A uniformly valid model for the limiting behaviour of voltage potentials in the presence of thin inhomogeneities II. A local energy approximation result

Abstract

In this second part of a two part paper, we establish a local, uniform energy-approximation estimate for the solutions to a simplified model of thin inhomogeneities with open mid-curves. This local result plays a crucial role in the proof of the global, uniform approximation results established in the first part of this paper (Asymptotic Analysis (2019)). For more details about the model we also refer the reader to (Chinese Annals of Mathematics, Series B 38 (2017) 293–344).

Keywords

Elliptic boundary-value problem thin inhomogeneity uniform approximation

1. Introduction

Let σ denote the line segment ${- 1 < x < 1, y = 0}$ and let $\tilde{Ω}$ denote the symmetric domain $\begin{matrix} \tilde{Ω} = ω_{δ_{0}^{'}} = {(x, y) \in R^{2} : dist ((x, y), σ) < δ_{0}^{'}} . \end{matrix}$ For any $0 < ϵ < δ_{0}^{'}$ , and arbitrary constants $0 < a_{ϵ}$ , we introduce the following two energy expressions $\begin{array}{l} (1) & \begin{matrix} E_{σ, \tilde{Ω}}^{ϵ} (v) & = \frac{1}{2} \int_{\tilde{Ω} ∖ σ} | \nabla v |^{2} d x d y + \frac{ϵ a_{ϵ}}{2} \int_{σ} ({(\frac{\partial v^{+}}{\partial τ})}^{2} + {(\frac{\partial v^{-}}{\partial τ})}^{2}) d s \\ + \frac{a_{ϵ}}{4 ϵ} \int_{σ} {(v^{+} - v^{-})}^{2} d s, \end{matrix} \\ (2) & E_{\tilde{Ω}}^{ϵ} (v) = \frac{1}{2} \int_{\tilde{Ω}} γ_{ϵ} | \nabla v |^{2} d x d y . \end{array}$ Here $\begin{matrix} γ_{ϵ} (x, y) = \{\begin{matrix} 1 & if (x, y) is
in \tilde{Ω} ∖ ω_{ϵ}, \\ a_{ϵ} & if (x, y) is
in ω_{ϵ}, \end{matrix} \end{matrix}$ and $\begin{matrix} ω_{ϵ} = {(x, y) \in R^{2} : dist ((x, y), σ) < ϵ} . \end{matrix}$ Note that these are exactly the energy expressions introduced in the first part of this paper, with the proviso that f vanishes in $\tilde{Ω}$ ; they are defined on ${v \in H^{1} (\tilde{Ω} ∖ σ) : v^{\pm} \in H^{1} (σ)}$ and $H^{1} (\tilde{Ω})$ , respectively. This second part of the paper is entirely devoted to the proof of the following local, uniform energy convergence result.

Theorem 1.
Let $\tilde{u}$ and ${\tilde{u}}_{ϵ}$ denote the minimizers of the energies ( 1 ) and ( 2 ), respectively, given Dirichlet boundary data $\tilde{φ} \in H^{1 / 2} (\partial \tilde{Ω})$ . Then $\begin{matrix} sup_{a_{ϵ} > 0, \tilde{φ} \neq 0} \frac{| E_{σ, \tilde{Ω}}^{ϵ} (\tilde{u}) - E_{\tilde{Ω}}^{ϵ} ({\tilde{u}}_{ϵ}) |}{‖ \tilde{φ} ‖_{H^{1 / 2} (\partial \tilde{Ω})}^{2}} \to 0, \end{matrix}$ as $ϵ \to 0$ .

Due to the symmetric nature of $\tilde{Ω}$ , we may decompose the solutions, $\tilde{u}$ and ${\tilde{u}}_{ϵ}$ , into their even and odd parts $\begin{matrix} \tilde{u} = u^{ev} + u^{odd}, and {\tilde{u}}_{ϵ} = u_{ϵ}^{ev} + u_{ϵ}^{odd}, \end{matrix}$ with $\begin{matrix} u^{ev} = \frac{1}{2} (\tilde{u} (x, y) + \tilde{u} (x, - y)), and u^{odd} (x, y) = \frac{1}{2} (\tilde{u} (x, y) - \tilde{u} (x, - y)) \end{matrix}$ and similarly for $u_{ϵ}^{ev}$ , $u_{ϵ}^{odd}$ . A simple calculation yields that $\begin{matrix} E_{σ, \tilde{Ω}}^{ϵ} (v) = E_{σ, \tilde{Ω}}^{ϵ} (v^{odd} + v^{ev}) = E_{σ, \tilde{Ω}}^{ϵ} (v^{odd}) + E_{σ, \tilde{Ω}}^{ϵ} (v^{ev}) \end{matrix}$ for any $v \in H^{1} (\tilde{Ω} ∖ σ)$ , $v^{\pm} \in H^{1} (σ)$ , and $\begin{matrix} E_{\tilde{Ω}}^{ϵ} (v) = E_{\tilde{Ω}}^{ϵ} (v^{odd} + v^{ev}) = E_{\tilde{Ω}}^{ϵ} (v^{odd}) + E_{\tilde{Ω}}^{ϵ} (v^{ev}) \end{matrix}$ for any $v \in H^{1} (\tilde{Ω})$ . If we combine this with the fact that $\begin{matrix} {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ⩽ ‖ \tilde{φ} ‖_{H^{1 / 2} (\partial \tilde{Ω})}, and {‖ {\tilde{φ}}^{odd} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ⩽ ‖ \tilde{φ} ‖_{H^{1 / 2} (\partial \tilde{Ω})}, \end{matrix}$ then Theorem 1 follows as an immediate consequence of the following two lemmata.
Lemma 1.
$\begin{matrix} sup_{a_{ϵ} > 0, {\tilde{φ}}^{ev} \neq 0} \frac{| E_{σ, \tilde{Ω}}^{ϵ} (u^{ev}) - E_{\tilde{Ω}}^{ϵ} (u_{ϵ}^{ev}) |}{‖ {\tilde{φ}}^{ev} ‖_{H^{1 / 2} (\partial \tilde{Ω})}^{2}} \to 0, \end{matrix}$

as $ϵ \to 0$ .
Lemma 2.
$\begin{matrix} sup_{a_{ϵ} > 0, {\tilde{φ}}^{odd} \neq 0} \frac{| E_{σ, \tilde{Ω}}^{ϵ} (u^{odd}) - E_{\tilde{Ω}}^{ϵ} (u_{ϵ}^{odd}) |}{‖ {\tilde{φ}}^{odd} ‖_{H^{1 / 2} (\partial \tilde{Ω})}^{2}} \to 0, \end{matrix}$

as $ϵ \to 0$ .

