [ τ w ] w d − 3 w^{d-3} w d − 3 10 d − 4 10^{d-4} 1 0 d − 4 d = 3 d=3 d = 3 d = 4 d=4 d = 4 E [ τ w ] \mathbb{E}[\tau_w] E [ τ w ] log ( 1 / w ) \log(1/w) log ( 1/ w )
Acknowledgments and AI statement
Part of this work is funded by the Agence Nationale de la Recherche through the projects IPSO (ANR-23-CE40-0027) and DySLoS (ANR-25-CE40-6875). The author has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 810367), project EMC2.
LLMs (Claude Opus 5.5) have been used to improve the general writing and presentation of the manuscript and correct typographical errors. Through this process a simpler yet more elegant proof of the results presented in § 5.1 was suggested by the LLM. The tikz figures were generated using Claude Opus. The author has verified the correctness of the results and proofs presented in this manuscript.
On the epsilon-uniform Poincaré–Wirtinger inequality
In this section, we prove that the Poincaré–Wirtinger inequality holds for all geometries considered in this work with a constant independent of ε \varepsilon ε [48 Theorem 1.2] , as the domains are bounded and satisfy the uniform cone condition. This is not the case for the tubular holes geometry (Type D). However, the uniformity of the constant in the Poincaré–Wirtinger inequality on domains with thin channels has been studied in the literature (see [34 ] [3]). In particular, the convergence of the Neumann spectrum of the operator defined on Ω ε \Omega_\varepsilon Ω ε Ω \Omega Ω max { μ 2 ( Ω ) − 1 / 2 , 2 ℓ / π } \max\{\mu_2(\Omega)^{-1/2},2\ell/\pi\} max { μ 2 ( Ω ) − 1/2 , 2 ℓ / π } ℓ \ell ℓ μ 2 ( Ω ) \mu_2(\Omega) μ 2 ( Ω ) Ω \Omega Ω Ω ε \Omega_\varepsilon Ω ε § 5 . We therefore provide a self-contained proof of the uniform Poincaré–Wirtinger inequality in this setting with a constant independent of ε \varepsilon ε
Lemma A.1. Consider the geometry of § 5 , introduced in (Type D) with the domains Ω ε \Omega_\varepsilon Ω ε (13) . There is C > 0 C>0 C > 0 ε \varepsilon ε u ∈ H 1 ( Ω ε ) u\in H^1(\Omega_\varepsilon) u ∈ H 1 ( Ω ε )
∥ u − 1 ∣ Ω ε ∣ ∫ Ω ε u ∥ L 2 ( Ω ε ) ≤ C ∥ ∇ u ∥ L 2 ( Ω ε ) . \left\|u-\frac{1}{|\Omega_\varepsilon|}\int_{\Omega_\varepsilon}u\right\|_{L^2(\Omega_\varepsilon)}\le C\|\nabla u\|_{L^2(\Omega_\varepsilon)}. u − ∣ Ω ε ∣ 1 ∫ Ω ε u L 2 ( Ω ε ) ≤ C ∥∇ u ∥ L 2 ( Ω ε ) . Proof. Fix u ∈ H 1 ( Ω ε ) u\in H^1(\Omega_\varepsilon) u ∈ H 1 ( Ω ε ) u ˉ = 1 ∣ Ω ε ∣ ∫ Ω ε u \bar{u}=\frac{1}{|\Omega_\varepsilon|}\int_{\Omega_\varepsilon}u u ˉ = ∣ Ω ε ∣ 1 ∫ Ω ε u Ω ε \Omega_\varepsilon Ω ε k ↦ ∥ u − k ∥ L 2 ( Ω ε ) k\mapsto\|u-k\|_{L^2(\Omega_\varepsilon)} k ↦ ∥ u − k ∥ L 2 ( Ω ε ) ∥ u − u ˉ ∥ L 2 ( Ω ε ) \|u-\bar{u}\|_{L^2(\Omega_\varepsilon)} ∥ u − u ˉ ∥ L 2 ( Ω ε ) T k , ε T_{k,\varepsilon} T k , ε Ω ε \Omega_\varepsilon Ω ε (13) , we have
∥ v ∥ L 2 ( Ω ε ) 2 = ∥ v ∥ L 2 ( Ω ) 2 + ∑ k = 1 N ∥ v ∥ L 2 ( T k , ε ) 2 ( v ∈ L 2 ( Ω ε ) ) . (85) \|v\|_{L^2(\Omega_\varepsilon)}^2=\|v\|_{L^2(\Omega)}^2+\sum_{k=1}^{N}\|v\|_{L^2(T_{k,\varepsilon})}^2\qquad(v\in L^2(\Omega_\varepsilon)). \tag*{(85)} ∥ v ∥ L 2 ( Ω ε ) 2 = ∥ v ∥ L 2 ( Ω ) 2 + k = 1 ∑ N ∥ v ∥ L 2 ( T k , ε ) 2 ( v ∈ L 2 ( Ω ε )) . (85) and the same identity holds for ∇ v \nabla v ∇ v
Step 1 (the fixed domain). Since Ω \Omega Ω ε \varepsilon ε C P > 0 C_P>0 C P > 0
∥ u − u ˉ ∥ L 2 ( Ω ) ≤ C P ∥ ∇ u ∥ L 2 ( Ω ) . (86) \|u-\bar{u}\|_{L^2(\Omega)}\le C_P\|\nabla u\|_{L^2(\Omega)}. \tag*{(86)} ∥ u − u ˉ ∥ L 2 ( Ω ) ≤ C P ∥∇ u ∥ L 2 ( Ω ) . (86) For the same reason, there is a constant C t r > 0 C_{\mathrm{tr}}>0 C tr > 0 v ∈ H 1 ( Ω ) v\in H^1(\Omega) v ∈ H 1 ( Ω )
∥ v ∥ L 2 ( ∂ Ω ) 2 ≤ C t r ( ∥ v ∥ L 2 ( Ω ) 2 + ∥ ∇ v ∥ L 2 ( Ω ) 2 ) , (87) \|v\|_{L^2(\partial\Omega)}^2\le C_{\mathrm{tr}}\left(\|v\|_{L^2(\Omega)}^2+\|\nabla v\|_{L^2(\Omega)}^2\right), \tag*{(87)} ∥ v ∥ L 2 ( ∂ Ω ) 2 ≤ C tr ( ∥ v ∥ L 2 ( Ω ) 2 + ∥∇ v ∥ L 2 ( Ω ) 2 ) , (87) which is useful in the next step.
Step 2 (the tubes). Fix k ∈ { 1 , … , N } k\in\{1,\ldots,N\} k ∈ { 1 , … , N } ℓ k = L k + sup ξ ∈ S ε ( k ) ℓ \ell_k=L_k+\sup_{\xi\in S_\varepsilon^{(k)}}\ell ℓ k = L k + sup ξ ∈ S ε ( k ) ℓ k k k
∥ u − u ˉ ∥ L 2 ( T k , ε ) 2 ≤ 2 ℓ k ∥ u − u ˉ ∥ L 2 ( Σ k , ε ) 2 + ℓ k 2 ∥ ∂ s u ∥ L 2 ( T k , ε ) 2 , (88) \|u-\bar{u}\|_{L^2(T_{k,\varepsilon})}^2\le2\ell_k\|u-\bar{u}\|_{L^2(\Sigma_{k,\varepsilon})}^2+\ell_k^2\|\partial_su\|_{L^2(T_{k,\varepsilon})}^2, \tag*{(88)} ∥ u − u ˉ ∥ L 2 ( T k , ε ) 2 ≤ 2 ℓ k ∥ u − u ˉ ∥ L 2 ( Σ k , ε ) 2 + ℓ k 2 ∥ ∂ s u ∥ L 2 ( T k , ε ) 2 , (88) where ∂ s u = ∇ u ⋅ n k \partial_su=\nabla u\cdot n_k ∂ s u = ∇ u ⋅ n k
For fixed ε \varepsilon ε T k , ε T_{k,\varepsilon} T k , ε C 1 ( T k , ε ‾ ) C^1(\overline{T_{k,\varepsilon}}) C 1 ( T k , ε ) H 1 ( T k , ε ) H^1(T_{k,\varepsilon}) H 1 ( T k , ε ) T k , ε T_{k,\varepsilon} T k , ε u − u ‾ u-\overline{u} u − u w ∈ C 1 ( T k , ε ‾ ) w\in C^{1}(\overline{T_{k,\varepsilon}}) w ∈ C 1 ( T k , ε ) x = x ( k ) + ρ + s n k x=x^{(k)}+\rho+sn_k x = x ( k ) + ρ + s n k (11) , and write w ( ρ , s ) = w ( x ( k ) + ρ + s n k ) w(\rho,s)=w(x^{(k)}+\rho+sn_k) w ( ρ , s ) = w ( x ( k ) + ρ + s n k ) ρ ∈ D k , ε \rho\in D_{k,\varepsilon} ρ ∈ D k , ε s ∈ ( h k ( ρ ) , L k ) s\in(h_k(\rho),L_k) s ∈ ( h k ( ρ ) , L k )
w ( ρ , s ) = w ( ρ , h k ( ρ ) ) + ∫ h k ( ρ ) s ∂ s w ( ρ , σ ) d σ . w(\rho,s)=w(\rho,h_k(\rho))+\int_{h_k(\rho)}^{s}\partial_s w(\rho,\sigma)\,\mathrm{d}\sigma. w ( ρ , s ) = w ( ρ , h k ( ρ )) + ∫ h k ( ρ ) s ∂ s w ( ρ , σ ) d σ . By ( a + b ) 2 ≤ 2 a 2 + 2 b 2 (a+b)^2\leq2a^2+2b^2 ( a + b ) 2 ≤ 2 a 2 + 2 b 2
w ( ρ , s ) 2 ≤ 2 w ( ρ , h k ( ρ ) ) 2 + 2 ( s − h k ( ρ ) ) ∫ h k ( ρ ) L k ∣ ∂ s w ( ρ , σ ) ∣ 2 d σ . w(\rho,s)^2\leq2w(\rho,h_k(\rho))^2+2(s-h_k(\rho))\int_{h_k(\rho)}^{L_k}|\partial_s w(\rho,\sigma)|^2\,\mathrm{d}\sigma. w ( ρ , s ) 2 ≤ 2 w ( ρ , h k ( ρ ) ) 2 + 2 ( s − h k ( ρ )) ∫ h k ( ρ ) L k ∣ ∂ s w ( ρ , σ ) ∣ 2 d σ . We integrate in s s s ( h k ( ρ ) , L k ) (h_k(\rho),L_k) ( h k ( ρ ) , L k ) ℓ k \ell_k ℓ k ∫ h k ( ρ ) L k 2 ( s − h k ( ρ ) ) d s = ( L k − h k ( ρ ) ) 2 ≤ ℓ k 2 \int_{h_k(\rho)}^{L_k}2(s-h_k(\rho))\,\mathrm{d}s=(L_k-h_k(\rho))^2\leq\ell_k^2 ∫ h k ( ρ ) L k 2 ( s − h k ( ρ )) d s = ( L k − h k ( ρ ) ) 2 ≤ ℓ k 2 ρ \rho ρ D k , ε D_{k,\varepsilon} D k , ε
∥ w ∥ L 2 ( T k , ε ) 2 ≤ 2 ℓ k ∫ D k , ε w ( ρ , h k ( ρ ) ) 2 d ρ + ℓ k 2 ∥ ∂ s w ∥ L 2 ( T k , ε ) 2 . \lVert w\rVert_{L^2(T_{k,\varepsilon})}^2\leq2\ell_k\int_{D_{k,\varepsilon}}w(\rho,h_k(\rho))^2\,\mathrm{d}\rho+\ell_k^2\lVert\partial_s w\rVert_{L^2(T_{k,\varepsilon})}^2. ∥ w ∥ L 2 ( T k , ε ) 2 ≤ 2 ℓ k ∫ D k , ε w ( ρ , h k ( ρ ) ) 2 d ρ + ℓ k 2 ∥ ∂ s w ∥ L 2 ( T k , ε ) 2 . Finally, Σ k , ε \Sigma_{k,\varepsilon} Σ k , ε ρ ↦ x ( k ) + ρ + h k ( ρ ) n k \rho\mapsto x^{(k)}+\rho+h_k(\rho)n_k ρ ↦ x ( k ) + ρ + h k ( ρ ) n k D k , ε D_{k,\varepsilon} D k , ε
d σ = 1 + ∣ ∇ h k ( ρ ) ∣ 2 d ρ ≥ d ρ . \mathrm{d}\sigma=\sqrt{1+|\nabla h_k(\rho)|^2}\,\mathrm{d}\rho\geq\mathrm{d}\rho. d σ = 1 + ∣∇ h k ( ρ ) ∣ 2 d ρ ≥ d ρ . Therefore ∫ D k , ε w ( ρ , h k ( ρ ) ) 2 d ρ ≤ ∥ w ∥ L 2 ( Σ k , ε ) 2 \int_{D_{k,\varepsilon}}w(\rho,h_k(\rho))^2\,\mathrm{d}\rho\leq\lVert w\rVert_{L^2(\Sigma_{k,\varepsilon})}^2 ∫ D k , ε w ( ρ , h k ( ρ ) ) 2 d ρ ≤ ∥ w ∥ L 2 ( Σ k , ε ) 2
Step 3 (conclusion). From their definition (12) , the mouths Σ k , ε \Sigma_{k,\varepsilon} Σ k , ε ∂ Ω \partial\Omega ∂ Ω ∑ k ∥ u − u ‾ ∥ L 2 ( Σ k , ε ) 2 ≤ ∥ u − u ‾ ∥ L 2 ( ∂ Ω ) 2 \sum_k\lVert u-\overline{u}\rVert_{L^2(\Sigma_{k,\varepsilon})}^2\leq\lVert u-\overline{u}\rVert_{L^2(\partial\Omega)}^2 ∑ k ∥ u − u ∥ L 2 ( Σ k , ε ) 2 ≤ ∥ u − u ∥ L 2 ( ∂ Ω ) 2 k k k ∣ ∂ s u ∣ ≤ ∣ ∇ u ∣ |\partial_su|\leq|\nabla u| ∣ ∂ s u ∣ ≤ ∣∇ u ∣