The proofs of Lemma 1 and Lemma 2 (which complete the proof of Theorem 1) are the focus of the remainder of this paper. The proofs proceed in five steps. In Sections 2–3 we establish estimates for $\begin{matrix} | E_{σ, \tilde{Ω}}^{ϵ} (u^{ev}) - E_{\tilde{Ω}}^{ϵ} (u_{ϵ}^{ev}) |, \end{matrix}$ separately, when $ϵ a_{ϵ} > m > 0$ (Section 2), and when $ϵ a_{ϵ} < 1$ (Section 3). In Sections 4–5 we establish estimates for $\begin{matrix} | E_{σ, \tilde{Ω}}^{ϵ} (u^{odd}) - E_{\tilde{Ω}}^{ϵ} (u_{ϵ}^{odd}) |, \end{matrix}$ separately, when $a_{ϵ} < M ϵ$ (Section 4), and when $a_{ϵ} > ϵ$ (Section 5). In Section 6 we combine the estimates from the previous four sections to obtain proofs of Lemma 1 and Lemma 2.
2. Local convergence – Even symmetry, $ϵ a_{ϵ} > m > 0$

In this section we shall show that, in the regime $ϵ a_{ϵ} > m > 0$ , $\begin{matrix} (3) & | E_{σ, \tilde{Ω}}^{ϵ} (u^{ev}) - E_{\tilde{Ω}}^{ϵ} (u_{ϵ}^{ev}) | ⩽ C_{γ, m} {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ^{γ / 4} \end{matrix}$ for any $γ < 1$ . The essential fact is that the constant $C_{γ, m}$ depends on the lower bound m, but is otherwise independent of $a_{ϵ}$ and ϵ. Here $E_{σ, \tilde{Ω}}^{ϵ}$ and $E_{\tilde{Ω}}^{ϵ}$ are the energy expressions introduced in (1) and (2). When restricted to even functions, these are given by $\begin{matrix} (4) & E_{σ, \tilde{Ω}}^{ϵ} (v^{ev}) = \frac{1}{2} \int_{\tilde{Ω} ∖ σ} {| \nabla v^{ev} |}^{2} d x d y + ϵ a_{ϵ} \int_{σ} {(\frac{\partial v^{ev}}{\partial x})}^{2} d x, \end{matrix}$ and $\begin{matrix} (5) & E_{\tilde{Ω}}^{ϵ} (v^{ev}) = \frac{1}{2} \int_{\tilde{Ω}} γ_{ϵ} {| \nabla v^{ev} |}^{2} d x d y . \end{matrix}$ $u^{ev}$ and $u_{ϵ}^{ev}$ are the respective minimizers with Dirichlet boundary data ${\tilde{φ}}^{ev}$ . The proof of (3) is fairly involved and proceeds in two stages; in the first stage we verify that $\begin{matrix} (6) & E_{\tilde{Ω}}^{ϵ} (u_{ϵ}^{ev}) - E_{σ, \tilde{Ω}}^{ϵ} (u^{ev}) ⩽ C_{γ, m} {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ^{γ / 2}, \end{matrix}$ and in the second stage we verify that $\begin{matrix} (7) & E_{σ, \tilde{Ω}}^{ϵ} (u^{ev}) - E_{\tilde{Ω}}^{ϵ} (u_{ϵ}^{ev}) ⩽ C_{γ, m} {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ^{γ / 4}, \end{matrix}$ for any $γ < 1$ . To accomplish the second stage, we introduce a dual energy principle. The idea of using a combination of primal and dual variational principles to obtain bounds on absolute energy differences has a long history in the theoretical mechanics literature, we refer the reader to [4] for an early example.

Let K be a fixed compact set inside $\tilde{Ω}$ , which contains σ in its interior. Since our analysis and estimates are asymptotic in $ϵ \to 0$ , we shall without loss of generality throughout this paper assume that ϵ is sufficiently small that $ω_{ϵ} \subset K$ . In the remainder of this section we shall for simplicity of notation drop the subscripts m and γ from the constant $C_{γ, m}$ . For simplicity of notation we shall also delete the explicit appearance of $\tilde{Ω}$ in the energy symbols, in other words we shall use $E^{ϵ}$ and $E_{σ}^{ϵ}$ in place of $E_{\tilde{Ω}}^{ϵ}$ and $E_{σ, \tilde{Ω}}^{ϵ}$ .

2.1. Proof of (6)

For any $(x, y) \in R^{2}$ , let $p_{σ} (x, y)$ denote the nearest point to $(x, y)$ on σ. We may define a natural map $Φ_{ϵ}$ by $\begin{matrix} ω_{1} ∋ (x, y) \to Φ_{ϵ} (x, y) = p_{σ} (x, y) + ϵ ((x, y) - p_{σ} (x, y)) \in ω_{ϵ} . \end{matrix}$ This map Φ also maps $\partial ω_{1}$ 1-to-1 onto $\partial ω_{ϵ}$ . For an arbitrary $u \in H_{ev}^{1} (ω_{ϵ})$ , define $v \in H_{ev}^{1} (ω_{1})$ by $v (x, y) = u \circ Φ_{ϵ} (x, y)$ , then a simple calculation gives $\begin{array}{rcl} \int_{ω_{ϵ}} a_{ϵ} | \nabla u |^{2} d x d y = \int_{ω_{1}^{ends}} a_{ϵ} | \nabla v |^{2} d x d y + \int_{ω_{1}^{int}} (ϵ a_{ϵ} {(\frac{\partial v}{\partial x})}^{2} + \frac{a_{ϵ}}{ϵ} {(\frac{\partial v}{\partial y})}^{2}) d x d y . \end{array}$ Here $ω_{1}^{int}$ denotes the square $\begin{matrix} ω_{1}^{int} = σ \times (- 1, 1) = (- 1, 1) \times (- 1, 1), \end{matrix}$ and $ω_{1}^{ends}$ denotes the complement of $ω_{1}^{int}$ , i.e., $ω_{1}^{ends} = ω_{1} ∖ ω_{1}^{int}$ . Let $V_{ϵ}$ denote the vector space $\begin{matrix} V_{ϵ} = {(u, v) \in H_{ev}^{1} (\tilde{Ω} ∖ {\overline{ω}}_{ϵ}) \times H_{ev}^{1} (ω_{1}), s.t. u \circ Φ_{ϵ} = v on \partial ω_{1}} . \end{matrix}$ Let $F^{ϵ}$ denote the energy $\begin{matrix} F^{ϵ} (u, v) = \frac{1}{2} \int_{\tilde{Ω} ∖ \overline{ω_{ϵ}}} | \nabla u |^{2} + \frac{ϵ a_{ϵ}}{2} \int_{ω_{1}^{int}} {(\frac{\partial v}{\partial x})}^{2} + \frac{a_{ϵ}}{2 ϵ} \int_{ω_{1}^{int}} {(\frac{\partial v}{\partial y})}^{2} + \frac{a_{ϵ}}{2} \int_{ω_{1}^{ends}} | \nabla v |^{2} . \end{matrix}$ The earlier calculation shows that $\begin{matrix} E^{ϵ} (u_{ϵ}^{ev}) = min_{\begin{array}{c} u \in H_{ev}^{1} (\tilde{Ω}) \\ u = {\tilde{φ}}^{ev} on \partial \tilde{Ω} \end{array}} E^{ϵ} (u) = min_{\begin{array}{c} (u, v) \in V_{ϵ} \\ u = {\tilde{φ}}^{ev} on \partial \tilde{Ω} \end{array}} F^{ϵ} (u, v), \end{matrix}$ and as a consequence, $\begin{matrix} (8) & E^{ϵ} (u_{ϵ}^{ev}) - E_{σ}^{ϵ} (u^{ev}) ⩽ F^{ϵ} (u, v) - E_{σ}^{ϵ} (u^{ev}) \end{matrix}$ for any pair of functions $(u, v) \in V_{ϵ}$ , with $u = {\tilde{φ}}^{ev}$ on $\partial \tilde{Ω}$ . We now construct an element in $V_{ϵ}$ that makes the right hand side of (8) of the order $C ‖ {\tilde{φ}}^{ev} ‖_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ^{γ / 2}$ . For that purpose let us try $u^{test} = u^{ev}$ , $v^{test} (x, y) = u^{ev} (p_{σ} (x, y))$ $\begin{matrix} F^{ϵ} (u^{test}, v^{test}) = \frac{1}{2} \int_{\tilde{Ω} ∖ \overline{ω_{ϵ}}} {| \nabla u^{ev} |}^{2} + ϵ a_{ϵ} \int_{σ} {(\frac{\partial u^{ev}}{\partial x})}^{2} ⩽ E_{σ}^{ϵ} (u^{ev}), \end{matrix}$ but $(u^{test}, v^{test})$ is not quite in $V_{ϵ}$ , since it fails to satisfy the condition $u^{test} \circ Φ_{ϵ} = v^{test}$ on $\partial ω_{1}$ . To remedy this, we introduce $z_{ϵ} \in H^{1} (\tilde{Ω} ∖ \overline{ω_{ϵ}})$ as the unique solution to $\begin{matrix} (9) & \{\begin{matrix} - Δ z_{ϵ} = 0 & in \tilde{Ω} ∖ \overline{ω_{ϵ}}, \\ z_{ϵ} = 0 & on \partial \tilde{Ω}, \\ z_{ϵ} = u^{ev} \circ p_{σ} - u^{ev} & on \partial ω_{ϵ} . \end{matrix} \end{matrix}$ In Appendix A it is shown that

Lemma 3.
The corrector $z_{ϵ}$ defined by ( 9 ) satisfies $\begin{matrix} ‖ \nabla z_{ϵ} ‖_{L^{2} (\tilde{Ω} ∖ \overline{ω_{ϵ}})} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{γ / 2}, \end{matrix}$ for any $γ < 1$ .

By construction, the pair $(u^{test} + z_{ϵ}, v^{test})$ belongs to $V_{ϵ}$ and $u^{test} + z_{ϵ} = {\tilde{φ}}^{ev}$ on $\partial \tilde{Ω}$ . Inserting the pair $(u^{test} + z_{ϵ}, v^{test})$ as a “test function” in $F^{ϵ}$ , and using Lemma 3, we now calculate: $\begin{array}{rcl} F^{ϵ} (u^{test} + z_{ϵ}, v^{test}) & = & F^{ϵ} (u^{test}, v^{test}) + \frac{1}{2} \int_{\tilde{Ω} ∖ \overline{ω_{ϵ}}} | \nabla z_{ϵ} |^{2} + \int_{\tilde{Ω} ∖ \overline{ω_{ϵ}}} \nabla z_{ϵ} \cdot \nabla u^{ev} \\ ⩽ & E_{σ}^{ϵ} (u^{ev}) + C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ^{γ} + C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{γ / 2} {‖ u^{ev} ‖}_{H^{1} (\tilde{Ω} ∖ \overline{ω_{ϵ}})} \\ ⩽ & E_{σ}^{ϵ} (u^{ev}) + C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ^{γ / 2}, \end{array}$ and so by insertion into (8) we obtain $\begin{matrix} E^{ϵ} (u_{ϵ}^{ev}) - E_{σ}^{ϵ} (u^{ev}) ⩽ F^{ϵ} (u^{test} + z_{ϵ}, v^{test}) - E_{σ}^{ϵ} (u^{ev}) ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ^{γ / 2}, \end{matrix}$ as desired. Remark.
We note that $u^{test} = u^{ev}$ has singularities of the form $r^{1 / 2} sin θ / 2$ near the endpoints of σ (see Proposition 1 of [1]), and that $\begin{matrix} a_{ϵ} \int_{r < ϵ} [{(\frac{\partial}{\partial r} r^{1 / 2} sin θ / 2)}^{2} + {(\frac{1}{r} \frac{\partial}{\partial θ} r^{1 / 2} sin θ / 2)}^{2}] r d r d θ = ϵ a_{ϵ} \frac{π}{2} . \end{matrix}$ Since this appears to be of the same order as $\begin{matrix} \frac{ϵ a_{ϵ}}{2} \int_{σ} ({(\frac{\partial u^{e v +}}{\partial τ})}^{2} + {(\frac{\partial u^{e v -}}{\partial τ})}^{2}) d s = ϵ a_{ϵ} \int_{σ} {(\frac{\partial u^{*}}{\partial τ})}^{2} d s, \end{matrix}$ (see Proposition 1 of [1]) one might expect that the singularities would contribute to the “limiting energy” $E_{σ, \tilde{Ω}}^{ϵ}$ . The analysis in this section has shown that, if instead of accounting for the boundary discrepancy in $u^{test} - v^{test} \circ Φ_{ϵ}^{- 1}$ by adding a corrector inside $ω_{ϵ}$ , one adds a corrector $z_{ϵ}$ outside $ω_{ϵ}$ , one may indeed do so in a way that is of the order $C ‖ {\tilde{φ}}^{ev} ‖_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{γ / 2}$ , and thus doesn’t contribute to the “limiting energy”.