∑ k = 1 N ∥ u − u ‾ ∥ L 2 ( T k , ε ) 2 ≤ 2 ℓ C t r ( 1 + C P 2 ) ∥ ∇ u ∥ L 2 ( Ω ) 2 + ℓ 2 ∑ k = 1 N ∥ ∇ u ∥ L 2 ( T k , ε ) 2 . (89) \sum_{k=1}^{N}\lVert u-\overline{u}\rVert_{L^2(T_{k,\varepsilon})}^2\leq2\ell C_{\mathrm{tr}}(1+C_P^2)\lVert\nabla u\rVert_{L^2(\Omega)}^2+\ell^2\sum_{k=1}^{N}\lVert\nabla u\rVert_{L^2(T_{k,\varepsilon})}^2. \tag*{(89)} k = 1 ∑ N ∥ u − u ∥ L 2 ( T k , ε ) 2 ≤ 2 ℓ C tr ( 1 + C P 2 ) ∥ ∇ u ∥ L 2 ( Ω ) 2 + ℓ 2 k = 1 ∑ N ∥ ∇ u ∥ L 2 ( T k , ε ) 2 . (89) where ℓ = max k ℓ k \ell=\max_k\ell_k ℓ = max k ℓ k
∥ u − u ‾ ∥ L 2 ( Ω ε ) 2 ≤ K ∥ ∇ u ∥ L 2 ( Ω ε ) 2 , K = max { C P 2 + 2 ℓ C t r ( 1 + C P 2 ) , ℓ 2 } . \lVert u-\overline{u}\rVert_{L^2(\Omega_\varepsilon)}^2\leq K\lVert\nabla u\rVert_{L^2(\Omega_\varepsilon)}^2,\qquad K=\max\left\{C_P^2+2\ell C_{\mathrm{tr}}(1+C_P^2),\ell^2\right\}. ∥ u − u ∥ L 2 ( Ω ε ) 2 ≤ K ∥ ∇ u ∥ L 2 ( Ω ε ) 2 , K = max { C P 2 + 2 ℓ C tr ( 1 + C P 2 ) , ℓ 2 } . The lemma holds with C = K 1 / 2 C=K^{1/2} C = K 1/2
Appendix B. Proof of the main theorem
We provide here the proof of Theorem 2.2 in the general case, following the steps made in [13, Section 3]. Throughout this section, ( ⋅ , ⋅ ) Ω ε (\cdot,\cdot)_{\Omega_\varepsilon} ( ⋅ , ⋅ ) Ω ε L 2 ( Ω ε ) L^2(\Omega_\varepsilon) L 2 ( Ω ε ) ⟨ ⋅ , ⋅ ⟩ \langle\cdot,\cdot\rangle ⟨ ⋅ , ⋅ ⟩ H − 1 / 2 ( ∂ Ω ε ) H^{-1/2}(\partial\Omega_\varepsilon) H − 1/2 ( ∂ Ω ε ) H 1 / 2 ( ∂ Ω ε ) H^{1/2}(\partial\Omega_\varepsilon) H 1/2 ( ∂ Ω ε ) H 1 ( Ω ε ) H^1(\Omega_\varepsilon) H 1 ( Ω ε ) ∂ Ω ε \partial\Omega_\varepsilon ∂ Ω ε
Let us first give a meaning to the normal derivatives introduced in (4) and (PDE). For u ∈ H 1 ( Ω ε ) u\in H^1(\Omega_\varepsilon) u ∈ H 1 ( Ω ε ) Δ u ∈ L 2 ( Ω ε ) \Delta u\in L^2(\Omega_\varepsilon) Δ u ∈ L 2 ( Ω ε ) ∂ n u \partial_nu ∂ n u H − 1 / 2 ( ∂ Ω ε ) H^{-1/2}(\partial\Omega_\varepsilon) H − 1/2 ( ∂ Ω ε ) v ∈ H 1 / 2 ( ∂ Ω ε ) v\in H^{1/2}(\partial\Omega_\varepsilon) v ∈ H 1/2 ( ∂ Ω ε )
⟨ ∂ n u , v ⟩ = ∫ Ω ε ∇ u ⋅ ∇ w + ( Δ u ) w . (90) \langle\partial_nu,v\rangle=\int_{\Omega_\varepsilon}\nabla u\cdot\nabla w+(\Delta u)w. \tag*{(90)} ⟨ ∂ n u , v ⟩ = ∫ Ω ε ∇ u ⋅ ∇ w + ( Δ u ) w . (90) where w w w v v v H 1 ( Ω ε ) H^1(\Omega_\varepsilon) H 1 ( Ω ε ) tr ∂ Ω ε w = v \operatorname{tr}_{\partial\Omega_\varepsilon}w=v tr ∂ Ω ε w = v w w w ∂ n u = 0 \partial_nu=0 ∂ n u = 0 Γ N ε \Gamma^\varepsilon_N Γ N ε ⟨ ∂ n u , v ⟩ = 0 \langle\partial_nu,v\rangle=0 ⟨ ∂ n u , v ⟩ = 0 v ∈ H 1 / 2 ( ∂ Ω ε ) v\in H^{1/2}(\partial\Omega_\varepsilon) v ∈ H 1/2 ( ∂ Ω ε ) Γ D ε \Gamma^\varepsilon_D Γ D ε u 1 , u 2 ∈ H 1 ( Ω ε ) u_1,u_2\in H^1(\Omega_\varepsilon) u 1 , u 2 ∈ H 1 ( Ω ε ) Δ u 1 , Δ u 2 ∈ L 2 ( Ω ε ) \Delta u_1,\Delta u_2\in L^2(\Omega_\varepsilon) Δ u 1 , Δ u 2 ∈ L 2 ( Ω ε )
⟨ ∂ n u 1 , u 2 ⟩ − ⟨ ∂ n u 2 , u 1 ⟩ = ( Δ u 1 , u 2 ) Ω ε − ( u 1 , Δ u 2 ) Ω ε . (91) \langle\partial_nu_1,u_2\rangle-\langle\partial_nu_2,u_1\rangle=(\Delta u_1,u_2)_{\Omega_\varepsilon}-(u_1,\Delta u_2)_{\Omega_\varepsilon}. \tag*{(91)} ⟨ ∂ n u 1 , u 2 ⟩ − ⟨ ∂ n u 2 , u 1 ⟩ = ( Δ u 1 , u 2 ) Ω ε − ( u 1 , Δ u 2 ) Ω ε . (91) In particular, if u 1 = 0 u_1=0 u 1 = 0 Γ D ε \Gamma^\varepsilon_D Γ D ε ∂ n u 2 = 0 \partial_nu_2=0 ∂ n u 2 = 0 Γ N ε \Gamma^\varepsilon_N Γ N ε ⟨ ∂ n u 2 , u 1 ⟩ = 0 \langle\partial_nu_2,u_1\rangle=0 ⟨ ∂ n u 2 , u 1 ⟩ = 0
We now provide a technical lemma.
Lemma B.1. Under Hypothesis 2.1 , consider the quasi-mode φ ε \varphi_\varepsilon φ ε (16) and the density of the quasi-stationary distribution v ε v_\varepsilon v ε (6) . Then, the following estimates hold
( v ε , φ ε ) Ω ε = 1 + O ( E 2 ( ( K ε ( j ) ) j = 1 N ) ) , (92) (v_\varepsilon,\varphi_\varepsilon)_{\Omega_\varepsilon}=1+O\left(\mathcal{E}_2\left((K_\varepsilon^{(j)})_{j=1}^{N}\right)\right), \tag*{(92)} ( v ε , φ ε ) Ω ε = 1 + O ( E 2 ( ( K ε ( j ) ) j = 1 N ) ) , (92) and
⟨ ∂ n v ε , φ ε ⟩ = O ( λ ε E 1 ( ( K ε ( j ) ) j = 1 N ) ) . (93) \langle\partial_n v_\varepsilon,\varphi_\varepsilon\rangle=O\left(\lambda_\varepsilon\mathcal{E}_1\left((K_\varepsilon^{(j)})_{j=1}^{N}\right)\right). \tag*{(93)} ⟨ ∂ n v ε , φ ε ⟩ = O ( λ ε E 1 ( ( K ε ( j ) ) j = 1 N ) ) . (93) Proof. We start by showing that the L 2 L^2 L 2 v ε v_\varepsilon v ε ε \varepsilon ε ε \varepsilon ε Ω ε \Omega_\varepsilon Ω ε [48 Theorem 1.2] for (Type A)–(Type C) and Lemma A.1 for (Type D)), there exists C P W > 0 C_{\mathrm{PW}}>0 C PW > 0 ε \varepsilon ε
∥ v ε − 1 ∣ Ω ε ∣ ( v ε , 1 ) Ω ε ∥ L 2 ( Ω ε ) 2 ≤ C P W ∥ ∇ v ε ∥ L 2 ( Ω ε ) 2 = C P W λ ε ∥ v ε ∥ L 2 ( Ω ε ) 2 . (94) \left\|v_\varepsilon-\frac{1}{|\Omega_\varepsilon|}(v_\varepsilon,1)_{\Omega_\varepsilon}\right\|_{L^2(\Omega_\varepsilon)}^2\leq C_{\mathrm{PW}}\|\nabla v_\varepsilon\|_{L^2(\Omega_\varepsilon)}^2=C_{\mathrm{PW}}\lambda_\varepsilon\|v_\varepsilon\|_{L^2(\Omega_\varepsilon)}^2. \tag*{(94)} v ε − ∣ Ω ε ∣ 1 ( v ε , 1 ) Ω ε L 2 ( Ω ε ) 2 ≤ C PW ∥∇ v ε ∥ L 2 ( Ω ε ) 2 = C PW λ ε ∥ v ε ∥ L 2 ( Ω ε ) 2 . (94) where we used (6) to obtain the equality. Since ∥ v ε ∥ L 1 ( Ω ε ) = 1 \|v_\varepsilon\|_{L^1(\Omega_\varepsilon)}=1 ∥ v ε ∥ L 1 ( Ω ε ) = 1
∥ v ε − 1 ∣ Ω ε ∣ ( v ε , 1 ) Ω ε ∥ L 2 ( Ω ε ) 2 = ∥ v ε ∥ L 2 ( Ω ε ) 2 − 1 ∣ Ω ε ∣ . (95) \left\|v_\varepsilon-\frac{1}{|\Omega_\varepsilon|}(v_\varepsilon,1)_{\Omega_\varepsilon}\right\|_{L^2(\Omega_\varepsilon)}^2=\|v_\varepsilon\|_{L^2(\Omega_\varepsilon)}^2-\frac{1}{|\Omega_\varepsilon|}. \tag*{(95)} v ε − ∣ Ω ε ∣ 1 ( v ε , 1 ) Ω ε L 2 ( Ω ε ) 2 = ∥ v ε ∥ L 2 ( Ω ε ) 2 − ∣ Ω ε ∣ 1 . (95) Plugging (B.5) into (B.6), one obtains
∥ v ε ∥ L 2 ( Ω ε ) 2 ( 1 − C P W λ ε ) ≤ ∣ Ω ε ∣ − 1 . (96) \|v_\varepsilon\|_{L^2(\Omega_\varepsilon)}^2(1-C_{\mathrm{PW}}\lambda_\varepsilon)\leq|\Omega_\varepsilon|^{-1}. \tag*{(96)} ∥ v ε ∥ L 2 ( Ω ε ) 2 ( 1 − C PW λ ε ) ≤ ∣ Ω ε ∣ − 1 . (96) Since, for all kinds of holes it holds
∣ Ω ∣ = ∣ Ω ε ∣ + o ( 1 ) , |\Omega|=|\Omega_\varepsilon|+o(1), ∣Ω∣ = ∣ Ω ε ∣ + o ( 1 ) , and since 1 − C P W λ ε ≥ 1 2 1-C_{\mathrm{PW}}\lambda_\varepsilon\geq\frac{1}{2} 1 − C PW λ ε ≥ 2 1 ε \varepsilon ε
∥ v ε ∥ L 2 ( Ω ε ) ≤ M , (97) \|v_\varepsilon\|_{L^2(\Omega_\varepsilon)}\leq M, \tag*{(97)} ∥ v ε ∥ L 2 ( Ω ε ) ≤ M , (97) for some constant M > 0 M>0 M > 0 ε \varepsilon ε
Let us now move on to the estimates claimed by this lemma. From the Cauchy–Schwarz inequality, (B.8), the L 1 L^1 L 1 v ε v_\varepsilon v ε (17) , we have
∣ ( v ε , φ ε ) Ω ε − 1 ∣ = ∣ ( v ε , φ ε − 1 ) Ω ε ∣ ≤ ∥ v ε ∥ L 2 ( Ω ε ) ∥ φ ε − 1 ∥ L 2 ( Ω ε ) = O ( E 2 ( ( K ε ( j ) ) j = 1 N ) ) . \left|(v_\varepsilon,\varphi_\varepsilon)_{\Omega_\varepsilon}-1\right|=\left|(v_\varepsilon,\varphi_\varepsilon-1)_{\Omega_\varepsilon}\right|\leq\|v_\varepsilon\|_{L^2(\Omega_\varepsilon)}\|\varphi_\varepsilon-1\|_{L^2(\Omega_\varepsilon)}=O\left(\mathcal{E}_2\left((K_\varepsilon^{(j)})_{j=1}^{N}\right)\right). ∣ ( v ε , φ ε ) Ω ε − 1 ∣ = ∣ ( v ε , φ ε − 1 ) Ω ε ∣ ≤ ∥ v ε ∥ L 2 ( Ω ε ) ∥ φ ε − 1 ∥ L 2 ( Ω ε ) = O ( E 2 ( ( K ε ( j ) ) j = 1 N ) ) . For the normal derivative, recall from [13, Proposition 1.3] that − λ ε − 1 ∂ n v ε -\lambda_\varepsilon^{-1}\partial_n v_\varepsilon − λ ε − 1 ∂ n v ε X τ ε X_{\tau_\varepsilon} X τ ε ∂ n v ε \partial_n v_\varepsilon ∂ n v ε Γ D ε \Gamma_D^\varepsilon Γ D ε ∣ ∂ n v ε ( Γ D ε ) ∣ = λ ε \left|\partial_n v_\varepsilon(\Gamma_D^\varepsilon)\right|=\lambda_\varepsilon ∣ ∂ n v ε ( Γ D ε ) ∣ = λ ε g ∈ C 0 ( Γ D ε ‾ ) g\in C^0(\overline{\Gamma_D^\varepsilon}) g ∈ C 0 ( Γ D ε )
∣ ∫ Γ D ε g d ( ∂ n v ε ) ∣ ≤ λ ε ∥ g ∥ L ∞ ( Γ D ε ) . (98) \left|\int_{\Gamma_D^\varepsilon}g\,\mathrm{d}(\partial_n v_\varepsilon)\right|\leq\lambda_\varepsilon\|g\|_{L^\infty(\Gamma_D^\varepsilon)}. \tag*{(98)} ∫ Γ D ε g d ( ∂ n v ε ) ≤ λ ε ∥ g ∥ L ∞ ( Γ D ε ) . (98) Applying (B.9) to g = φ ε g=\varphi_\varepsilon g = φ ε (17) , gives (B.4). □
We can now move on to the proof of Theorem 2.2 .