2.2. Proof of (7)

It is well known that $\begin{array}{rcl} E^{ϵ} (u_{ϵ}^{ev}) & = & min_{\begin{array}{c} u \in H_{ev}^{1} (\tilde{Ω}) \\ u = {\tilde{φ}}^{ev} on \partial \tilde{Ω} \end{array}} E^{ϵ} (u) \\ (10) & = & min_{\begin{array}{c} u \in H^{1} (\tilde{Ω}) \\ u = {\tilde{φ}}^{ev} on \partial \tilde{Ω} \end{array}} E^{ϵ} (u) = max_{\begin{array}{c} ξ \in L^{2} {(\tilde{Ω})}^{2} \\ - \nabla \cdot ξ = 0 \end{array}} E_{c}^{ϵ} (ξ), \end{array}$ where the complementary energy expression $E_{c}^{ϵ}$ is given by $\begin{matrix} E_{c}^{ϵ} (ξ) = \int_{\partial \tilde{Ω}} ξ \cdot n {\tilde{φ}}^{ev} d s - \frac{1}{2} \int_{\tilde{Ω}} γ_{ϵ}^{- 1} | ξ |^{2} d x d y . \end{matrix}$ We may split the volume integral as $\begin{matrix} \int_{\tilde{Ω}} γ_{ϵ}^{- 1} | ξ |^{2} d x = \int_{\tilde{Ω} ∖ ω_{ϵ}} | ξ |^{2} d x d y + a_{ϵ}^{- 1} \int_{ω_{ϵ}} | ξ |^{2} d x d y . \end{matrix}$ Consistent with this splitting, the maximization in (10) may be rewritten as taking place over $\begin{array}{l} ξ_{+} \in L^{2} {(\tilde{Ω} ∖ ω_{ϵ})}^{2} with - \nabla \cdot ξ_{+} = 0 in \tilde{Ω} ∖ ω_{ϵ}, \\ ξ_{-} \in L^{2} {(ω_{ϵ})}^{2} with - \nabla \cdot ξ_{-} = 0 in ω_{ϵ}, \\ subject to the
boundary condition ξ_{+} \cdot n = ξ_{-} \cdot n on \partial ω_{ϵ} . \end{array}$ From now on n denotes the outward normal to $ω_{ϵ}$ at $\partial ω_{ϵ}$ (or to $ω_{1}$ at $\partial ω_{1}$ ). Let $Φ_{ϵ}$ denote the previously defined 1-to-1 mapping of $ω_{1}$ onto $ω_{ϵ}$ . As $ξ^{*}$ ranges over the set $\begin{matrix} {ξ \in L^{2} {(ω_{ϵ})}^{2} : \nabla \cdot ξ = 0 in ω_{ϵ} and ξ \cdot n = ξ_{+} \cdot n on \partial ω_{ϵ}}, \end{matrix}$ for a fixed $ξ_{+}$ , then the vector field $η^{*} = | det \nabla Φ_{ϵ} | {(\nabla Φ_{ϵ})}^{- 1} ξ^{*} \circ Φ_{ϵ}$ ranges over the set $\begin{matrix} {η \in L^{2} {(ω_{1})}^{2} : \nabla \cdot η = 0 in ω_{1} and η \cdot n = | \frac{\partial}{\partial τ} Φ_{ϵ} | (ξ_{+} \cdot n) \circ Φ_{ϵ} on \partial ω_{1}}, \end{matrix}$ and $\begin{matrix} \int_{ω_{ϵ}} {| ξ^{*} |}^{2} d x d y = \int_{ω_{1}} \frac{1}{| det \nabla Φ_{ϵ} |} (\nabla Φ_{ϵ}^{T} \nabla Φ_{ϵ}) η^{*} \cdot η^{*} d x d y . \end{matrix}$ From these calculations it follows that $\begin{matrix} (11) & E^{ϵ} (u_{ϵ}^{ev}) = max_{(ξ, η) \in W_{ϵ}} F_{c}^{ϵ} (ξ, η), \end{matrix}$ where $F_{c}^{ϵ}$ is given by $\begin{array}{rcl} F_{c}^{ϵ} (ξ, η) & = & \int_{\partial \tilde{Ω}} ξ \cdot n {\tilde{φ}}^{ev} d s - \frac{1}{2} \int_{\tilde{Ω} ∖ ω_{ϵ}} | ξ |^{2} d x d y \\ (12) & - \frac{1}{2 a_{ϵ}} \int_{ω_{1}} \frac{1}{| det \nabla Φ_{ϵ} |} (\nabla Φ_{ϵ}^{T} \nabla Φ_{ϵ}) η \cdot η d x d y, \end{array}$ and the set $W_{ϵ}$ is given by $\begin{array}{rcl} W_{ϵ} & \equiv & {(ξ, η) \in L^{2} {(\tilde{Ω} ∖ ω_{ϵ})}^{2} \times L^{2} {(ω_{1})}^{2} : \nabla \cdot ξ = 0 in \tilde{Ω} ∖ ω_{ϵ}, \nabla \cdot η = 0 in ω_{1}, \\ (13) & and η \cdot n = | \frac{\partial}{\partial τ} Φ_{ϵ} | (ξ \cdot n) \circ Φ_{ϵ} on \partial ω_{1}} . \end{array}$ We now divide $ω_{1}$ into two parts as before $\begin{matrix} ω_{1}^{int} = σ \times (- 1, 1) = (- 1, 1) \times (- 1, 1) and ω_{1}^{ends} = ω_{1} ∖ ω_{1}^{int} . \end{matrix}$ and compute $\begin{matrix} (14) & \nabla Φ_{ϵ} = (\begin{matrix} 1 & 0 \\ 0 & ϵ \end{matrix}) in ω_{1}^{int}, | \frac{\partial}{\partial τ} Φ_{ϵ} | = 1 on \partial ω_{1} \cap \partial ω_{1}^{int}, \end{matrix}$ and $\begin{matrix} (15) & \nabla Φ_{ϵ} = ϵ I in ω_{1}^{ends}, | \frac{\partial}{\partial τ} Φ_{ϵ} | = ϵ on \partial ω_{1} \cap \partial ω_{1}^{ends} . \end{matrix}$ It thus follows that $\begin{array}{rcl} F_{c}^{ϵ} (ξ, η) & = & \int_{\partial \tilde{Ω}} ξ \cdot n {\tilde{φ}}^{ev} d s - \frac{1}{2} \int_{\tilde{Ω} ∖ ω_{ϵ}} | ξ |^{2} d x d y \\ - \frac{1}{2 ϵ a_{ϵ}} \int_{ω_{1}^{int}} {(η_{1})}^{2} d x d y - \frac{ϵ}{2 a_{ϵ}} \int_{ω_{1}^{int}} {(η_{2})}^{2} d x d y \\ - \frac{1}{2 a_{ϵ}} \int_{ω_{1}^{ends}} | η |^{2} d x d y \end{array}$ and that $W_{ϵ}$ alternatively may be characterized as $\begin{array}{rcl} W_{ϵ} & = & {(ξ, η) \in L^{2} {(\tilde{Ω} ∖ ω_{ϵ})}^{2} \times L^{2} {(ω_{1})}^{2} : \nabla \cdot ξ = 0 in \tilde{Ω} ∖ ω_{ϵ}, \nabla \cdot η = 0 in ω_{1}, \\ and η_{2} (x, y) = ξ_{2} (x, ϵ y) for - 1 ⩽ x ⩽ 1, y = \pm 1, \\ η \cdot n = ϵ (ξ \cdot n) \circ Φ_{ϵ} on \partial ω_{1} \cap \partial ω_{1}^{ends}} . \end{array}$ From (11) we have that $\begin{matrix} (16) & E_{σ}^{ϵ} (u^{ev}) - E^{ϵ} (u_{ϵ}^{ev}) ⩽ E_{σ}^{ϵ} (u^{ev}) - F_{c}^{ϵ} (ξ, η), \end{matrix}$ for any pair $(ξ, η) \in W_{ϵ}$ . We now proceed to construct a test pair from $W_{ϵ}$ , so that we may establish (7), in other words a test pair which satisfies $\begin{matrix} F_{c}^{ϵ} (ξ, η) ⩾ E_{σ}^{ϵ} (u^{ev}) - C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ^{γ / 4} . \end{matrix}$ From the representation formula for $u^{ev}$ in Proposition 1 of Appendix A in [1], and the remark immediately following that proposition, we know that $\begin{matrix} (17) & u^{ev} = b_{1} r^{1 / 2} sin (θ / 2) ϕ (r) + b_{1}^{'} r^{' 1 / 2} sin (θ^{'} / 2) ϕ (r^{'}) + u^{*}, \end{matrix}$ where $(r, θ)$ and $(r^{'}, θ^{'})$ are polar coordinates around the two endpoints of σ, and $u^{*}$ is bounded in $C_{ev}^{1, γ / 2} (K ∖ σ)$ , up to the curve σ, by $C ‖ {\tilde{φ}}^{ev} ‖_{H^{1 / 2} (\partial \tilde{Ω})}$ . Here we make use of the fact that $ϵ a_{ϵ} > m > 0$ . If we differentiate this representation with respect to y and evaluate it on σ, then we obtain $\begin{matrix} \frac{\partial u^{ev}}{\partial y} (x, 0_{\pm}) = \pm (\frac{b_{1}}{2} {(x + 1)}^{- 1 / 2} ϕ (x + 1) + \frac{b_{1}^{'}}{2} {(1 - x)}^{- 1 / 2} ϕ (1 - x) + \frac{\partial u^{*}}{\partial y} (x, 0_{+})), \end{matrix}$ for $- 1 < x < 1$ . Notice that these two functions are not in $L^{2} (- 1, 1)$ . However, we may define approximations that are in $L^{2}$ , as follows $\begin{array}{rcl} D U_{\pm} (x) & = & \pm D U_{+} (x) \\ = & \pm (\frac{b_{1}}{2} max {x + 1, ϵ}^{- 1 / 2} ϕ (x + 1) + \frac{b_{1}^{'}}{2} max {1 - x, ϵ}^{- 1 / 2} ϕ (1 - x) \\ + \frac{\partial u^{*}}{\partial y} (x, 0_{+}) + d_{ϵ}), \end{array}$ with the constant $d_{ϵ}$ chosen so that $\begin{matrix} \int_{- 1}^{1} D U_{\pm} (x) d x = \int_{- 1}^{1} \frac{\partial u^{ev}}{\partial y} (x, 0_{\pm}) d x . \end{matrix}$ This requires that $\begin{array}{rcl} d_{ϵ} & = & \frac{b_{1}}{4} \int_{- 1}^{- 1 + ϵ} [{(x + 1)}^{- 1 / 2} - ϵ^{- 1 / 2}] d x + \frac{b_{1}^{'}}{4} \int_{1 - ϵ}^{1} [{(1 - x)}^{- 1 / 2} - ϵ^{- 1 / 2}] d x \\ = & \sqrt{ϵ} (\frac{b_{1}}{4} + \frac{b_{1}^{'}}{4}) . \end{array}$ Since $ϵ a_{ϵ} > m$ , we observe that $| d_{ϵ} | ⩽ C ‖ {\tilde{φ}}^{ev} ‖_{H^{1 / 2} (\partial \tilde{Ω})} \sqrt{ϵ}$ . We also calculate $\begin{matrix} (18) & ‖ D U_{+} ‖_{L^{2} (- 1, 1)} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} | log ϵ |^{1 / 2} . \end{matrix}$ Define the function R by $\begin{matrix} R (x) = \int_{- 1}^{x} [\frac{\partial u^{ev}}{\partial y} (s, 0_{+}) - D U_{+} (s)] d s, - 1 < x < 1 . \end{matrix}$ A simple calculation shows that $\begin{matrix} (19) & ‖ R ‖_{L^{2} (- 1, 1)} ⩽ \sqrt{ϵ} C (| b_{1} | + | b_{1}^{'} |) + | d_{ϵ} | ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} \sqrt{ϵ} . \end{matrix}$ The function R is actually continuous with the same $C^{0}$ -bound of $C ‖ {\tilde{φ}}^{ev} ‖_{H^{1 / 2} (\partial \tilde{Ω})} \sqrt{ϵ}$ , and $\begin{matrix} (20) & R (- 1) = R (1) = 0 . \end{matrix}$ To construct a test pair for (16) we select $ξ^{*} = \nabla u^{ev}$ and define $η^{*}$ by $\begin{matrix} (\begin{matrix} η_{1}^{*} (x, y) \\ η_{2}^{*} (x, y) \end{matrix}) = (\begin{matrix} ϵ a_{ϵ} \frac{\partial}{\partial x} u^{ev} (x, 0) + R (x) \\ y D U_{+} (x) \end{matrix}) for (x, y) \in ω_{1}^{int}, \end{matrix}$ and, $\begin{matrix} (\begin{matrix} η_{1}^{*} (x, y) \\ η_{2}^{*} (x, y) \end{matrix}) = (\begin{matrix} 0 \\ 0 \end{matrix}) for (x, y) \in ω_{1}^{ends} . \end{matrix}$ The pair $(ξ^{*}, η^{*})$ clearly lies in $L^{2} {(Ω ∖ ω_{ϵ})}^{2} \times L^{2} {(ω_{1})}^{2}$ , and it satisfies the two divergence constraints prescribed in $W_{ϵ}$ : $\begin{matrix} \nabla \cdot ξ^{*} = 0 in \tilde{Ω} ∖ ω_{ϵ}, and \nabla \cdot η^{*} = 0 in ω_{1} . \end{matrix}$ Here we use the fact that $\begin{array}{rcl} \frac{\partial}{\partial x} (ϵ a_{ϵ} \frac{\partial}{\partial x} u^{ev} (x, 0) + R (x)) & = & - \frac{\partial u^{ev}}{\partial y} (x, 0_{+}) + (\frac{\partial u^{ev}}{\partial y} (x, 0_{+}) - D U_{+} (x)) \\ = & - \frac{\partial}{\partial y} (y D U_{+} (x)) \end{array}$ in $ω_{1}^{int}$ . We also use (20) and the fact that $\begin{matrix} \frac{\partial}{\partial x} u^{ev} (- 1_{+}, 0) = \frac{\partial}{\partial x} u^{*} (- 1, 0) = 0, and \frac{\partial}{\partial x} u^{ev} (1_{-}, 0) = \frac{\partial}{\partial x} u^{*} (1, 0) = 0, \end{matrix}$ which follows from the decomposition (17) and Lemma 2 in Appendix A of [1]. In general, this pair $(ξ^{*}, η^{*})$ fails to satisfy the “interface” conditions, instead it satisfies $\begin{matrix} (21) & η_{2}^{*} (x, y) = \pm D U_{+} (x) = D U_{\pm} (x), for - 1 < x < 1, y = \pm 1, \end{matrix}$ and $\begin{matrix} η^{*} \cdot n = 0 on \partial ω_{1} \cap {(x, y) : | x | > 1} . \end{matrix}$ To remedy this situation we seek $ξ_{ϵ} \in {(L^{2} (\tilde{Ω} ∖ ω_{ϵ}))}^{2}$ so that $\begin{array}{l} \nabla \cdot ξ_{ϵ} = 0 in \tilde{Ω} ∖ ω_{ϵ}, ξ_{ϵ} \cdot n = 0 on \partial \tilde{Ω}, \\ (22) & {(ξ_{ϵ})}_{2} (x, \pm ϵ) = D U_{\pm} (x) - \frac{\partial}{\partial y} u^{ev} (x, \pm ϵ), \\ ξ_{ϵ} \cdot n = - \frac{\partial}{\partial n} u^{ev} on \partial ω_{ϵ} \cap {(x, y) : | x | > 1} . \end{array}$ In Appendix B it is shown that

Lemma 4.
It is possible to find a field $ξ_{ϵ} \in {(L^{2} (\tilde{Ω} ∖ ω_{ϵ}))}^{2}$ , which satisfies the requirements ( 22 ), together with the estimate $\begin{matrix} (23) & ‖ ξ_{ϵ} ‖_{L^{2} (\tilde{Ω} ∖ ω_{ϵ})} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{γ / 4}, \end{matrix}$ for any $γ < 1$ .

Given such a $ξ_{ϵ}$ , the pair $(ξ^{} + ξ_{ϵ}, η^{})$ is in $W_{ϵ}$ . We now calculate $\begin{array}{rcl} F_{c}^{ϵ} (ξ^{} + ξ_{ϵ}, η^{}) & = & \int_{\partial \tilde{Ω}} \frac{\partial u^{ev}}{\partial n} {\tilde{φ}}^{ev} d s - \frac{1}{2} \int_{\tilde{Ω} ∖ ω_{ϵ}} {| \nabla u^{ev} + ξ_{ϵ} |}^{2} d x d y \\ - \frac{1}{2 ϵ a_{ϵ}} \int_{ω_{1}^{int}} {| ϵ a_{ϵ} \frac{\partial u^{ev}}{\partial x} (x, 0) + R (x) |}^{2} d x d y \\ - \frac{ϵ}{2 a_{ϵ}} \int_{ω_{1}^{int}} {| y D U_{+} (x) |}^{2} d x d y \\ = & \int_{\partial \tilde{Ω}} \frac{\partial u^{ev}}{\partial n} {\tilde{φ}}^{ev} d s - \frac{1}{2} \int_{\tilde{Ω} ∖ ω_{ϵ}} {| \nabla u^{ev} + ξ_{ϵ} |}^{2} d x d y \\ - \frac{1}{ϵ a_{ϵ}} \int_{σ} {| ϵ a_{ϵ} \frac{\partial u^{ev}}{\partial x} (x, 0) + R (x) |}^{2} d x \\ - \frac{ϵ}{3 a_{ϵ}} \int_{σ} {| D U_{+} (x) |}^{2} d x \\ ⩾ & \int_{\partial \tilde{Ω}} \frac{\partial u^{ev}}{\partial n} {\tilde{φ}}^{ev} d s - \frac{1}{2} \int_{\tilde{Ω} ∖ ω_{ϵ}} {| \nabla u^{ev} |}^{2} d x d y - ϵ a_{ϵ} \int_{σ} {| \frac{\partial u^{ev}}{\partial x} (x, 0) |}^{2} d x \\ - {‖ \nabla u^{ev} ‖}_{L^{2} (\tilde{Ω} ∖ ω_{ϵ})} ‖ ξ_{ϵ} ‖_{L^{2} (\tilde{Ω} ∖ ω_{ϵ})} - \frac{1}{2} ‖ ξ_{ϵ} ‖_{L^{2} (\tilde{Ω} ∖ ω_{ϵ})}^{2} \\ - \frac{2}{ϵ a_{ϵ}} ({‖ ϵ a_{ϵ} \frac{\partial u^{ev}}{\partial x} ‖}_{L^{2} (σ)} ‖ R ‖_{L^{2} (σ)} + ‖ R ‖_{L^{2} (σ)}^{2}) \\ (24) & - \frac{ϵ}{3 a_{ϵ}} ‖ D U_{+} ‖_{L^{2} (σ)}^{2} . \end{array}$ From a standard energy estimate we have $\begin{matrix} {‖ \nabla u^{ev} ‖}_{L^{2} (\tilde{Ω} ∖ ω_{ϵ})} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}, and {(ϵ a_{ϵ})}^{1 / 2} {‖ \frac{\partial u^{ev}}{\partial x} ‖}_{L^{2} (σ)} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}, \end{matrix}$ and by insertion of this and the estimates (18), (19) and (23) into (24), we arrive at $\begin{array}{rcl} F_{c}^{ϵ} (ξ^{} + ξ_{ϵ}, η^{}) & ⩾ & \int_{\partial \tilde{Ω}} \frac{\partial u^{ev}}{\partial n} {\tilde{φ}}^{ev} d s - \frac{1}{2} \int_{\tilde{Ω} ∖ ω_{ϵ}} {| \nabla u^{ev} |}^{2} d x d y - ϵ a_{ϵ} \int_{σ} {| \frac{\partial u^{ev}}{\partial x} (x, 0) |}^{2} d x \\ - C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ^{γ / 4} - C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ^{γ / 2} - \frac{C ‖ {\tilde{φ}}^{ev} ‖_{H^{1 / 2} (\partial \tilde{Ω})}^{2} \sqrt{ϵ}}{{(ϵ a_{ϵ})}^{1 / 2}} \\ - \frac{C ‖ {\tilde{φ}}^{ev} ‖_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ}{ϵ a_{ϵ}} - \frac{C ‖ {\tilde{φ}}^{ev} ‖_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ^{2} | log ϵ |}{ϵ a_{ϵ}} \\ ⩾ & \int_{\partial \tilde{Ω}} \frac{\partial u^{ev}}{\partial n} {\tilde{φ}}^{ev} d s - \frac{1}{2} \int_{\tilde{Ω} ∖ σ} {| \nabla u^{ev} |}^{2} d x d y - ϵ a_{ϵ} \int_{σ} {| \frac{\partial u^{ev}}{\partial x} (x, 0) |}^{2} d x \\ - C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ^{γ / 4} \\ = & E_{σ}^{ϵ} (u^{ev}) - C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ^{γ / 4}, \end{array}$ since by assumption $ϵ a_{ϵ}$ is bounded uniformly away from zero. The exponent γ may be chosen arbitrarily close to 1. Insertion of this inequality into (16) now finally yields $\begin{matrix} E_{σ}^{ϵ} (u^{ev}) - E^{ϵ} (u_{ϵ}^{ev}) ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ^{γ / 4}, \end{matrix}$ which completes the proof of (7).
3. Local convergence – Even symmetry, $ϵ a_{ϵ} < 1$