Proof of Theorem 2.2 . From [13, Proposition 1.3], starting from ν ε \nu_\varepsilon ν ε τ ε \tau_\varepsilon τ ε λ ε \lambda_\varepsilon λ ε k ∈ { 1 , … , N } k\in\{1,\ldots,N\} k ∈ { 1 , … , N }
P ν ε ( κ ε = k ) = − 1 λ ε ∫ Γ D ε 1 Γ D ε , k d ( ∂ n v ε ) , (99) \mathbb{P}_{\nu_\varepsilon}(\kappa_\varepsilon=k)=-\frac{1}{\lambda_\varepsilon}\int_{\Gamma_D^\varepsilon}\mathbb{1}_{\Gamma_{D}^{\varepsilon,k}}\,\mathrm{d}(\partial_n v_\varepsilon), \tag*{(99)} P ν ε ( κ ε = k ) = − λ ε 1 ∫ Γ D ε 1 Γ D ε , k d ( ∂ n v ε ) , (99) where 1 Γ D ε ∈ C 0 ( Γ D ε ‾ ) \mathbb{1}_{\Gamma_D^\varepsilon}\in C^0(\overline{\Gamma_D^\varepsilon}) 1 Γ D ε ∈ C 0 ( Γ D ε ) λ ε \lambda_\varepsilon λ ε E = E 1 ( ( K ε ( j ) ) j = 1 N ) + E 2 ( ( K ε ( j ) ) j = 1 N ) \mathcal{E}=\mathcal{E}_1\left((K_\varepsilon^{(j)})_{j=1}^{N}\right)+\mathcal{E}_2\left((K_\varepsilon^{(j)})_{j=1}^{N}\right) E = E 1 ( ( K ε ( j ) ) j = 1 N ) + E 2 ( ( K ε ( j ) ) j = 1 N ) E → 0 \mathcal{E}\to0 E → 0 ε → 0 \varepsilon\to0 ε → 0
Mean exit time. Apply (B.2) with u 1 = v ε u_1=v_\varepsilon u 1 = v ε u 2 = φ ε u_2=\varphi_\varepsilon u 2 = φ ε v ε = 0 v_\varepsilon=0 v ε = 0 Γ D ε \Gamma_D^\varepsilon Γ D ε ∂ n φ ε = 0 \partial_n\varphi_\varepsilon=0 ∂ n φ ε = 0 Γ N ε \Gamma_N^\varepsilon Γ N ε ⟨ ∂ n φ ε , v ε ⟩ = 0 \langle\partial_n\varphi_\varepsilon,v_\varepsilon\rangle=0 ⟨ ∂ n φ ε , v ε ⟩ = 0 (6) , (17) and ∫ Ω ε v ε = 1 \int_{\Omega_\varepsilon}v_\varepsilon=1 ∫ Ω ε v ε = 1
⟨ ∂ n v ε , φ ε ⟩ = ( Δ v ε , φ ε ) Ω ε − ( v ε , Δ φ ε ) Ω ε = − λ ε ( v ε , φ ε ) Ω ε + C K ‾ ε . (100) \langle\partial_n v_\varepsilon,\varphi_\varepsilon\rangle=(\Delta v_\varepsilon,\varphi_\varepsilon)_{\Omega_\varepsilon}-(v_\varepsilon,\Delta\varphi_\varepsilon)_{\Omega_\varepsilon}=-\lambda_\varepsilon(v_\varepsilon,\varphi_\varepsilon)_{\Omega_\varepsilon}+C\overline{K}_\varepsilon. \tag*{(100)} ⟨ ∂ n v ε , φ ε ⟩ = ( Δ v ε , φ ε ) Ω ε − ( v ε , Δ φ ε ) Ω ε = − λ ε ( v ε , φ ε ) Ω ε + C K ε . (100) Plugging (B.3) and (B.4) into (B.11) leads to
λ ε ( 1 + O ( E 2 ( ( K ε ( j ) ) j = 1 N ) ) ) + O ( λ ε E 1 ( ( K ε ( j ) ) j = 1 N ) ) = λ ε ( 1 + O ( E ) ) = C K ‾ ε . \lambda_\varepsilon\left(1+O\left(\mathcal{E}_2\left((K_\varepsilon^{(j)})_{j=1}^{N}\right)\right)\right)+O\left(\lambda_\varepsilon\mathcal{E}_1\left((K_\varepsilon^{(j)})_{j=1}^{N}\right)\right)=\lambda_\varepsilon(1+O(\mathcal{E}))=C\overline{K}_\varepsilon. λ ε ( 1 + O ( E 2 ( ( K ε ( j ) ) j = 1 N ) ) ) + O ( λ ε E 1 ( ( K ε ( j ) ) j = 1 N ) ) = λ ε ( 1 + O ( E )) = C K ε . Isolating λ ε \lambda_\varepsilon λ ε ε \varepsilon ε
λ ε = C K ‾ ε ( 1 + O ( E ) ) , E v ε [ τ ε ] = 1 λ ε = 1 C K ‾ ε ( 1 + O ( E ) ) , (101) \lambda_\varepsilon=C\overline{K}_\varepsilon(1+O(\mathcal{E})),\qquad\mathbb{E}_{v_\varepsilon}[\tau_\varepsilon]=\frac{1}{\lambda_\varepsilon}=\frac{1}{C\overline{K}_\varepsilon}(1+O(\mathcal{E})), \tag*{(101)} λ ε = C K ε ( 1 + O ( E )) , E v ε [ τ ε ] = λ ε 1 = C K ε 1 ( 1 + O ( E )) , (101) which is the first claim.
Exit position. Fix k ∈ { 1 , … , N } k\in\{1,\ldots,N\} k ∈ { 1 , … , N } v ε v_\varepsilon v ε k k k
φ ε ( − k ) = 1 − ∑ j ≠ k ψ ε ( j ) = φ ε + ψ ε ( k ) , \varphi_\varepsilon^{(-k)}=1-\sum_{j\ne k}\psi_\varepsilon^{(j)}=\varphi_\varepsilon+\psi_\varepsilon^{(k)}, φ ε ( − k ) = 1 − j = k ∑ ψ ε ( j ) = φ ε + ψ ε ( k ) , with ψ ε ( j ) \psi_\varepsilon^{(j)} ψ ε ( j ) j ≠ k j\ne k j = k (17) , − Δ φ ε ( − k ) = C ( K ‾ ε − K ε ( k ) ) -\Delta\varphi_\varepsilon^{(-k)}=C(\overline{K}_\varepsilon-K_\varepsilon^{(k)}) − Δ φ ε ( − k ) = C ( K ε − K ε ( k ) ) Ω ε \Omega_\varepsilon Ω ε ∂ n φ ε ( − k ) = 0 \partial_n\varphi_\varepsilon^{(-k)}=0 ∂ n φ ε ( − k ) = 0 Γ ε N \Gamma_\varepsilon^N Γ ε N ∥ φ ε ( − k ) − 1 ∥ L 2 ( Ω ε ) = O ( E 2 ( ( K ε ( j ) ) j = 1 N ) ) \|\varphi_\varepsilon^{(-k)}-1\|_{L^2(\Omega_\varepsilon)}=O\left(\mathcal{E}_2\left((K_\varepsilon^{(j)})_{j=1}^{N}\right)\right) ∥ φ ε ( − k ) − 1 ∥ L 2 ( Ω ε ) = O ( E 2 ( ( K ε ( j ) ) j = 1 N ) ) 1 − 1 Γ ε k = ∑ j ≠ k 1 Γ ε j 1-\mathbb{1}_{\Gamma_\varepsilon^k}=\sum_{j\ne k}\mathbb{1}_{\Gamma_\varepsilon^j} 1 − 1 Γ ε k = ∑ j = k 1 Γ ε j Γ ε D \Gamma_\varepsilon^D Γ ε D
∥ φ ε ( − k ) − 1 Γ ε k ∥ L ∞ ( Γ ε D ) = ∥ ∑ j ≠ k ( 1 Γ ε j − ψ ε ( j ) ) ∥ L ∞ ( Γ ε D ) = O ( E 1 ( ( K ε ( j ) ) j = 1 N ) ) . (102) \left\|\varphi_\varepsilon^{(-k)}-\mathbb{1}_{\Gamma_\varepsilon^k}\right\|_{L^\infty(\Gamma_\varepsilon^D)}
=\left\|\sum_{j\ne k}\left(\mathbb{1}_{\Gamma_\varepsilon^j}-\psi_\varepsilon^{(j)}\right)\right\|_{L^\infty(\Gamma_\varepsilon^D)}
=O\left(\mathcal{E}_1\left((K_\varepsilon^{(j)})_{j=1}^{N}\right)\right). \tag*{(102)} φ ε ( − k ) − 1 Γ ε k L ∞ ( Γ ε D ) = j = k ∑ ( 1 Γ ε j − ψ ε ( j ) ) L ∞ ( Γ ε D ) = O ( E 1 ( ( K ε ( j ) ) j = 1 N ) ) . (102) Proceeding as in (B.11) with φ ε ( − k ) \varphi_\varepsilon^{(-k)} φ ε ( − k ) φ ε \varphi_\varepsilon φ ε
⟨ ∂ n v ε , φ ε ( − k ) ⟩ = − λ ε ( v ε , φ ε ( − k ) ) Ω ε + C ( K ‾ ε − K ε ( k ) ) = − C K ε ( k ) + ( C K ‾ ε − λ ε ) − λ ε ( v ε , φ ε ( − k ) − 1 ) Ω ε . (103) \langle\partial_n v_\varepsilon,\varphi_\varepsilon^{(-k)}\rangle=-\lambda_\varepsilon(v_\varepsilon,\varphi_\varepsilon^{(-k)})_{\Omega_\varepsilon}+C(\overline{K}_\varepsilon-K_\varepsilon^{(k)})=-CK_\varepsilon^{(k)}+(C\overline{K}_\varepsilon-\lambda_\varepsilon)-\lambda_\varepsilon(v_\varepsilon,\varphi_\varepsilon^{(-k)}-1)_{\Omega_\varepsilon}. \tag*{(103)} ⟨ ∂ n v ε , φ ε ( − k ) ⟩ = − λ ε ( v ε , φ ε ( − k ) ) Ω ε + C ( K ε − K ε ( k ) ) = − C K ε ( k ) + ( C K ε − λ ε ) − λ ε ( v ε , φ ε ( − k ) − 1 ) Ω ε . (103) The second term in the right-hand side of (B.14) is a O ( K ‾ ε E ) O(\overline{K}_\varepsilon\mathcal{E}) O ( K ε E ) M λ ε ∥ φ ε ( − k ) − 1 ∥ L 2 ( Ω ε ) = O ( K ‾ ε E 2 ( ( K ε ( j ) ) j = 1 N ) ) M\lambda_\varepsilon\|\varphi_\varepsilon^{(-k)}-1\|_{L^2(\Omega_\varepsilon)}=O\left(\overline{K}_\varepsilon\mathcal{E}_2\left((K_\varepsilon^{(j)})_{j=1}^{N}\right)\right) M λ ε ∥ φ ε ( − k ) − 1 ∥ L 2 ( Ω ε ) = O ( K ε E 2 ( ( K ε ( j ) ) j = 1 N ) )
Therefore, decomposing 1 Γ ε D = φ ε ( − k ) + ( 1 Γ ε D − φ ε ( − k ) ) \mathbb{1}_{\Gamma_\varepsilon^D}=\varphi_\varepsilon^{(-k)}+(\mathbb{1}_{\Gamma_\varepsilon^D}-\varphi_\varepsilon^{(-k)}) 1 Γ ε D = φ ε ( − k ) + ( 1 Γ ε D − φ ε ( − k ) )
∫ Γ ε D 1 Γ ε k d ( ∂ n v ε ) = ⟨ ∂ n v ε , φ ε ( − k ) ⟩ + O ( λ ε E ) = − C K ε ( k ) + O ( K ‾ ε E ) . \int_{\Gamma_\varepsilon^D}\mathbb{1}_{\Gamma_\varepsilon^k}\,\mathrm{d}(\partial_n v_\varepsilon)=\langle\partial_n v_\varepsilon,\varphi_\varepsilon^{(-k)}\rangle+O(\lambda_\varepsilon\mathcal{E})=-CK_\varepsilon^{(k)}+O(\overline{K}_\varepsilon\mathcal{E}). ∫ Γ ε D 1 Γ ε k d ( ∂ n v ε ) = ⟨ ∂ n v ε , φ ε ( − k ) ⟩ + O ( λ ε E ) = − C K ε ( k ) + O ( K ε E ) . Finally, plugging this estimate and (B.12) into (B.10), and since K ε ( k ) ≤ K ‾ ε K_\varepsilon^{(k)}\le\overline{K}_\varepsilon K ε ( k ) ≤ K ε
P v ε ( κ ε = k ) = C K ε ( k ) + O ( K ‾ ε E ) C K ‾ ε ( 1 + O ( E ) ) = K ε ( k ) K ‾ ε + O ( E ) , \mathbb{P}_{v_\varepsilon}(\kappa_\varepsilon=k)=\frac{CK_\varepsilon^{(k)}+O(\overline{K}_\varepsilon\mathcal{E})}{C\overline{K}_\varepsilon(1+O(\mathcal{E}))}=\frac{K_\varepsilon^{(k)}}{\overline{K}_\varepsilon}+O(\mathcal{E}), P v ε ( κ ε = k ) = C K ε ( 1 + O ( E )) C K ε ( k ) + O ( K ε E ) = K ε K ε ( k ) + O ( E ) , which concludes the proof. □
Appendix C. On the Green function of the Neumann problem
Several quasi-modes in this work depend on the Green function of the Laplacian with Neumann boundary conditions. In this section, we recall its definition and some of its properties.
Definition C.1. For any y ∈ ∂ Ω y\in\partial\Omega y ∈ ∂ Ω p ∈ ( 1 , 3 / 2 ) p\in(1,3/2) p ∈ ( 1 , 3/2 ) G ( ⋅ , y ) G(\cdot,y) G ( ⋅ , y ) W 1 , p ( Ω ) W^{1,p}(\Omega) W 1 , p ( Ω )
{ Δ x G ( ⋅ , y ) = 1 ∣ Ω ∣ in Ω , ∂ n G ( ⋅ , y ) = δ y on ∂ Ω . (104) \begin{cases}
\Delta_xG(\cdot,y)=\frac{1}{|\Omega|} & \text{in }\Omega,\\
\partial_nG(\cdot,y)=\delta_y & \text{on }\partial\Omega. \tag*{(104)}
\end{cases} { Δ x G ( ⋅ , y ) = ∣Ω∣ 1 ∂ n G ( ⋅ , y ) = δ y in Ω , on ∂ Ω. (104) The existence, uniqueness and regularity of G G G G G G g ∈ C 0 ( ∂ Ω ) g\in C^0(\partial\Omega) g ∈ C 0 ( ∂ Ω )
{ Δ u = 1 ∣ Ω ∣ ∫ ∂ Ω g in Ω , ∂ n u = g on ∂ Ω , (105) \begin{cases}
\Delta u=\frac{1}{|\Omega|}\int_{\partial\Omega}g & \text{in }\Omega,\\
\partial_nu=g & \text{on }\partial\Omega, \tag*{(105)}
\end{cases} { Δ u = ∣Ω∣ 1 ∫ ∂ Ω g ∂ n u = g in Ω , on ∂ Ω , (105) can be expressed as
u ( x ) = ∫ ∂ Ω G ( x , y ) g ( y ) d σ ( y ) . (106) u(x)=\int_{\partial\Omega}G(x,y)g(y)\,\mathrm{d}\sigma(y). \tag*{(106)} u ( x ) = ∫ ∂ Ω G ( x , y ) g ( y ) d σ ( y ) . (106) This function has been studied in the context of the narrow escape problem in [56 ] [15, 13] and as itself in [26 , 40 ] . In particular, let us recall the following result from [26 ] .
Lemma C.2. There exists K > 0 K>0 K > 0 x ∈ Ω x\in\Omega x ∈ Ω y ∈ ∂ Ω y\in\partial\Omega y ∈ ∂ Ω G G G Definition C.1 and its derivatives can be bounded as
∣ G ( x , y ) ∣ ≤ K ∣ x − y ∣ − 1 and ∣ ∇ x G ( x , y ) ∣ ≤ K ∣ x − y ∣ − 2 . \lvert G(x,y)\rvert\le K\lvert x-y\rvert^{-1}\qquad\text{and}\qquad\lvert\nabla_xG(x,y)\rvert\le K\lvert x-y\rvert^{-2}. ∣ G ( x , y )∣ ≤ K ∣ x − y ∣ − 1 and ∣ ∇ x G ( x , y )∣ ≤ K ∣ x − y ∣ − 2 . Proof. Direct consequence of [26 ] , Equation 4.3, as Ω × ∂ Ω ⊂ Ω ‾ × Ω ‾ ∖ { x = y } \Omega\times\partial\Omega\subset\overline{\Omega}\times\overline{\Omega}\setminus\{x=y\} Ω × ∂ Ω ⊂ Ω × Ω ∖ { x = y }
We also recall the main result of [40 ] .