In this section we shall show that, in the regime $ϵ a_{ϵ} < 1$ , $\begin{matrix} (25) & | E_{σ, \tilde{Ω}}^{ϵ} (u^{ev}) - E_{\tilde{Ω}}^{ϵ} (u_{ϵ}^{ev}) | ⩽ C_{γ} {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} (ϵ a_{ϵ} + ϵ^{γ}), \end{matrix}$ for any $γ < 1$ . Here $E_{σ, \tilde{Ω}}^{ϵ}$ and $E_{\tilde{Ω}}^{ϵ}$ are the energy expressions introduced in (1) and (2), which when restricted to even functions are given by (4) and (5). We shall prove (25) in two steps: first we prove $\begin{matrix} (26) & | E_{σ, \tilde{Ω}}^{ϵ} (u^{ev}) - E_{\tilde{Ω}}^{0} (u_{0}^{ev}) | ⩽ C_{γ} {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ a_{ϵ}, \end{matrix}$ and then we prove $\begin{matrix} (27) & | E_{\tilde{Ω}}^{0} (u_{0}^{ev}) - E_{\tilde{Ω}}^{ϵ} (u_{ϵ}^{ev}) | ⩽ C_{γ} {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} (ϵ^{γ} + ϵ a_{ϵ}) . \end{matrix}$ Here $u_{0}^{ev}$ denotes the solution to $\begin{matrix} - Δ u_{0}^{ev} = 0 in \tilde{Ω}, u_{0}^{ev} = {\tilde{φ}}^{ev} on \partial \tilde{Ω}, \end{matrix}$ which is obtained through the minimization $\begin{matrix} E_{\tilde{Ω}}^{0} (u_{0}^{ev}) = min_{\begin{array}{c} u \in H^{1} (\tilde{Ω}) \\ u = {\tilde{φ}}^{ev} on \partial \tilde{Ω} \end{array}} E_{\tilde{Ω}}^{0} (u) = min_{\begin{array}{c} u \in H_{ev}^{1} (\tilde{Ω}) \\ u = {\tilde{φ}}^{ev} on \partial \tilde{Ω} \end{array}} E_{\tilde{Ω}}^{0} (u), \end{matrix}$ with $\begin{matrix} E_{\tilde{Ω}}^{0} (u) = \frac{1}{2} \int_{\tilde{Ω}} | \nabla u |^{2} d x d y . \end{matrix}$ The estimate (25) follows immediately by combination of (26) and (27). For simplicity of notation we shall in the following delete the explicit appearance of $\tilde{Ω}$ in the energy expression symbols, in other words we shall use $E_{σ}^{ϵ}$ , $E^{ϵ}$ , and $E^{0}$ in place of $E_{σ, \tilde{Ω}}^{ϵ}$ , $E_{\tilde{Ω}}^{ϵ}$ and $E_{\tilde{Ω}}^{0}$ . We also drop the subscript γ on the constant $C_{γ}$ .

3.1. Proof of (26)

Due to the minimizing properties of $u_{0}^{ev}$ and $u^{ev}$ , and the positivity of $ϵ a_{ϵ}$ , $\begin{array}{rcl} \frac{1}{2} \int_{\tilde{Ω}} {| \nabla u_{0}^{ev} |}^{2} d x d y & ⩽ & \frac{1}{2} \int_{\tilde{Ω}} {| \nabla u^{ev} |}^{2} d x d y \\ ⩽ & \frac{1}{2} \int_{\tilde{Ω} ∖ σ} {| \nabla u^{ev} |}^{2} d x d y + ϵ a_{ϵ} \int_{σ} {| \frac{\partial u^{ev}}{\partial x} |}^{2} d s \\ ⩽ & \frac{1}{2} \int_{\tilde{Ω} ∖ σ} {| \nabla u_{0}^{ev} |}^{2} d x d y + ϵ a_{ϵ} \int_{σ} {| \frac{\partial u_{0}^{ev}}{\partial x} |}^{2} d s \\ ⩽ & \frac{1}{2} \int_{\tilde{Ω}} {| \nabla u_{0}^{ev} |}^{2} d x d y + C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ a_{ϵ} . \end{array}$ Here we have used that $u^{ev}$ is indeed in $H^{1} (\tilde{Ω})$ (not just in $H^{1} (\tilde{Ω} ∖ σ)$ ) due to its even symmetry. We have also used that, due to elliptic regularity, $\begin{matrix} \int_{σ} {| \frac{\partial u_{0}^{ev}}{\partial x} |}^{2} d s ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} . \end{matrix}$ The previous set of inequalities yields that $\begin{matrix} E^{0} (u_{0}^{ev}) ⩽ E_{σ}^{ϵ} (u^{ev}) ⩽ E^{0} (u_{0}^{ev}) + C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ a_{ϵ}, \end{matrix}$ which immediately leads to the estimate (26).