Lemma C.3. In the limit ∣ x − y ∣ → 0 \lvert x-y\rvert\to0 ∣ x − y ∣ → 0 G G G Definition C.1 can be written as
G ( x , y ) = 1 2 π 1 ∣ x − y ∣ + O ( ∣ log ∣ x − y ∣ ∣ ) , G(x,y)=\frac{1}{2\pi}\frac{1}{\lvert x-y\rvert}+O\bigl(\lvert\log\lvert x-y\rvert\rvert\bigr), G ( x , y ) = 2 π 1 ∣ x − y ∣ 1 + O ( ∣ log ∣ x − y ∣∣ ) , where the O O O ∣ x − y ∣ → 0 \lvert x-y\rvert\to0 ∣ x − y ∣ → 0 y ∈ ∂ Ω y\in\partial\Omega y ∈ ∂ Ω x ∈ Ω ‾ ∖ { y } x\in\overline{\Omega}\setminus\{y\} x ∈ Ω ∖ { y }
With these estimates, we now extend (C.2) and (C.3) to the case of g ∈ L p ( ∂ Ω ) g\in L^p(\partial\Omega) g ∈ L p ( ∂ Ω ) p ∈ ( 1 , ∞ ) p\in(1,\infty) p ∈ ( 1 , ∞ )
Lemma C.4. Fix p ∈ ( 1 , ∞ ) p\in(1,\infty) p ∈ ( 1 , ∞ ) g ∈ L p ( ∂ Ω ) g\in L^p(\partial\Omega) g ∈ L p ( ∂ Ω ) φ \varphi φ Ω \Omega Ω
φ ( x ) = ∫ ∂ Ω G ( x , y ) g ( y ) d σ ( y ) . \varphi(x)=\int_{\partial\Omega}G(x,y)g(y)\,\mathrm{d}\sigma(y). φ ( x ) = ∫ ∂ Ω G ( x , y ) g ( y ) d σ ( y ) . Then φ \varphi φ W 1 , p ( Ω ) W^{1,p}(\Omega) W 1 , p ( Ω )
{ Δ φ = 1 ∣ Ω ∣ ∫ ∂ Ω g in Ω , ∂ n φ = g on ∂ Ω , (107) \begin{cases}
\Delta\varphi=\dfrac{1}{\lvert\Omega\rvert}\int_{\partial\Omega}g & \text{in }\Omega,\\
\partial_n\varphi=g & \text{on }\partial\Omega, \tag*{(107)}
\end{cases} ⎩ ⎨ ⎧ Δ φ = ∣ Ω ∣ 1 ∫ ∂ Ω g ∂ n φ = g in Ω , on ∂ Ω , (107) that is, for every v ∈ C 1 ( Ω ‾ ) v\in C^1(\overline{\Omega}) v ∈ C 1 ( Ω )
∫ Ω ∇ φ ⋅ ∇ v = − 1 ∣ Ω ∣ ( ∫ ∂ Ω g d σ ) ( ∫ Ω v ) + ∫ ∂ Ω g v d σ . \int_{\Omega}\nabla\varphi\cdot\nabla v=-\frac{1}{\lvert\Omega\rvert}\left(\int_{\partial\Omega}g\,\mathrm{d}\sigma\right)\left(\int_{\Omega}v\right)+\int_{\partial\Omega}gv\,\mathrm{d}\sigma. ∫ Ω ∇ φ ⋅ ∇ v = − ∣ Ω ∣ 1 ( ∫ ∂ Ω g d σ ) ( ∫ Ω v ) + ∫ ∂ Ω g v d σ . Moreover, for almost every x ∈ Ω x\in\Omega x ∈ Ω
∇ φ ( x ) = ∫ ∂ Ω ∇ x G ( x , y ) g ( y ) d σ ( y ) . (108) \nabla\varphi(x)=\int_{\partial\Omega}\nabla_xG(x,y)g(y)\,\mathrm{d}\sigma(y). \tag*{(108)} ∇ φ ( x ) = ∫ ∂ Ω ∇ x G ( x , y ) g ( y ) d σ ( y ) . (108) Finally, if p ≥ 4 / 3 p\ge4/3 p ≥ 4/3 φ ∈ H 1 ( Ω ) \varphi\in H^1(\Omega) φ ∈ H 1 ( Ω ) v ∈ H 1 ( Ω ) v\in H^1(\Omega) v ∈ H 1 ( Ω )
Proof. First notice that φ \varphi φ Ω \Omega Ω x ∈ Ω x\in\Omega x ∈ Ω Lemma C.2 gives ∣ G ( x , ⋅ ) ∣ ≤ K dist ( x , ∂ Ω ) − 1 \lvert G(x,\cdot)\rvert\le K\operatorname{dist}(x,\partial\Omega)^{-1} ∣ G ( x , ⋅ )∣ ≤ K dist ( x , ∂ Ω ) − 1 ∂ Ω \partial\Omega ∂ Ω g ∈ L p ( ∂ Ω ) ⊂ L 1 ( ∂ Ω ) g\in L^p(\partial\Omega)\subset L^1(\partial\Omega) g ∈ L p ( ∂ Ω ) ⊂ L 1 ( ∂ Ω )
Weak solution. From [2, Proposition 2.3], since g ∈ L p ( ∂ Ω ) g\in L^p(\partial\Omega) g ∈ L p ( ∂ Ω ) φ ^ ∈ W 1 , p ( Ω ) \widehat{\varphi}\in W^{1,p}(\Omega) φ ∈ W 1 , p ( Ω )
∥ φ ^ ∥ W 1 , p ( Ω ) ≤ C ∥ g ∥ L p ( ∂ Ω ) . (109) \|\widehat{\varphi}\|_{W^{1,p}(\Omega)}\le C\|g\|_{L^p(\partial\Omega)}. \tag*{(109)} ∥ φ ∥ W 1 , p ( Ω ) ≤ C ∥ g ∥ L p ( ∂ Ω ) . (109) It remains to show that φ = φ ^ \varphi=\widehat{\varphi} φ = φ p < ∞ p<\infty p < ∞ ( g n ) n ∈ N (g_n)_{n\in\mathbb{N}} ( g n ) n ∈ N ∂ Ω \partial\Omega ∂ Ω g n → g g_n\to g g n → g L p ( ∂ Ω ) L^p(\partial\Omega) L p ( ∂ Ω ) n ∈ N n\in\mathbb{N} n ∈ N φ n ( x ) = ∫ ∂ Ω G ( x , y ) g n ( y ) d σ ( y ) \varphi_n(x)=\int_{\partial\Omega}G(x,y)g_n(y)\,\mathrm{d}\sigma(y) φ n ( x ) = ∫ ∂ Ω G ( x , y ) g n ( y ) d σ ( y ) φ n \varphi_n φ n g n g_n g n g g g W 1 , p ( Ω ) W^{1,p}(\Omega) W 1 , p ( Ω ) g n g_n g n g n − g g_n-g g n − g
∥ φ n − φ ^ ∥ W 1 , p ( Ω ) ≤ C ∥ g n − g ∥ L p ( ∂ Ω ) → n → ∞ 0 , \|\varphi_n-\widehat{\varphi}\|_{W^{1,p}(\Omega)}
\le C\|g_n-g\|_{L^p(\partial\Omega)}
\xrightarrow[n\to\infty]{}0, ∥ φ n − φ ∥ W 1 , p ( Ω ) ≤ C ∥ g n − g ∥ L p ( ∂ Ω ) n → ∞ 0 , so that, up to a subsequence, φ n → φ ^ \varphi_n \to\widehat{\varphi} φ n → φ Ω \Omega Ω x ∈ Ω x \in\Omega x ∈ Ω G ( x , ⋅ ) G(x,\cdot) G ( x , ⋅ ) ∂ Ω \partial\Omega ∂ Ω
∣ φ n ( x ) − φ ( x ) ∣ ≤ ∥ G ( x , ⋅ ) ∥ L p ′ ( ∂ Ω ) ∥ g n − g ∥ L p ( ∂ Ω ) → n → ∞ 0. (110) \left|\varphi_n(x)-\varphi(x)\right| \le\left\|G(x,\cdot)\right\|_{L^{p'}(\partial\Omega)}\left\|g_n-g\right\|_{L^p(\partial\Omega)} \xrightarrow[n\to\infty]{} 0. \tag*{(110)} ∣ φ n ( x ) − φ ( x ) ∣ ≤ ∥ G ( x , ⋅ ) ∥ L p ′ ( ∂ Ω ) ∥ g n − g ∥ L p ( ∂ Ω ) n → ∞ 0. (110) where p ′ = p p − 1 p'=\frac{p}{p-1} p ′ = p − 1 p p p p φ = φ ^ \varphi=\widehat{\varphi} φ = φ Ω \Omega Ω
Gradient. Let u ∈ C c ∞ ( Ω ) u\in C^\infty_c(\Omega) u ∈ C c ∞ ( Ω )
⟨ ∇ φ , u ⟩ D ′ ( Ω ) , C c ∞ ( Ω ) = − ∫ Ω φ ∇ u = − ∫ Ω ( ∫ ∂ Ω G ( x , y ) g ( y ) d σ ( y ) ) ∇ u ( x ) d x . (111) \langle\nabla\varphi,u\rangle_{D'(\Omega),C^\infty_c(\Omega)}
=-\int_\Omega\varphi\nabla u
=-\int_\Omega\left(\int_{\partial\Omega}G(x,y)g(y)\,\mathrm{d}\sigma(y)\right)\nabla u(x)\,\mathrm{d}x. \tag*{(111)} ⟨ ∇ φ , u ⟩ D ′ ( Ω ) , C c ∞ ( Ω ) = − ∫ Ω φ ∇ u = − ∫ Ω ( ∫ ∂ Ω G ( x , y ) g ( y ) d σ ( y ) ) ∇ u ( x ) d x . (111) To exchange the integrals, notice that, as Ω ⊂ R 3 \Omega\subset\mathbb{R}^3 Ω ⊂ R 3 M = sup y ∈ ∂ Ω ∫ Ω ( ∣ x − y ∣ − 1 + ∣ x − y ∣ − 2 ) d x < ∞ M=\sup_{y\in\partial\Omega}\int_\Omega\left(|x-y|^{-1}+|x-y|^{-2}\right)\,\mathrm{d}x<\infty M = sup y ∈ ∂ Ω ∫ Ω ( ∣ x − y ∣ − 1 + ∣ x − y ∣ − 2 ) d x < ∞
∫ ∂ Ω ∫ Ω ( ∣ G ( x , y ) ∣ + ∣ ∇ x G ( x , y ) ∣ ) ∣ g ( y ) ∣ d x d σ ( y ) ≤ K M ∥ g ∥ L 1 ( ∂ Ω ) < ∞ . (112) \int_{\partial\Omega}\int_\Omega\left(|G(x,y)|+|\nabla_xG(x,y)|\right)|g(y)|\,\mathrm{d}x\,\mathrm{d}\sigma(y)\le KM\|g\|_{L^1(\partial\Omega)}<\infty. \tag*{(112)} ∫ ∂ Ω ∫ Ω ( ∣ G ( x , y ) ∣ + ∣ ∇ x G ( x , y ) ∣ ) ∣ g ( y ) ∣ d x d σ ( y ) ≤ K M ∥ g ∥ L 1 ( ∂ Ω ) < ∞. (112) Moreover, for a fixed y ∈ ∂ Ω y\in\partial\Omega y ∈ ∂ Ω Δ x G ( ⋅ , y ) \Delta_xG(\cdot,y) Δ x G ( ⋅ , y ) Ω \Omega Ω G ( ⋅ , y ) G(\cdot,y) G ( ⋅ , y ) Ω \Omega Ω u u u Ω \Omega Ω − ∫ Ω G ( x , y ) ∇ u ( x ) d x = ∫ Ω ∇ x G ( x , y ) u ( x ) d x -\int_\Omega G(x,y)\nabla u(x)\,\mathrm{d}x=\int_\Omega\nabla_xG(x,y)u(x)\,\mathrm{d}x − ∫ Ω G ( x , y ) ∇ u ( x ) d x = ∫ Ω ∇ x G ( x , y ) u ( x ) d x u u u ∇ u \nabla u ∇ u
⟨ ∇ φ , u ⟩ D ′ ( Ω ) , C c ∞ ( Ω ) = − ∫ ∂ Ω ( ∫ Ω G ( x , y ) ∇ u ( x ) d x ) g ( y ) d σ ( y ) = ∫ ∂ Ω ( ∫ Ω ∇ x G ( x , y ) u ( x ) d x ) g ( y ) d σ ( y ) = ∫ Ω ( ∫ ∂ Ω ∇ x G ( x , y ) g ( y ) d σ ( y ) ) u ( x ) d x . \begin{aligned}
\langle\nabla\varphi,u\rangle_{D'(\Omega),C^\infty_c(\Omega)}
=-\int_{\partial\Omega}\left(\int_\Omega G(x,y)\nabla u(x)\,\mathrm{d}x\right)g(y)\,\mathrm{d}\sigma(y) \\
&=\int_{\partial\Omega}\left(\int_\Omega\nabla_xG(x,y)u(x)\,\mathrm{d}x\right)g(y)\,\mathrm{d}\sigma(y) \\
&=\int_\Omega\left(\int_{\partial\Omega}\nabla_xG(x,y)g(y)\,\mathrm{d}\sigma(y)\right)u(x)\,\mathrm{d}x.