3.2. Proof of (27)

In this subsection we shall prove that $\begin{matrix} | E^{0} (u_{0}^{ev}) - E^{ϵ} (u_{ϵ}^{ev}) | ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} (ϵ^{γ} + ϵ a_{ϵ}) . \end{matrix}$ We achieve this in two steps, by proving that $\begin{matrix} (28) & E^{ϵ} (u_{ϵ}^{ev}) - E^{0} (u_{0}^{ev}) ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ a_{ϵ}, \end{matrix}$ and $\begin{matrix} (29) & E^{0} (u_{0}^{ev}) - E^{ϵ} (u_{ϵ}^{ev}) ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ^{γ}, \end{matrix}$ for any $γ < 1$ . We calculate $\begin{array}{rcl} E^{ϵ} (u_{ϵ}^{ev}) & ⩽ & E^{ϵ} (u_{0}^{ev}) = \frac{1}{2} \int_{\tilde{Ω} ∖ ω_{ϵ}} {| \nabla u_{0}^{ev} |}^{2} d x d y + \frac{1}{2} \int_{ω_{ϵ}} a_{ϵ} {| \nabla u_{0}^{ev} |}^{2} d x d y \\ ⩽ & \frac{1}{2} \int_{\tilde{Ω} ∖ ω_{ϵ}} {| \nabla u_{0}^{ev} |}^{2} d x d y + C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ a_{ϵ} \\ ⩽ & \frac{1}{2} \int_{\tilde{Ω}} {| \nabla u_{0}^{ev} |}^{2} d x d y + C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ a_{ϵ} \\ = & E^{0} (u_{0}^{ev}) + C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ a_{ϵ}, \end{array}$ to establish (28). By duality we know that $\begin{matrix} E^{ϵ} (u_{ϵ}^{ev}) = max_{ξ \in W} [\int_{\partial \tilde{Ω}} {\tilde{φ}}^{ev} ξ \cdot n d s - \frac{1}{2} \int_{\tilde{Ω} ∖ ω_{ϵ}} | ξ |^{2} d x d y - \frac{1}{2 a_{ϵ}} \int_{ω_{ϵ}} | ξ |^{2} d x d y], \end{matrix}$ where $W \subset L^{2} {(\tilde{Ω})}^{2}$ denotes the following set of vector fields $\begin{matrix} W = {ξ \in L^{2} {(\tilde{Ω})}^{2} : \nabla \cdot ξ = 0 in \tilde{Ω}} . \end{matrix}$ As a test field, we define $\begin{matrix} (30) & ξ^{test} (x, y) = \{\begin{matrix} \nabla u_{0}^{ev} (x, y) + ξ_{ϵ} (x, y) & if (x, y) is in \tilde{Ω} ∖ ω_{ϵ}, \\ 0 & if (x, y) is
in ω_{ϵ}, \end{matrix} \end{matrix}$ where $ξ_{ϵ}$ is selected so that $\begin{matrix} (31) & \nabla \cdot ξ_{ϵ} = 0 in \tilde{Ω} ∖ ω_{ϵ}, \end{matrix}$ and $\begin{matrix} (32) & ξ_{ϵ} \cdot n = 0 on \partial \tilde{Ω}, ξ_{ϵ} \cdot n = - \nabla u_{0}^{ev} \cdot n on \partial ω_{ϵ} . \end{matrix}$ To construct $ξ_{ϵ}$ , we use a simplified version of the approach we use in Appendix B to prove Lemma 4. We first construct a function V on $\partial ω_{ϵ}$ by counterclockwise integration of $- \nabla u_{0}^{ev} \cdot n$ along $\partial ω_{ϵ}$ : $V (\cdot) = - \int_{0}^{\cdot} \nabla u_{0}^{ev} \cdot n d s$ on $\partial ω_{ϵ}$ (with s denoting arclength, and $s = 0$ corresponding to the point $(- 1, ϵ)$ ). We then extend V to $\tilde{Ω} ∖ ω_{ϵ}$ by setting it to be constant along normal rays emanating from $\partial ω_{ϵ}$ . Since $u_{0}^{ev}$ is $C^{2}$ strictly inside $\tilde{Ω}$ (with $C^{2}$ -norm bounded by $C ‖ {\tilde{φ}}^{ev} ‖_{H^{1 / 2} (\partial \tilde{Ω})}$ ), and since $\frac{\partial}{\partial y} u_{0}^{ev} = 0$ on $y = 0$ , this extension, V, is in $H^{1} (\tilde{Ω} ∖ ω_{ϵ})$ , with $\begin{matrix} ‖ V ‖_{H^{1} (\tilde{Ω} ∖ ω_{ϵ})} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{γ}, \end{matrix}$ for any $γ < 1$ . We cut V off near the boundary, so that $V = 0$ near $\partial \tilde{Ω}$ , and then define $\begin{matrix} ξ_{ϵ} = - \nabla V^{⊥} . \end{matrix}$ This field satisfies $\begin{matrix} ‖ ξ_{ϵ} ‖_{L^{2} (\tilde{Ω} ∖ ω_{ϵ})} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{γ} . \end{matrix}$ Furthermore $ξ_{ϵ}$ satisfies (31) and (32), and so $ξ^{test}$ defined by (30) lies in the set W. We now calculate $\begin{array}{rcl} E^{ϵ} (u_{ϵ}^{ev}) & ⩾ & \int_{\partial \tilde{Ω}} {\tilde{φ}}^{ev} ξ^{test} \cdot n d s - \frac{1}{2} \int_{\tilde{Ω} ∖ ω_{ϵ}} {| ξ^{test} |}^{2} d x d y - \frac{1}{2 a_{ϵ}} \int_{ω_{ϵ}} {| ξ^{test} |}^{2} d x d y \\ = & \int_{\partial \tilde{Ω}} {\tilde{φ}}^{ev} \frac{\partial u_{0}^{ev}}{\partial n} d s - \frac{1}{2} \int_{\tilde{Ω} ∖ ω_{ϵ}} {| \nabla u_{0}^{ev} |}^{2} d x d y \\ - \frac{1}{2} \int_{\tilde{Ω} ∖ ω_{ϵ}} | ξ_{ϵ} |^{2} d x d y - \int_{\tilde{Ω} ∖ ω_{ϵ}} \nabla u_{0}^{ev} \cdot ξ_{ϵ} d x d y \\ = & E^{0} (u_{0}^{ev}) + \frac{1}{2} \int_{ω_{ϵ}} {| \nabla u_{0}^{ev} |}^{2} d x d y - \frac{1}{2} \int_{\tilde{Ω} ∖ ω_{ϵ}} | ξ_{ϵ} |^{2} d x d y - \int_{Ω ∖ ω_{ϵ}} \nabla u_{0}^{ev} \cdot ξ_{ϵ} d x d y \\ ⩾ & E^{0} (u_{0}^{ev}) - C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ^{γ}, \end{array}$ which establishes (29).