\end{aligned} ⟨ ∇ φ , u ⟩ D ′ ( Ω ) , C c ∞ ( Ω ) = − ∫ ∂ Ω ( ∫ Ω G ( x , y ) ∇ u ( x ) d x ) g ( y ) d σ ( y ) = ∫ ∂ Ω ( ∫ Ω ∇ x G ( x , y ) u ( x ) d x ) g ( y ) d σ ( y ) = ∫ Ω ( ∫ ∂ Ω ∇ x G ( x , y ) g ( y ) d σ ( y ) ) u ( x ) d x . Hence (C.5) holds in the sense of distributions. As both sides of (C.5) belong to L 1 ( Ω ) L^1(\Omega) L 1 ( Ω ) φ ∈ W 1 , p ( Ω ) \varphi\in W^{1,p}(\Omega) φ ∈ W 1 , p ( Ω ) Ω \Omega Ω
H 1 H^1 H 1 p ≥ 4 / 3 p\ge4/3 p ≥ 4/3 g ∈ L 4 / 3 ( ∂ Ω ) g\in L^{4/3}(\partial\Omega) g ∈ L 4/3 ( ∂ Ω ) ∂ Ω \partial\Omega ∂ Ω ∂ Ω \partial\Omega ∂ Ω H 1 / 2 ( ∂ Ω ) ⊂ L 4 ( ∂ Ω ) H^{1/2}(\partial\Omega)\subset L^4(\partial\Omega) H 1/2 ( ∂ Ω ) ⊂ L 4 ( ∂ Ω )
∣ ∫ ∂ Ω g v d σ ∣ ≤ ∥ g ∥ L 4 / 3 ( ∂ Ω ) ∥ v ∥ L 4 ( ∂ Ω ) ≤ C ∥ g ∥ L 4 / 3 ( ∂ Ω ) ∥ v ∥ H 1 ( Ω ) , v ∈ H 1 ( Ω ) , \left|\int_{\partial\Omega}gv\,\mathrm{d}\sigma\right|
\le\|g\|_{L^{4/3}(\partial\Omega)}\|v\|_{L^4(\partial\Omega)}
\le C\|g\|_{L^{4/3}(\partial\Omega)}\|v\|_{H^1(\Omega)},\qquad v\in H^1(\Omega), ∫ ∂ Ω g v d σ ≤ ∥ g ∥ L 4/3 ( ∂ Ω ) ∥ v ∥ L 4 ( ∂ Ω ) ≤ C ∥ g ∥ L 4/3 ( ∂ Ω ) ∥ v ∥ H 1 ( Ω ) , v ∈ H 1 ( Ω ) , so that the right-hand side of the weak formulation is a continuous linear form on H 1 ( Ω ) H^1(\Omega) H 1 ( Ω ) H 1 ( Ω ) H^1(\Omega) H 1 ( Ω ) Ω \Omega Ω φ ~ ∈ H 1 ( Ω ) \widetilde{\varphi}\in H^1(\Omega) φ ∈ H 1 ( Ω ) v ∈ H 1 ( Ω ) v\in H^1(\Omega) v ∈ H 1 ( Ω ) Ω \Omega Ω φ ~ ∈ W 1 , 4 / 3 ( Ω ) \widetilde{\varphi}\in W^{1,4/3}(\Omega) φ ∈ W 1 , 4/3 ( Ω ) φ ~ \widetilde{\varphi} φ W 1 , 4 / 3 ( Ω ) W^{1,4/3}(\Omega) W 1 , 4/3 ( Ω ) 4 / 3 4/3 4/3 φ = φ ~ ∈ H 1 ( Ω ) \varphi=\widetilde{\varphi}\in H^1(\Omega) φ = φ ∈ H 1 ( Ω )
Let us conclude this section with a uniform bound on the L 2 L^2 L 2 G G G
Lemma C.5. There exists a constant M M M Ω \Omega Ω
sup y ∈ ∂ Ω ∫ Ω G ( x , y ) 2 d x < M . (113) \sup_{y\in\partial\Omega}\int_\Omega G(x,y)^2\,\mathrm{d}x<M. \tag*{(113)} y ∈ ∂ Ω sup ∫ Ω G ( x , y ) 2 d x < M . (113) Proof. Let D > 0 D>0 D > 0 Ω ⊂ B ( y , D ) \Omega\subset B(y,D) Ω ⊂ B ( y , D ) y ∈ Ω ‾ y\in\overline{\Omega} y ∈ Ω
∫ Ω G ( x , y ) 2 d x ≤ K 2 ∫ Ω d x ∣ x − y ∣ 2 . \int_\Omega G(x,y)^2\,\mathrm{d}x\le K^2\int_\Omega\frac{\mathrm{d}x}{|x-y|^2}. ∫ Ω G ( x , y ) 2 d x ≤ K 2 ∫ Ω ∣ x − y ∣ 2 d x . Then, passing to spherical coordinates centered at y y y Ω ⊂ B ( y , D ) \Omega\subset B(y,D) Ω ⊂ B ( y , D )
∫ Ω d x ∣ x − y ∣ 2 ≤ ∫ B ( y , D ) d x ∣ x − y ∣ 2 = ∫ 0 2 π ∫ 0 π ∫ 0 D 1 r 2 r 2 sin θ d r d θ d ϕ = 4 π D < ∞ , \int_\Omega\frac{\mathrm{d}x}{|x-y|^2}
\le\int_{B(y,D)}\frac{\mathrm{d}x}{|x-y|^2}
=\int_0^{2\pi}\int_0^\pi\int_0^D\frac{1}{r^2}r^2\sin\theta\,\mathrm{d}r\,\mathrm{d}\theta\,\mathrm{d}\phi
=4\pi D<\infty, ∫ Ω ∣ x − y ∣ 2 d x ≤ ∫ B ( y , D ) ∣ x − y ∣ 2 d x = ∫ 0 2 π ∫ 0 π ∫ 0 D r 2 1 r 2 sin θ d r d θ d ϕ = 4 π D < ∞ , which is indeed bounded uniformly in y ∈ ∂ Ω y\in\partial\Omega y ∈ ∂ Ω
:::
Computation of the asymptotics of the integrals
This section is devoted to the computation of the asymptotics of the integrals appearing in the proof of Proposition 3.3 and Lemma 4.1.
In the case of spherical holes from Section 3.2
D.1. In the case of spherical holes from Section 3.2. In this subsection, we go back to the notation of Section 3.2 and compute the asymptotics of the integrals appearing in the proof of Proposition 3.3. For this let us introduce a local parametrization of the boundary ∂ Ω \partial\Omega ∂ Ω r > 0 r>0 r > 0 k ∈ { 1 , … , N } k\in\{1,\ldots,N\} k ∈ { 1 , … , N } ∂ Ω \partial\Omega ∂ Ω x ( k ) x^{(k)} x ( k )
y ( ⋅ ) : B ( 0 , r ) ⊂ R 2 ⟶ ∂ Ω , y ( z ) : = exp x ( k ) ∂ Ω ( z ) , y(\cdot): B(0,r)\subset\mathbb{R}^{2}\longrightarrow\partial\Omega,\qquad y(z):=\exp_{x^{(k)}}^{\partial\Omega}(z), y ( ⋅ ) : B ( 0 , r ) ⊂ R 2 ⟶ ∂ Ω , y ( z ) := exp x ( k ) ∂ Ω ( z ) , where T x ( k ) ∂ Ω ≃ R 2 T_{x^{(k)}}\partial\Omega\simeq\mathbb{R}^{2} T x ( k ) ∂ Ω ≃ R 2 ( e 1 , e 2 ) (e_1,e_2) ( e 1 , e 2 ) n ( x ( k ) ) n(x^{(k)}) n ( x ( k ) ) Ω \Omega Ω x ( k ) x^{(k)} x ( k ) ∂ Ω \partial\Omega ∂ Ω r > 0 r>0 r > 0 k k k r ε ( k ) < r r_{\varepsilon}^{(k)}<r r ε ( k ) < r y ( ⋅ ) y(\cdot) y ( ⋅ ) 0 0 0 y y y
Gauss’s lemma. Radial lines through the origin are unit-speed geodesics issued from x ( k ) x^{(k)} x ( k )
d ∂ Ω ( y ( z ) , x ( k ) ) = ∣ z ∣ , hence y − 1 ( Γ ε k ) = B ( 0 , r ε ( k ) ) exactly . (114) d_{\partial\Omega}(y(z),x^{(k)})=|z|,\qquad\text{hence}\qquad y^{-1}(\Gamma_{\varepsilon}^{k})=B(0,r_{\varepsilon}^{(k)})\text{ exactly}. \tag*{(114)} d ∂ Ω ( y ( z ) , x ( k ) ) = ∣ z ∣ , hence y − 1 ( Γ ε k ) = B ( 0 , r ε ( k ) ) exactly . (114) Jacobian. Writing the surface measure on Γ ε k \Gamma_{\varepsilon}^{k} Γ ε k d σ ( y ) = J k ( z ) d z d\sigma(y)=J_k(z)\,dz d σ ( y ) = J k ( z ) d z k k k
∣ J k ( z ) − 1 ∣ ≤ C ∣ z ∣ 2 , C = C ( Ω ) > 0. (115) |J_k(z)-1|\leq C|z|^2,\qquad C=C(\Omega)>0. \tag*{(115)} ∣ J k ( z ) − 1∣ ≤ C ∣ z ∣ 2 , C = C ( Ω ) > 0. (115) Extrinsic distance. The map exp x ( k ) ∂ Ω \exp_{x^{(k)}}^{\partial\Omega} exp x ( k ) ∂ Ω I I x ( k ) \mathrm{II}_{x^{(k)}} II x ( k ) ∂ Ω \partial\Omega ∂ Ω x ( k ) x^{(k)} x ( k )
y ( z ) = x ( k ) + z + 1 2 I I x ( k ) ( z , z ) n ( x ( k ) ) + O ( ∣ z ∣ 3 ) . (116) y(z)=x^{(k)}+z+\frac{1}{2}\mathrm{II}_{x^{(k)}}(z,z)n(x^{(k)})+O(|z|^3). \tag*{(116)} y ( z ) = x ( k ) + z + 2 1 II x ( k ) ( z , z ) n ( x ( k ) ) + O ( ∣ z ∣ 3 ) . (116) Notice that in (D.3), the quadratic correction is normal to T x ( k ) ∂ Ω T_{x^{(k)}}\partial\Omega T x ( k ) ∂ Ω w 0 , w ∈ B ( 0 , 1 ) ‾ w_0,w\in\overline{B(0,1)} w 0 , w ∈ B ( 0 , 1 )
∣ y ( r ε ( k ) w 0 ) − y ( r ε ( k ) w ) ∣ = r ε ( k ) ∣ w 0 − w ∣ ( 1 + O ( ( r ε ( k ) ) 2 ) ) , uniformly . (117) \left|y(r_{\varepsilon}^{(k)}w_0)-y(r_{\varepsilon}^{(k)}w)\right|=r_{\varepsilon}^{(k)}|w_0-w|\left(1+O((r_{\varepsilon}^{(k)})^2)\right),
\qquad\text{uniformly}. \tag*{(117)} y ( r ε ( k ) w 0 ) − y ( r ε ( k ) w ) = r ε ( k ) ∣ w 0 − w ∣ ( 1 + O (( r ε ( k ) ) 2 ) ) , uniformly . (117) Let us now state the main result of this subsection, which is a direct computation of the integrals needed in Section 3.2.
Lemma D.1. Fix k ∈ { 1 , … , N } k\in\{1,\ldots,N\} k ∈ { 1 , … , N } ρ ε ( k ) \rho_{\varepsilon}^{(k)} ρ ε ( k ) Γ ε k \Gamma_{\varepsilon}^{k} Γ ε k
ρ ε ( k ) : y ↦ ( ( r ε ( k ) ) 2 − d ∂ Ω ( y , x ( k ) ) 2 ) − 1 / 2 . (118) \rho_{\varepsilon}^{(k)}:y\mapsto\left((r_{\varepsilon}^{(k)})^2-d_{\partial\Omega}(y,x^{(k)})^2\right)^{-1/2}. \tag*{(118)} ρ ε ( k ) : y ↦ ( ( r ε ( k ) ) 2 − d ∂ Ω ( y , x ( k ) ) 2 ) − 1/2 . (118) Then, as ε → 0 \varepsilon\to0 ε → 0
(a) (Integrability) for every p ∈ [ 1 , 2 ) p\in[1,2) p ∈ [ 1 , 2 )
∫ Γ ε k ( ρ ε ( k ) ( y ) ) p d σ ( y ) = 2 π 2 − p ( r ε ( k ) ) 2 − p ( 1 + O ( ( r ε ( k ) ) 2 ) ) , \int_{\Gamma_{\varepsilon}^{k}}\left(\rho_{\varepsilon}^{(k)}(y)\right)^p\,d\sigma(y)=\frac{2\pi}{2-p}(r_{\varepsilon}^{(k)})^{2-p}\left(1+O((r_{\varepsilon}^{(k)})^2)\right), ∫ Γ ε k ( ρ ε ( k ) ( y ) ) p d σ ( y ) = 2 − p 2 π ( r ε ( k ) ) 2 − p ( 1 + O (( r ε ( k ) ) 2 ) ) , while ρ ε ( k ) ∉ L 2 ( Γ ε k ) \rho_{\varepsilon}^{(k)}\notin L^2(\Gamma_{\varepsilon}^{k}) ρ ε ( k ) ∈ / L 2 ( Γ ε k )
(b) (Mass) in particular, for p = 1 p=1 p = 1
∫ Γ ε k ρ ε ( k ) ( y ) d σ ( y ) = 2 π r ε ( k ) ( 1 + O ( ( r ε ( k ) ) 2 ) ) ; \int_{\Gamma_{\varepsilon}^{k}}\rho_{\varepsilon}^{(k)}(y)\,d\sigma(y)=2\pi r_{\varepsilon}^{(k)}\left(1+O((r_{\varepsilon}^{(k)})^2)\right); ∫ Γ ε k ρ ε ( k ) ( y ) d σ ( y ) = 2 π r ε ( k ) ( 1 + O (( r ε ( k ) ) 2 ) ) ; (c) (Newtonian potential) for every x ∈ Γ ε k x\in\Gamma_{\varepsilon}^{k} x ∈ Γ ε k
∫ Γ ε k ρ ε ( k ) ( y ) 2 π ∣ x − y ∣ d σ ( y ) = π 2 + O ( r ε ( k ) ) ; \int_{\Gamma_{\varepsilon}^{k}}\frac{\rho_{\varepsilon}^{(k)}(y)}{2\pi|x-y|}\,d\sigma(y)=\frac{\pi}{2}+O(r_{\varepsilon}^{(k)}); ∫ Γ ε k 2 π ∣ x − y ∣ ρ ε ( k ) ( y ) d σ ( y ) = 2 π + O ( r ε ( k ) ) ; (d) (Logarithmic potential) for every x ∈ Γ ε k x\in\Gamma_{\varepsilon}^{k} x ∈ Γ ε k
∫ Γ ε k ρ ε ( k ) ( y ) log ∣ x − y ∣ d σ ( y ) = O ( r ε ( k ) ∣ log r ε ( k ) ∣ ) . \int_{\Gamma_{\varepsilon}^{k}}\rho_{\varepsilon}^{(k)}(y)\log|x-y|\,d\sigma(y)=O\left(r_{\varepsilon}^{(k)}\left|\log r_{\varepsilon}^{(k)}\right|\right). ∫ Γ ε k ρ ε ( k ) ( y ) log ∣ x − y ∣ d σ ( y ) = O ( r ε ( k ) log r ε ( k ) ) . Proof. Fix k ∈ { 1 , … , N } k\in\{1,\ldots,N\} k ∈ { 1 , … , N } y ∈ Γ ε k y\in\Gamma_{\varepsilon}^{k} y ∈ Γ ε k ρ ε ( k ) \rho_{\varepsilon}^{(k)} ρ ε ( k ) y = y ( z ) y=y(z) y = y ( z ) z ∈ B ( 0 , r ε ( k ) ) z\in B(0,r_{\varepsilon}^{(k)}) z ∈ B ( 0 , r ε ( k ) ) z = r ε ( k ) w z=r_{\varepsilon}^{(k)}w z = r ε ( k ) w
ρ ε ( k ) ( y ( r ε ( k ) w ) ) = ( ( r ε ( k ) ) 2 − ∣ r ε ( k ) w ∣ 2 ) − 1 / 2 = 1 r ε ( k ) 1 1 − ∣ w ∣ 2 . (119) \rho_{\varepsilon}^{(k)}\left(y(r_{\varepsilon}^{(k)}w)\right)=\left((r_{\varepsilon}^{(k)})^2-|r_{\varepsilon}^{(k)}w|^2\right)^{-1/2}=\frac{1}{r_{\varepsilon}^{(k)}}\frac{1}{\sqrt{1-|w|^2}}. \tag*{(119)} ρ ε ( k ) ( y ( r ε ( k ) w ) ) = ( ( r ε ( k ) ) 2 − ∣ r ε ( k ) w ∣ 2 ) − 1/2 = r ε ( k ) 1 1 − ∣ w ∣ 2 1 . (119) Thus, for any F : Γ ε k → R F:\Gamma_{\varepsilon}^{k}\to\mathbb{R} F : Γ ε k → R
∫ Γ ε k ρ ε ( k ) ( y ) F ( y ) d σ ( y ) = r ε ( k ) ∫ B ( 0 , 1 ) J k ( r ε ( k ) w ) 1 − ∣ w ∣ 2 F ( y ( r ε ( k ) w ) ) d w . (120) \int_{\Gamma_{\varepsilon}^{k}}\rho_{\varepsilon}^{(k)}(y)F(y)\,\mathrm{d}\sigma(y)=r_{\varepsilon}^{(k)}\int_{B(0,1)}\frac{J_k(r_{\varepsilon}^{(k)}w)}{\sqrt{1-|w|^2}}F(y(r_{\varepsilon}^{(k)}w))\,\mathrm{d}w. \tag*{(120)} ∫ Γ ε k ρ ε ( k ) ( y ) F ( y ) d σ ( y ) = r ε ( k ) ∫ B ( 0 , 1 ) 1 − ∣ w ∣ 2 J k ( r ε ( k ) w ) F ( y ( r ε ( k ) w )) d w . (120) Using (D.2), we can expand the integrand in (D.7) as
∫ Γ ε k ρ ε ( k ) ( y ) F ( y ) d σ ( y ) = r ε ( k ) ∫ B ( 0 , 1 ) F ( y ( r ε ( k ) w ) ) 1 − ∣ w ∣ 2 d w + O ( ( r ε ( k ) ) 3 ) ∫ B ( 0 , 1 ) ∣ F ( y ( r ε ( k ) w ) ) ∣ 1 − ∣ w ∣ 2 d w . (121) \int_{\Gamma_{\varepsilon}^{k}}\rho_{\varepsilon}^{(k)}(y)F(y)\,\mathrm{d}\sigma(y)=r_{\varepsilon}^{(k)}\int_{B(0,1)}\frac{F(y(r_{\varepsilon}^{(k)}w))}{\sqrt{1-|w|^2}}\,\mathrm{d}w+O((r_{\varepsilon}^{(k)})^3)\int_{B(0,1)}\frac{|F(y(r_{\varepsilon}^{(k)}w))|}{\sqrt{1-|w|^2}}\,\mathrm{d}w. \tag*{(121)} ∫ Γ ε k ρ ε ( k ) ( y ) F ( y ) d σ ( y ) = r ε ( k ) ∫ B ( 0 , 1 ) 1 − ∣ w ∣ 2 F ( y ( r ε ( k ) w )) d w + O (( r ε ( k ) ) 3 ) ∫ B ( 0 , 1 ) 1 − ∣ w ∣ 2 ∣ F ( y ( r ε ( k ) w )) ∣ d w . (121) (a) Integrability. For p ∈ [ 1 , 2 ) p\in[1,2) p ∈ [ 1 , 2 ) y = y ( r ε ( k ) w ) y=y(r_{\varepsilon}^{(k)}w) y = y ( r ε ( k ) w )
∫ Γ ε k ρ ε ( k ) ( y ) p d σ ( y ) = ( r ε ( k ) ) 2 − p ∫ B ( 0 , 1 ) J k ( r ε ( k ) w ) ( 1 − ∣ w ∣ 2 ) p / 2 d w = ( r ε ( k ) ) 2 − p ( 1 + O ( ( r ε ( k ) ) 2 ) ) 2 π ∫ 0 1 s d s ( 1 − s 2 ) p / 2 . \int_{\Gamma_{\varepsilon}^{k}}\rho_{\varepsilon}^{(k)}(y)^p\,\mathrm{d}\sigma(y)=(r_{\varepsilon}^{(k)})^{2-p}\int_{B(0,1)}\frac{J_k(r_{\varepsilon}^{(k)}w)}{(1-|w|^2)^{p/2}}\,\mathrm{d}w=(r_{\varepsilon}^{(k)})^{2-p}(1+O((r_{\varepsilon}^{(k)})^2))\,2\pi\int_0^1\frac{s\,\mathrm{d}s}{(1-s^2)^{p/2}}. ∫ Γ ε k ρ ε ( k ) ( y ) p d σ ( y ) = ( r ε ( k ) ) 2 − p ∫ B ( 0 , 1 ) ( 1 − ∣ w ∣ 2 ) p /2 J k ( r ε ( k ) w ) d w = ( r ε ( k ) ) 2 − p ( 1 + O (( r ε ( k ) ) 2 )) 2 π ∫ 0 1 ( 1 − s 2 ) p /2 s d s . The last integral equals 1 / ( 2 − p ) 1/(2-p) 1/ ( 2 − p ) p = 2 p=2 p = 2 ρ ε ( k ) ∉ L 2 ( Γ ε k ) \rho_{\varepsilon}^{(k)}\notin L^2(\Gamma_{\varepsilon}^{k}) ρ ε ( k ) ∈ / L 2 ( Γ ε k )
(b) Mass. This is (a) with p = 1 p=1 p = 1
For the two remaining estimates, we apply (D.8) with F ( y ) = ∣ x − y ∣ − 1 F(y)=|x-y|^{-1} F ( y ) = ∣ x − y ∣ − 1 F ( y ) = log ∣ x − y ∣ F(y)=\log|x-y| F ( y ) = log ∣ x − y ∣ x = y ( r ε ( k ) w 0 ) ∈ Γ ε k x=y(r_{\varepsilon}^{(k)}w_0)\in\Gamma_{\varepsilon}^{k} x = y ( r ε ( k ) w 0 ) ∈ Γ ε k w 0 ∈ B ( 0 , 1 ) w_0\in B(0,1) w 0 ∈ B ( 0 , 1 )
(c) Newtonian potential. By (D.4), ∣ x − y ( r ε ( k ) w ) ∣ − 1 = 1 r ε ( k ) ∣ w 0 − w ∣ ( 1 + O ( ( r ε ( k ) ) 2 ) ) |x-y(r_{\varepsilon}^{(k)}w)|^{-1}=\dfrac{1}{r_{\varepsilon}^{(k)}|w_0-w|}(1+O((r_{\varepsilon}^{(k)})^2)) ∣ x − y ( r ε ( k ) w ) ∣ − 1 = r ε ( k ) ∣ w 0 − w ∣ 1 ( 1 + O (( r ε ( k ) ) 2 )) w ≠ w 0 w\ne w_0 w = w 0
∫ Γ ε k ρ ε ( k ) ( y ) d σ ( y ) 2 π ∣ x − y ∣ = ( 1 + O ( ( r ε ( k ) ) 2 ) ) ∫ B ( 0 , 1 ) d w 2 π 1 − ∣ w ∣ 2 ∣ w 0 − w ∣ . (122) \int_{\Gamma_{\varepsilon}^{k}}\rho_{\varepsilon}^{(k)}(y)\frac{\mathrm{d}\sigma(y)}{2\pi|x-y|}=(1+O((r_{\varepsilon}^{(k)})^2))\int_{B(0,1)}\frac{\mathrm{d}w}{2\pi\sqrt{1-|w|^2}|w_0-w|}. \tag*{(122)} ∫ Γ ε k ρ ε ( k ) ( y ) 2 π ∣ x − y ∣ d σ ( y ) = ( 1 + O (( r ε ( k ) ) 2 )) ∫ B ( 0 , 1 ) 2 π 1 − ∣ w ∣ 2 ∣ w 0 − w ∣ d w . (122) The integral in the right-hand side of (D.9) is well known in potential theory, and its value is independent of w 0 ∈ B ( 0 , 1 ) w_0\in B(0,1) w 0 ∈ B ( 0 , 1 ) [45 Equation 4.6] ). Hence, the integral can be evaluated as
∫ Γ ε k ρ ε ( k ) ( y ) d σ ( y ) 2 π ∣ x − y ∣ = π 2 ( 1 + O ( ( r ε ( k ) ) 2 ) ) , uniformly in x ∈ Γ ε k . \int_{\Gamma_{\varepsilon}^{k}}\rho_{\varepsilon}^{(k)}(y)\frac{\mathrm{d}\sigma(y)}{2\pi|x-y|}=\frac{\pi}{2}\left(1+O((r_{\varepsilon}^{(k)})^2)\right),\qquad\text{uniformly in }x\in\Gamma_{\varepsilon}^{k}. ∫ Γ ε k ρ ε ( k ) ( y ) 2 π ∣ x − y ∣ d σ ( y ) = 2 π ( 1 + O (( r ε ( k ) ) 2 ) ) , uniformly in x ∈ Γ ε k . (d) Logarithmic potential. By (D.4), log ∣ x − y ( r ε ( k ) w ) ∣ = log r ε ( k ) + log ∣ w 0 − w ∣ + O ( ( r ε ( k ) ) 2 ) \log|x-y(r_{\varepsilon}^{(k)}w)|=\log r_{\varepsilon}^{(k)}+\log|w_0-w|+O((r_{\varepsilon}^{(k)})^2) log ∣ x − y ( r ε ( k ) w ) ∣ = log r ε ( k ) + log ∣ w 0 − w ∣ + O (( r ε ( k ) ) 2 ) w ≠ w 0 w\ne w_0 w = w 0
∫ Γ ε k ρ ε ( k ) ( y ) log ∣ x − y ∣ d σ ( y ) = r ε ( k ) log r ε ( k ) ∫ B ( 0 , 1 ) d w 1 − ∣ w ∣ 2 ⏟ = 2 π + r ε ( k ) ∫ B ( 0 , 1 ) log ∣ w 0 − w ∣ 1 − ∣ w ∣ 2 d w + O ( ( r ε ( k ) ) 3 ) . \int_{\Gamma_{\varepsilon}^{k}}\rho_{\varepsilon}^{(k)}(y)\log|x-y|\,\mathrm{d}\sigma(y)=r_{\varepsilon}^{(k)}\log r_{\varepsilon}^{(k)}\underbrace{\int_{B(0,1)}\frac{\mathrm{d}w}{\sqrt{1-|w|^2}}}_{=2\pi}+r_{\varepsilon}^{(k)}\int_{B(0,1)}\frac{\log|w_0-w|}{\sqrt{1-|w|^2}}\,\mathrm{d}w+O((r_{\varepsilon}^{(k)})^3). ∫ Γ ε k ρ ε ( k ) ( y ) log ∣ x − y ∣ d σ ( y ) = r ε ( k ) log r ε ( k ) = 2 π ∫ B ( 0 , 1 ) 1 − ∣ w ∣ 2 d w + r ε ( k ) ∫ B ( 0 , 1 ) 1 − ∣ w ∣ 2 log ∣ w 0 − w ∣ d w + O (( r ε ( k ) ) 3 ) . The last integral is finite and bounded uniformly in w 0 ∈ B ( 0 , 1 ) w_0\in B(0,1) w 0 ∈ B ( 0 , 1 ) ( 1 − ∣ w ∣ 2 ) − 1 / 2 (1-|w|^2)^{-1/2} ( 1 − ∣ w ∣ 2 ) − 1/2 log ∣ w 0 − w ∣ \log|w_0-w| log ∣ w 0 − w ∣ L l o c p L^p_{\mathrm{loc}} L loc p p < 2 p<2 p < 2 p < ∞ p<\infty p < ∞ w 0 w_0 w 0
∫ Γ ε k ρ ε ( k ) ( y ) log ∣ x − y ∣ d σ ( y ) = 2 π r ε ( k ) log r ε ( k ) + O ( r ε ( k ) ) = O ( r ε ( k ) ∣ log r ε ( k ) ∣ ) , \int_{\Gamma_{\varepsilon}^{k}}\rho_{\varepsilon}^{(k)}(y)\log|x-y|\,\mathrm{d}\sigma(y)=2\pi r_{\varepsilon}^{(k)}\log r_{\varepsilon}^{(k)}+O(r_{\varepsilon}^{(k)})=O(r_{\varepsilon}^{(k)}|\log r_{\varepsilon}^{(k)}|), ∫ Γ ε k ρ ε ( k ) ( y ) log ∣ x − y ∣ d σ ( y ) = 2 π r ε ( k ) log r ε ( k ) + O ( r ε ( k ) ) = O ( r ε ( k ) ∣ log r ε ( k ) ∣ ) , which is the claimed estimate.
The previous estimates also give the continuity of the quasi-mode on the holes, required in Hypothesis 2.1 .
Lemma D.2. Fix k ∈ { 1 , … , N } k\in\{1,\ldots,N\} k ∈ { 1 , … , N } ψ ε ( k ) \psi_{\varepsilon}^{(k)} ψ ε ( k ) (21) is continuous on Γ D ε \Gamma_{\mathrm{D}}^{\varepsilon} Γ D ε
Proof. Recall that, by local elliptic regularity, G ( ⋅ , y ) G(\cdot,y) G ( ⋅ , y ) Ω ‾ ∖ { y } \overline{\Omega}\setminus\{y\} Ω ∖ { y } y ∈ ∂ Ω y\in\partial\Omega y ∈ ∂ Ω j ≠ k j\ne k j = k Γ j ε ‾ \overline{\Gamma_j^\varepsilon} Γ j ε Γ k ε ‾ \overline{\Gamma_k^\varepsilon} Γ k ε ψ ε ( k ) \psi_\varepsilon^{(k)} ψ ε ( k ) Γ j ε ‾ \overline{\Gamma_j^\varepsilon} Γ j ε ρ ε ∈ L 1 ( Γ k ε ‾ ) \rho_\varepsilon\in L^1(\overline{\Gamma_k^\varepsilon}) ρ ε ∈ L 1 ( Γ k ε ) Γ k ε ‾ \overline{\Gamma_k^\varepsilon} Γ k ε y ( ⋅ ) y(\cdot) y ( ⋅ ) ∂ Ω \partial\Omega ∂ Ω x ( k ) x^{(k)} x ( k ) y − 1 ( Γ k ε ‾ ) = B ( 0 , r ε ( k ) ) y^{-1}(\overline{\Gamma_k^\varepsilon})=B(0,r_\varepsilon^{(k)}) y − 1 ( Γ k ε ) = B ( 0 , r ε ( k ) ) G = 1 2 π ∣ x − y ∣ + g ( x , y ) G=\frac{1}{2\pi|x-y|}+g(x,y) G = 2 π ∣ x − y ∣ 1 + g ( x , y ) y ( ⋅ ) y(\cdot) y ( ⋅ ) d y ( 0 ) = I dy(0)=I d y ( 0 ) = I ∣ y ( z 0 ) − y ( z ) ∣ = ∣ z 0 − z ∣ Q ( z 0 , z ) 1 / 2 |y(z_0)-y(z)|=|z_0-z|Q(z_0,z)^{1/2} ∣ y ( z 0 ) − y ( z ) ∣ = ∣ z 0 − z ∣ Q ( z 0 , z ) 1/2 Q Q Q B ( 0 , r ε ( k ) ) 2 B(0,r_\varepsilon^{(k)})^2 B ( 0 , r ε ( k ) ) 2 ρ f l a t ( z ) = ( ( r ε ( k ) ) 2 − ∣ z ∣ 2 ) − 1 / 2 \rho_{\mathrm{flat}}(z)=((r_\varepsilon^{(k)})^2-|z|^2)^{-1/2} ρ flat ( z ) = (( r ε ( k ) ) 2 − ∣ z ∣ 2 ) − 1/2 K ( z 0 , z ) = J k ( z ) Q ( z 0 , z ) − 1 / 2 \mathcal{K}(z_0,z)=J_k(z)Q(z_0,z)^{-1/2} K ( z 0 , z ) = J k ( z ) Q ( z 0 , z ) − 1/2 ψ ε ( k ) \psi_\varepsilon^{(k)} ψ ε ( k ) x = y ( z 0 ) x=y(z_0) x = y ( z 0 ) 1 / π 2 1/\pi^2 1/ π 2
∫ B ( 0 , r ε ( k ) ) ρ f l a t ( z ) K ( z 0 , z ) ∣ z 0 − z ∣ d z = K ( z 0 , z 0 ) ∫ B ( 0 , r ε ( k ) ) ρ f l a t ( z ) ∣ z 0 − z ∣ d z + ∫ B ( 0 , r ε ( k ) ) ρ f l a t ( z ) ( K ( z 0 , z ) − K ( z 0 , z 0 ) ) ∣ z 0 − z ∣ d z . \int_{B(0,r_\varepsilon^{(k)})}\frac{\rho_{\mathrm{flat}}(z)\mathcal{K}(z_0,z)}{|z_0-z|}\,dz
=
\mathcal{K}(z_0,z_0)\int_{B(0,r_\varepsilon^{(k)})}\frac{\rho_{\mathrm{flat}}(z)}{|z_0-z|}\,dz
+
\int_{B(0,r_\varepsilon^{(k)})}\frac{\rho_{\mathrm{flat}}(z)\bigl(\mathcal{K}(z_0,z)-\mathcal{K}(z_0,z_0)\bigr)}{|z_0-z|}\,dz. ∫ B ( 0 , r ε ( k ) ) ∣ z 0 − z ∣ ρ flat ( z ) K ( z 0 , z ) d z = K ( z 0 , z 0 ) ∫ B ( 0 , r ε ( k ) ) ∣ z 0 − z ∣ ρ flat ( z ) d z + ∫ B ( 0 , r ε ( k ) ) ∣ z 0 − z ∣ ρ flat ( z ) ( K ( z 0 , z ) − K ( z 0 , z 0 ) ) d z . The first integral equals π 2 \pi^2 π 2 z 0 ∈ B ( 0 , r ε ( k ) ) z_0\in B(0,r_\varepsilon^{(k)}) z 0 ∈ B ( 0 , r ε ( k ) ) [45 Equation 4.6] ), and z 0 ↦ K ( z 0 , z 0 ) z_0\mapsto\mathcal{K}(z_0,z_0) z 0 ↦ K ( z 0 , z 0 ) K \mathcal{K} K z 0 z_0 z 0 C ρ f l a t ∈ L 1 ( B ( 0 , r ε ( k ) ) ) C\rho_{\mathrm{flat}}\in L^1(B(0,r_\varepsilon^{(k)})) C ρ flat ∈ L 1 ( B ( 0 , r ε ( k ) )) z 0 z_0 z 0 z ≠ z 0 z\ne z_0 z = z 0 z 0 z_0 z 0 x ↦ g ( x , y ) x\mapsto g(x,y) x ↦ g ( x , y ) Ω ‾ ∖ { y } \overline{\Omega}\setminus\{y\} Ω ∖ { y } ∣ g ( x , y ) ∣ ≤ C ( 1 + ∣ log ∣ x − y ∣ ∣ ) |g(x,y)|\le C(1+|\log|x-y||) ∣ g ( x , y ) ∣ ≤ C ( 1 + ∣ log ∣ x − y ∣∣ ) ρ ε ∈ L p ( Γ k ε ‾ ) \rho_\varepsilon\in L^p(\overline{\Gamma_k^\varepsilon}) ρ ε ∈ L p ( Γ k ε ) p ∈ ( 1 , 2 ) p\in(1,2) p ∈ ( 1 , 2 ) y ↦ log ∣ x − y ∣ y\mapsto\log|x-y| y ↦ log ∣ x − y ∣ L p ′ ( Γ k ε ‾ ) L^{p'}(\overline{\Gamma_k^\varepsilon}) L p ′ ( Γ k ε ) x ∈ Γ k ε ‾ x\in\overline{\Gamma_k^\varepsilon} x ∈ Γ k ε p ′ = p / ( p − 1 ) p'=p/(p-1) p ′ = p / ( p − 1 ) ( ρ ε ( k ) g ( x , ⋅ ) ) x ∈ Γ k ε ‾ (\rho_\varepsilon^{(k)}g(x,\cdot))_{x\in\overline{\Gamma_k^\varepsilon}} ( ρ ε ( k ) g ( x , ⋅ ) ) x ∈ Γ k ε x ↦ ∫ Γ k ε ρ ε ( k ) ( y ) g ( x , y ) d σ ( y ) x\mapsto\int_{\Gamma_k^\varepsilon}\rho_\varepsilon^{(k)}(y)g(x,y)\,d\sigma(y) x ↦ ∫ Γ k ε ρ ε ( k ) ( y ) g ( x , y ) d σ ( y ) Γ k ε ‾ \overline{\Gamma_k^\varepsilon} Γ k ε ψ ε ( k ) \psi_\varepsilon^{(k)} ψ ε ( k ) Γ k ε ‾ \overline{\Gamma_k^\varepsilon} Γ k ε
In the case of the slit holes from Section 4
In this subsection, we go back to the notation of § 4 and prove a technical lemma needed in the proof of Lemma 4.1.
Lemma D.3. For a fixed k ∈ { 1 , … , N } k\in\{1,\ldots,N\} k ∈ { 1 , … , N } ε > 0 \varepsilon>0 ε > 0 Γ k ε \Gamma_k^\varepsilon Γ k ε § 4 , and G G G Definition C.1 . Then for any x ∈ Γ k ε x\in\Gamma_k^\varepsilon x ∈ Γ k ε ε → 0 \varepsilon\to0 ε → 0
∫ H k G ( x , y ) d σ ( y ) = − 1 π log ( w ε ( k ) ) + O ( 1 ) . \int_{\mathcal{H}_k}G(x,y)\,d\sigma(y)=-\frac{1}{\pi}\log\left(w_\varepsilon^{(k)}\right)+O(1). ∫ H k G ( x , y ) d σ ( y ) = − π 1 log ( w ε ( k ) ) + O ( 1 ) . Proof. In this proof we fix k ∈ { 1 , … , N } k\in\{1,\ldots,N\} k ∈ { 1 , … , N } x ∈ Γ k ε x\in\Gamma_k^\varepsilon x ∈ Γ k ε k k k u u u G G G
u ( x ) = ∫ H G ( x , y ) d σ ( y ) = ∫ H [ 1 2 π ∣ x − y ∣ + r ( x , y ) ] d σ ( y ) , (123) u(x)=\int_{\mathcal{H}}G(x,y)\,d\sigma(y)=\int_{\mathcal{H}}\left[\frac{1}{2\pi|x-y|}+r(x,y)\right]\,d\sigma(y), \tag*{(123)} u ( x ) = ∫ H G ( x , y ) d σ ( y ) = ∫ H [ 2 π ∣ x − y ∣ 1 + r ( x , y ) ] d σ ( y ) , (123) with r : Γ k ε × H → R r:\Gamma_k^\varepsilon\times\mathcal{H}\to\mathbb{R} r : Γ k ε × H → R r r r
u 0 ( x ) = 1 2 π ∫ H d σ ( y ) ∣ x − y ∣ = 1 2 π ∫ − ℓ ℓ d s ∣ x − γ ( s ) ∣ , (124) u_0(x)=\frac{1}{2\pi}\int_{\mathcal{H}}\frac{d\sigma(y)}{|x-y|}=\frac{1}{2\pi}\int_{-\ell}^{\ell}\frac{ds}{|x-\gamma(s)|}, \tag*{(124)} u 0 ( x ) = 2 π 1 ∫ H ∣ x − y ∣ d σ ( y ) = 2 π 1 ∫ − ℓ ℓ ∣ x − γ ( s ) ∣ d s , (124) where we used the parametrization of H \mathcal{H} H § 2.2 . Notice that as H \mathcal{H} H γ \gamma γ s x ∈ [ − ℓ , ℓ ] s_x\in[-\ell,\ell] s x ∈ [ − ℓ , ℓ ] γ ( s x ) \gamma(s_x) γ ( s x ) H \mathcal{H} H x x x
γ ( s ) = γ ( s x ) + σ τ ( s x ) + E ( s , s x ) , with ∣ E ( s , s x ) ∣ ≤ K 1 σ 2 , (125) \gamma(s)=\gamma(s_x)+\sigma\tau(s_x)+E(s,s_x),\qquad\text{with }|E(s,s_x)|\le K_1\sigma^2, \tag*{(125)} γ ( s ) = γ ( s x ) + σ τ ( s x ) + E ( s , s x ) , with ∣ E ( s , s x ) ∣ ≤ K 1 σ 2 , (125) writing σ = s − s x \sigma=s-s_x σ = s − s x K 1 > 0 K_1>0 K 1 > 0 H \mathcal{H} H
x − γ ( s ) = ( t − σ ) τ ( s x ) + ρ − E ( s , s x ) . (126) x-\gamma(s)=(t-\sigma)\tau(s_x)+\rho-E(s,s_x). \tag*{(126)} x − γ ( s ) = ( t − σ ) τ ( s x ) + ρ − E ( s , s x ) . (126) It is now natural to introduce the flat slit distance, defined as (D.14)
b ( s ) = ∣ ( t − σ ) τ ( s x ) + ρ ∣ = ρ 2 + ( t − σ ) 2 = ρ 2 + ( z − s ) 2 , (127) b(s)=|(t-\sigma)\tau(s_x)+\rho|=\sqrt{\rho^2+(t-\sigma)^2}=\sqrt{\rho^2+(z-s)^2}, \tag*{(127)} b ( s ) = ∣ ( t − σ ) τ ( s x ) + ρ ∣ = ρ 2 + ( t − σ ) 2 = ρ 2 + ( z − s ) 2 , (127) where z = t + s x z=t+s_x z = t + s x
u 0 ( x ) = 1 2 π ∫ − ℓ ℓ d s b ( s ) + 1 2 π ∫ − ℓ ℓ ( 1 ∣ x − γ ( s ) ∣ − 1 b ( s ) ) d s (128) u_0(x)=\frac{1}{2\pi}\int_{-\ell}^{\ell}\frac{\mathrm{d}s}{b(s)}+\frac{1}{2\pi}\int_{-\ell}^{\ell}\left(\frac{1}{\lvert x-\gamma(s)\rvert}-\frac{1}{b(s)}\right)\mathrm{d}s \tag*{(128)} u 0 ( x ) = 2 π 1 ∫ − ℓ ℓ b ( s ) d s + 2 π 1 ∫ − ℓ ℓ ( ∣ x − γ ( s )∣ 1 − b ( s ) 1 ) d s (128) and estimate the two terms separately.
The flat integral u f l a t u_{\mathrm{flat} } u flat By (D.14), notice that u f l a t u_{\mathrm{flat}} u flat ρ > 0 \rho>0 ρ > 0
∫ − ℓ ℓ d s ρ 2 + ( z − s ) 2 = arcsinh ( ℓ − z ρ ) + arcsinh ( ℓ + z ρ ) = log ( ℓ − z ) + r − ρ + log ( ℓ + z ) + r + ρ , (129) \int_{-\ell}^{\ell}\frac{\mathrm{d}s}{\sqrt{\rho^2+(z-s)^2}}=\operatorname{arcsinh}\left(\frac{\ell-z}{\rho}\right)+\operatorname{arcsinh}\left(\frac{\ell+z}{\rho}\right)=\log\frac{(\ell-z)+r_-}{\rho}+\log\frac{(\ell+z)+r_+}{\rho}, \tag*{(129)} ∫ − ℓ ℓ ρ 2 + ( z − s ) 2 d s = arcsinh ( ρ ℓ − z ) + arcsinh ( ρ ℓ + z ) = log ρ ( ℓ − z ) + r − + log ρ ( ℓ + z ) + r + , (129) where r − = ρ 2 + ( z − ℓ ) 2 r_-=\sqrt{\rho^2+(z-\ell)^2} r − = ρ 2 + ( z − ℓ ) 2 r + = ρ 2 + ( z + ℓ ) 2 r_+=\sqrt{\rho^2+(z+\ell)^2} r + = ρ 2 + ( z + ℓ ) 2 ( a − b ) ( a + b ) = a 2 − b 2 (a-b)(a+b)=a^2-b^2 ( a − b ) ( a + b ) = a 2 − b 2
( r ± ( ρ , z ) − ( ℓ ± z ) ) ( r ± ( ρ , z ) + ( ℓ ± z ) ) = ρ 2 . \left(r_{\pm}(\rho,z)-(\ell\pm z)\right)\left(r_{\pm}(\rho,z)+(\ell\pm z)\right)=\rho^2. ( r ± ( ρ , z ) − ( ℓ ± z ) ) ( r ± ( ρ , z ) + ( ℓ ± z ) ) = ρ 2 . Summing the previous equation for r + r_+ r + r − r_- r − ρ \rho ρ z z z
r + + r − − 2 ℓ = ρ 2 r + + ( ℓ + z ) + ρ 2 r − + ( ℓ − z ) = ρ 2 r + + r − + 2 ℓ ( r + + ( ℓ + z ) ) ( r − + ( ℓ − z ) ) . r_++r_--2\ell=\frac{\rho^2}{r_++(\ell+z)}+\frac{\rho^2}{r_-+(\ell-z)}=\rho^2\frac{r_++r_-+2\ell}{\left(r_++(\ell+z)\right)\left(r_-+(\ell-z)\right)}. r + + r − − 2 ℓ = r + + ( ℓ + z ) ρ 2 + r − + ( ℓ − z ) ρ 2 = ρ 2 ( r + + ( ℓ + z ) ) ( r − + ( ℓ − z ) ) r + + r − + 2 ℓ . Finally, the explicit expression of u f l a t u_{\mathrm{flat}} u flat
u f l a t ( x ) = log ( ( r + ( ρ , z ) + ( ℓ + z ) ) ( r − ( ρ , z ) + ( ℓ − z ) ) ρ 2 ) = − log ( r + ( ρ , z ) + r − ( ρ , z ) − 2 ℓ r + ( ρ , z ) + r − ( ρ , z ) + 2 ℓ ) (130) \begin{aligned}
u_{\mathrm{flat}}(x)=\log\left(\frac{\left(r_+(\rho,z)+(\ell+z)\right)\left(r_-(\rho,z)+(\ell-z)\right)}{\rho^2}\right) \\
&=-\log\left(\frac{r_+(\rho,z)+r_-(\rho,z)-2\ell}{r_+(\rho,z)+r_-(\rho,z)+2\ell}\right) \tag*{(130)}
\end{aligned} u flat ( x ) = log ( ρ 2 ( r + ( ρ , z ) + ( ℓ + z ) ) ( r − ( ρ , z ) + ( ℓ − z ) ) ) = − log ( r + ( ρ , z ) + r − ( ρ , z ) + 2 ℓ r + ( ρ , z ) + r − ( ρ , z ) − 2 ℓ ) (130) By Type C, the point x ∈ Γ k ε x\in\Gamma_k^\varepsilon x ∈ Γ k ε a = ℓ 2 + w 2 a=\sqrt{\ell^2+w^2} a = ℓ 2 + w 2 w w w ( z , ρ ) (z,\rho) ( z , ρ ) w = w ε ( k ) w=w_\varepsilon^{(k)} w = w ε ( k ) a = a ε ( k ) a=a_\varepsilon^{(k)} a = a ε ( k ) (8) . As a 2 − w 2 = ℓ 2 a^2-w^2=\ell^2 a 2 − w 2 = ℓ 2 z = ± ℓ z=\pm\ell z = ± ℓ
r + ( ρ , z ) + r − ( ρ , z ) = 2 a , for x ∈ Γ k ε . r_+(\rho,z)+r_-(\rho,z)=2a,\qquad\text{for }x\in\Gamma_k^\varepsilon. r + ( ρ , z ) + r − ( ρ , z ) = 2 a , for x ∈ Γ k ε . Thus, on Γ k ε \Gamma_k^\varepsilon Γ k ε (8) , we have
u f l a t ( x ) = − log ( 2 a − 2 ℓ 2 a + 2 ℓ ) = log ( ( a + ℓ ) 2 w 2 ) = − 2 log ( w ) + 2 log ( a + ℓ ) = − 2 log ( w ) + 2 log ( 2 ℓ ) + O ( w 2 ) , (131) \begin{aligned}
u_{\mathrm{flat}}(x)=-\log\left(\frac{2a-2\ell}{2a+2\ell}\right)=\log\left(\frac{(a+\ell)^2}{w^2}\right) \\
&=-2\log(w)+2\log(a+\ell)=-2\log(w)+2\log(2\ell)+O(w^2), \tag*{(131)}
\end{aligned} u flat ( x ) = − log ( 2 a + 2 ℓ 2 a − 2 ℓ ) = log ( w 2 ( a + ℓ ) 2 ) = − 2 log ( w ) + 2 log ( a + ℓ ) = − 2 log ( w ) + 2 log ( 2 ℓ ) + O ( w 2 ) , (131) which is the expected result. For the case ρ = 0 \rho=0 ρ = 0 x ∉ H x\notin\mathcal{H} x ∈ / H ρ = 0 \rho=0 ρ = 0 ∣ z ∣ > ℓ \lvert z\rvert>\ell ∣ z ∣ > ℓ
u f l a t ( x ) = ∫ − ℓ ℓ 1 ∣ z − s ∣ d s = log ∣ z + ℓ z − ℓ ∣ . \begin{aligned}
u_{\mathrm{flat}}(x)=\int_{-\ell}^{\ell}\frac{1}{\lvert z-s\rvert}\,\mathrm{d}s \\
&=\log\left\lvert\frac{z+\ell}{z-\ell}\right\rvert.
\end{aligned} u flat ( x ) = ∫ − ℓ ℓ ∣ z − s ∣ 1 d s = log z − ℓ z + ℓ . On Γ k ε \Gamma_k^\varepsilon Γ k ε ρ = 0 \rho=0 ρ = 0 ∣ z ∣ = a \lvert z\rvert=a ∣ z ∣ = a log a + ℓ a − ℓ = log ( a + ℓ ) 2 w 2 \log\frac{a+\ell}{a-\ell}=\log\frac{(a+\ell)^2}{w^2} log a − ℓ a + ℓ = log w 2 ( a + ℓ ) 2
The difference u r e s t u_{\mathrm{rest}} u rest By the very definition of the foot point s x s_x s x t t t σ \sigma σ
∣ t − σ ∣ ≥ ∣ σ ∣ for every admissible σ . (132) \lvert t-\sigma\rvert\geq\lvert\sigma\rvert\qquad\text{for every admissible }\sigma. \tag*{(132)} ∣ t − σ ∣ ≥ ∣ σ ∣ for every admissible σ . (132) Indeed, if γ ( s x ) \gamma(s_x) γ ( s x ) H \mathcal{H} H t = 0 t=0 t = 0 s x = ℓ s_x=\ell s x = ℓ σ = s − ℓ ≤ 0 \sigma=s-\ell\le0 σ = s − ℓ ≤ 0 t = ⟨ x − γ ( ℓ ) , τ ( ℓ ) ⟩ ≥ 0 t=\langle x-\gamma(\ell),\tau(\ell)\rangle\ge0 t = ⟨ x − γ ( ℓ ) , τ ( ℓ )⟩ ≥ 0 t − σ ≥ ∣ σ ∣ t-\sigma\ge|\sigma| t − σ ≥ ∣ σ ∣
b ( s ) ≥ ∣ t − σ ∣ ≥ ∣ σ ∣ . (133) b(s)\ge|t-\sigma|\ge|\sigma|. \tag*{(133)} b ( s ) ≥ ∣ t − σ ∣ ≥ ∣ σ ∣. (133) Moreover, by (D.13) and the reverse triangle inequality,
∣ ∣ x − γ ( s ) ∣ − b ( s ) ∣ ≤ ∣ E ( s , s x ) ∣ ≤ K 1 σ 2 . (134) \left||x-\gamma(s)|-b(s)\right|\le|E(s,s_x)|\le K_1\sigma^2. \tag*{(134)} ∣ ∣ x − γ ( s ) ∣ − b ( s ) ∣ ≤ ∣ E ( s , s x ) ∣ ≤ K 1 σ 2 . (134) Fix now σ 0 = ( 2 K 1 ) − 1 \sigma_0=(2K_1)^{-1} σ 0 = ( 2 K 1 ) − 1 H \mathcal{H} H ∣ σ ∣ > σ 0 |\sigma|>\sigma_0 ∣ σ ∣ > σ 0 b ( s ) ≥ ∣ σ ∣ > σ 0 b(s)\ge|\sigma|>\sigma_0 b ( s ) ≥ ∣ σ ∣ > σ 0 H \mathcal{H} H c H > 0 c_{\mathcal{H}}>0 c H > 0 ∣ γ ( s ) − γ ( s ′ ) ∣ ≥ c H ∣ s − s ′ ∣ |\gamma(s)-\gamma(s')|\ge c_{\mathcal{H}}|s-s'| ∣ γ ( s ) − γ ( s ′ ) ∣ ≥ c H ∣ s − s ′ ∣ ∣ x − γ ( s x ) ∣ ≤ w ε ( k ) → 0 |x-\gamma(s_x)|\le w_\varepsilon^{(k)}\to0 ∣ x − γ ( s x ) ∣ ≤ w ε ( k ) → 0 x ∈ Γ ε k x\in\Gamma_\varepsilon^k x ∈ Γ ε k ∣ x − γ ( s ) ∣ ≥ c H σ 0 / 2 |x-\gamma(s)|\ge c_{\mathcal{H}}\sigma_0/2 ∣ x − γ ( s ) ∣ ≥ c H σ 0 /2 ε \varepsilon ε c ( σ 0 ) > 0 c(\sigma_0)>0 c ( σ 0 ) > 0 2 / c ( σ 0 ) 2/c(\sigma_0) 2/ c ( σ 0 ) ∣ σ ∣ ≤ σ 0 |\sigma|\le\sigma_0 ∣ σ ∣ ≤ σ 0 K 1 σ 2 ≤ 1 2 ∣ σ ∣ ≤ 1 2 b ( s ) K_1\sigma^2\le\frac12|\sigma|\le\frac12b(s) K 1 σ 2 ≤ 2 1 ∣ σ ∣ ≤ 2 1 b ( s ) ∣ x − γ ( s ) ∣ ≥ 1 2 b ( s ) |x-\gamma(s)|\ge\frac12b(s) ∣ x − γ ( s ) ∣ ≥ 2 1 b ( s )
∣ 1 ∣ x − γ ( s ) ∣ − 1 b ( s ) ∣ = ∣ ∣ x − γ ( s ) ∣ − b ( s ) ∣ ∣ x − γ ( s ) ∣ b ( s ) ≤ K 1 σ 2 1 2 b ( s ) 2 ≤ 2 K 1 σ 2 σ 2 = 2 K 1 . (135) \left|\frac{1}{|x-\gamma(s)|}-\frac{1}{b(s)}\right|
=\frac{\left||x-\gamma(s)|-b(s)\right|}{|x-\gamma(s)|b(s)}
\le\frac{K_1\sigma^2}{\frac12b(s)^2}
\le\frac{2K_1\sigma^2}{\sigma^2}=2K_1. \tag*{(135)} ∣ x − γ ( s ) ∣ 1 − b ( s ) 1 = ∣ x − γ ( s ) ∣ b ( s ) ∣ ∣ x − γ ( s ) ∣ − b ( s ) ∣ ≤ 2 1 b ( s ) 2 K 1 σ 2 ≤ σ 2 2 K 1 σ 2 = 2 K 1 . (135) In either case the integrand of u r e s t u_{\mathrm{rest}} u rest x x x H \mathcal{H} H
u r e s t ( x ) = O ( 1 ) . (136) u_{\mathrm{rest}}(x)=O(1). \tag*{(136)} u rest ( x ) = O ( 1 ) . (136) Combining (D.15), (D.18) and (D.23),
u 0 ( x ) = − 1 π log ( w ) + O ( 1 ) . (137) u_0(x)=-\frac{1}{\pi}\log(w)+O(1). \tag*{(137)} u 0 ( x ) = − π 1 log ( w ) + O ( 1 ) . (137) The remainder r. From Lemma C.3, there exist constants δ > 0 \delta>0 δ > 0 C δ > 0 C_\delta>0 C δ > 0 y ∈ H y\in\mathcal{H} y ∈ H ∣ x − y ∣ < δ |x-y|<\delta ∣ x − y ∣ < δ
∣ r ( x , y ) ∣ ≤ C δ ( 1 + ∣ log ∣ x − y ∣ ∣ ) . |r(x,y)|\le C_\delta(1+|\log|x-y||). ∣ r ( x , y ) ∣ ≤ C δ ( 1 + ∣ log ∣ x − y ∣∣ ) . Furthermore, from Lemma C.2, for any y ∈ H y\in\mathcal{H} y ∈ H ∣ x − y ∣ ≥ δ |x-y|\ge\delta ∣ x − y ∣ ≥ δ
∣ r ( x , y ) ∣ ≤ ∣ G ( x , y ) ∣ + 1 2 π ∣ x − y ∣ ≤ ( K + 1 2 π ) δ − 1 . |r(x,y)|\le|G(x,y)|+\frac{1}{2\pi|x-y|}
\le\left(K+\frac{1}{2\pi}\right)\delta^{-1}. ∣ r ( x , y ) ∣ ≤ ∣ G ( x , y ) ∣ + 2 π ∣ x − y ∣ 1 ≤ ( K + 2 π 1 ) δ − 1 . Then, the integral of the function r r r H \mathcal{H} H
∫ H ∣ r ( x , y ) ∣ d σ ( y ) ≤ C δ ∫ H ∩ B ( x , δ ) ( 1 + ∣ log ∣ x − y ∣ ∣ ) d σ ( y ) + ( K + 1 2 π ) δ − 1 ∣ H ∣ . (138) \int_{\mathcal{H}}|r(x,y)|\,\mathrm{d}\sigma(y)
\le C_\delta\int_{\mathcal{H}\cap B(x,\delta)}(1+|\log|x-y||)\,\mathrm{d}\sigma(y)
+\left(K+\frac{1}{2\pi}\right)\delta^{-1}|\mathcal{H}|. \tag*{(138)} ∫ H ∣ r ( x , y ) ∣ d σ ( y ) ≤ C δ ∫ H ∩ B ( x , δ ) ( 1 + ∣ log ∣ x − y ∣∣ ) d σ ( y ) + ( K + 2 π 1 ) δ − 1 ∣ H ∣. (138) It remains to check that the right-hand side of (D.25) is O ( 1 ) O(1) O ( 1 ) ( K + 1 2 π ) δ − 1 ∣ H ∣ \left(K+\frac{1}{2\pi}\right)\delta^{-1}|\mathcal{H}| ( K + 2 π 1 ) δ − 1 ∣ H ∣ C δ ∣ H ∣ C_\delta|\mathcal{H}| C δ ∣ H ∣ δ \delta δ H \mathcal{H} H H ∩ B ( x , δ ) \mathcal{H}\cap B(x,\delta) H ∩ B ( x , δ ) s x s_x s x ∣ σ ∣ ≤ C δ < 1 |\sigma|\le C\delta<1 ∣ σ ∣ ≤ C δ < 1 ∣ x − γ ( s ) ∣ < δ < 1 |x-\gamma(s)|<\delta<1 ∣ x − γ ( s ) ∣ < δ < 1 ∣ log ∣ x − γ ( s ) ∣ ∣ = − log ∣ x − γ ( s ) ∣ |\log|x-\gamma(s)||=-\log|x-\gamma(s)| ∣ log ∣ x − γ ( s ) ∣∣ = − log ∣ x − γ ( s ) ∣ ∣ x − γ ( s ) ∣ ≥ 1 2 b ( s ) |x-\gamma(s)|\ge\frac12b(s) ∣ x − γ ( s ) ∣ ≥ 2 1 b ( s )
∫ H ∩ B ( x , δ ) ∣ log ∣ x − y ∣ ∣ d σ ( y ) ≤ log 2 ⋅ ∣ H ∣ − ∫ ∣ σ ∣ ≤ C δ log b ( s ) d s . \int_{\mathcal{H}\cap B(x,\delta)}|\log|x-y||\,\mathrm{d}\sigma(y)
\le\log2\cdot|\mathcal{H}|-\int_{|\sigma|\le C\delta}\log b(s)\,\mathrm{d}s. ∫ H ∩ B ( x , δ ) ∣ log ∣ x − y ∣∣ d σ ( y ) ≤ log 2 ⋅ ∣ H ∣ − ∫ ∣ σ ∣ ≤ C δ log b ( s ) d s . By (D.14) and the substitution η = t − σ \eta=t-\sigma η = t − σ
− ∫ log b ( s ) d s = − 1 2 ∫ log ( ρ 2 + η 2 ) d η , -\int\log b(s)\,\mathrm{d}s=-\frac12\int\log(\rho^2+\eta^2)\,\mathrm{d}\eta, − ∫ log b ( s ) d s = − 2 1 ∫ log ( ρ 2 + η 2 ) d η , whose explicit primitive η log ( ρ 2 + η 2 ) − 2 η + 2 ρ arctan ( η / ρ ) \eta\log(\rho^2+\eta^2)-2\eta+2\rho\arctan(\eta/\rho) η log ( ρ 2 + η 2 ) − 2 η + 2 ρ arctan ( η / ρ ) η \eta η ρ → 0 \rho\to0 ρ → 0 η log ( ρ 2 + η 2 ) \eta\log(\rho^2+\eta^2) η log ( ρ 2 + η 2 ) η = 0 \eta=0 η = 0 arctan \arctan arctan O ( 1 ) O(1) O ( 1 ) ∫ H r ( x , y ) d σ ( y ) = O ( 1 ) \int_{\mathcal{H}}r(x,y)\,\mathrm{d}\sigma(y)=O(1) ∫ H r ( x , y ) d σ ( y ) = O ( 1 )
Collecting (D.24) and this last bound in (D.10),
u ( x ) = − 1 π log ( w ) + O ( 1 ) , u(x)=-\frac{1}{\pi}\log(w)+O(1), u ( x ) = − π 1 log ( w ) + O ( 1 ) , which is the announced estimate. □
References 