4. Local convergence – Odd symmetry, $a_{ϵ} < M ϵ$

We now move to consider the case of odd symmetry. When restricted to functions with $v^{odd} \in H_{odd}^{1} (\tilde{Ω} ∖ σ)$ , $v^{odd \pm} \in H^{1} (σ)$ , the energy $E_{σ, \tilde{Ω}}^{ϵ}$ , given by (1), takes the form $\begin{matrix} E_{σ, \tilde{Ω}}^{ϵ} (v^{odd}) = \int_{{\tilde{Ω}}^{+}} {| \nabla v^{odd} |}^{2} d x d y + ϵ a_{ϵ} \int_{σ} {(\frac{\partial v^{odd +}}{\partial τ})}^{2} d s + \frac{a_{ϵ}}{ϵ} \int_{σ} {(v^{odd +})}^{2} d s, \end{matrix}$ where ${\tilde{Ω}}^{+}$ denotes the domain ${\tilde{Ω}}^{+} = \tilde{Ω} \cap {y > 0}$ . The minimizer of the energy $E_{σ, \tilde{Ω}}^{ϵ}$ in $H^{1} (\tilde{Ω} ∖ σ) \cap {v^{\pm} \in H^{1} (σ), v = {\tilde{φ}}^{odd} on \partial \tilde{Ω}}$ is thus entirely determined by $\begin{array}{l} - Δ u^{odd} = 0 in {\tilde{Ω}}^{+}, \\ (33) & \frac{\partial u^{odd}}{\partial y} + ϵ^{2} \frac{a_{ϵ}}{ϵ} \frac{\partial^{2} u^{odd}}{\partial x^{2}} - \frac{a_{ϵ}}{ϵ} u^{odd} = 0 on σ, \\ u^{odd} = 0 on ({y = 0} \cap \tilde{Ω}) ∖ σ, u^{odd} = {\tilde{φ}}^{odd} on {(\partial \tilde{Ω})}^{+} . \end{array}$ We introduce the auxiliary problem $\begin{array}{l} - Δ u_{*}^{odd} = 0 in {\tilde{Ω}}^{+}, \\ \frac{\partial u_{*}^{odd}}{\partial y} - \frac{a_{ϵ}}{ϵ} u_{*}^{odd} = 0 on σ, \\ u_{*}^{odd} = 0 on ({y = 0} \cap \tilde{Ω}) ∖ σ, u_{*}^{odd} = {\tilde{φ}}^{odd} on {(\partial \tilde{Ω})}^{+}, \end{array}$ and let $E_{σ, \tilde{Ω}}^{* ϵ}$ denote the associated energy $\begin{matrix} E_{σ, \tilde{Ω}}^{* ϵ} (w) = \frac{1}{2} \int_{\tilde{Ω} ∖ σ} | \nabla w | d x d y + \frac{a_{ϵ}}{4 ϵ} \int_{σ} {(w^{+} - w^{-})}^{2} d x, \end{matrix}$ defined on $H^{1} (\tilde{Ω} ∖ σ)$ . The function $u_{*}^{odd}$ is simply the restriction to ${\tilde{Ω}}^{+}$ of the minimizer of this energy in $H^{1} (\tilde{Ω} ∖ σ) \cap {w |_{\partial \tilde{Ω}} = {\tilde{φ}}^{odd}}$ ; we denote the minimizer (defined on all of $\tilde{Ω}$ ) by $u_{*}^{odd}$ as well. Note that $u_{*}^{odd}$ is exactly the solution, for which we studied the regularity in Appendix B of [1]. The important role played by $u_{*}^{odd}$ is due to the following lemma.

Lemma 5.
Under the assumption that $a_{ϵ} < M ϵ$ , for some fixed constant M, the following estimate holds $\begin{matrix} | E_{σ, \tilde{Ω}}^{ϵ} (u^{odd}) - E_{σ, \tilde{Ω}}^{* ϵ} (u_{}^{odd}) | ⩽ C_{M} {‖ {\tilde{φ}}^{odd} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ, \end{matrix}$ uniformly in* $0 < a_{ϵ}$ and $0 < ϵ < ϵ_{0}$ .
Proof of Lemma 5.
In order to prove this lemma we shall use the previously noted fact that $\begin{array}{rcl} E_{σ, \tilde{Ω}}^{ϵ} (u^{odd}) & = & 2 (\frac{1}{2} \int_{{\tilde{Ω}}^{+}} {| \nabla u^{odd} |}^{2} d x d y + \frac{ϵ a_{ϵ}}{2} \int_{σ} {(\frac{\partial u^{odd +}}{\partial τ})}^{2} d s \\ (34) & + \frac{a_{ϵ}}{2 ϵ} \int_{σ} {(u^{odd +})}^{2} d s), \end{array}$ as well as the fact that $\begin{matrix} (35) & E_{σ, \tilde{Ω}}^{* ϵ} (u_{}^{odd}) = 2 (\frac{1}{2} \int_{{\tilde{Ω}}^{+}} {| \nabla u_{}^{odd} |}^{2} d x d y + \frac{a_{ϵ}}{2 ϵ} \int_{σ} {(u_{}^{odd +})}^{2} d s) . \end{matrix}$ We shall rely on the decomposition of $u_{}^{odd}$ (in ${\tilde{Ω}}^{+}$ ) established by Proposition 2 in Appendix B of [1], namely $\begin{matrix} u_{}^{odd} (x, y) = b_{1} r^{1 / 2} cos (θ / 2) ϕ (r) + b_{1}^{'} r^{' 1 / 2} cos (θ^{'} / 2) ϕ (r^{'}) + w^{} (x, y), \end{matrix}$ where $(r, θ)$ and $(r^{'}, θ^{'})$ denote polar coordinates around the two endpoints of σ and the remainder $w^{}$ is in $C^{1, γ / 2} (K^{+})$ for any $γ < 1$ . Furthermore $\begin{matrix} | b_{1} | + | b_{1}^{'} | ⩽ C_{M} {‖ {\tilde{φ}}^{odd} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} and {‖ w^{} ‖}_{C^{1, γ / 2} (K^{+})} ⩽ C_{M} {‖ {\tilde{φ}}^{odd} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}, \end{matrix}$ uniformly in ϵ and $a_{ϵ}$ (since $a_{ϵ} < M ϵ$ ). K is a compact subset of $\tilde{Ω}$ , which contains σ in its interior. We notice that, due to the minimizing properties of $u^{odd}$ and $u_{}^{odd}$ (on ${\tilde{Ω}}^{+}$ ), and the positivity of $\frac{ϵ a_{ϵ}}{2}$ , $\begin{array}{l} \frac{1}{2} \int_{{\tilde{Ω}}^{+}} {| \nabla u_{}^{odd} |}^{2} d x d y + \frac{a_{ϵ}}{2 ϵ} \int_{σ} {| u_{}^{odd +} |}^{2} d s \\ ⩽ \frac{1}{2} \int_{{\tilde{Ω}}^{+}} {| \nabla u^{odd} |}^{2} d x d y + \frac{a_{ϵ}}{2 ϵ} \int_{σ} {| u^{odd +} |}^{2} d s \\ ⩽ \frac{1}{2} \int_{{\tilde{Ω}}^{+}} {| \nabla u^{odd} |}^{2} d x d y + \frac{ϵ a_{ϵ}}{2} \int_{σ} {| \frac{\partial u^{odd +}}{\partial τ} |}^{2} d s + \frac{a_{ϵ}}{2 ϵ} \int_{σ} {| u^{odd +} |}^{2} d s . \end{array}$ It follows from (34) and (35) that $\begin{matrix} (36) & E_{σ, \tilde{Ω}}^{ ϵ} (u_{}^{odd}) ⩽ E_{σ, \tilde{Ω}}^{ϵ} (u^{odd}) . \end{matrix}$ We now define the following (smoother) approximation to $u_{}^{odd}$ (on ${\tilde{Ω}}^{+}$ ) $\begin{array}{rcl} v_{ϵ} (x, y) & = & b_{1} max {ϵ, r}^{- δ} r^{\frac{1}{2} + δ} cos (θ / 2) ϕ (r) \\ + b_{1}^{'} max {ϵ, r^{'}}^{- δ} r^{' \frac{1}{2} + δ} cos (θ^{'} / 2) ϕ (r^{'}) + w^{} (x, y), \end{array}$ where $b_{1}$ , $b_{1}^{'}$ and $w^{}$ are as above, and δ is a fixed positive number. Notice that $v_{ϵ}^{\pm}$ lie in $H^{1} (σ)$ (in contrast to $u_{}^{odd \pm}$ , which do not). Also notice that $v_{ϵ}$ is odd in y and $v_{ϵ} = {\tilde{φ}}^{odd}$ at ${(\partial \tilde{Ω})}^{+}$ . With this definition it can be verified that $\begin{array}{l} \int_{{\tilde{Ω}}^{+}} | \nabla v_{ϵ} |^{2} d x d y ⩽ \int_{{\tilde{Ω}}^{+}} {| \nabla u_{}^{odd} |}^{2} d x d y + C_{M} {‖ {\tilde{φ}}^{odd} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ, \\ \begin{matrix} \int_{σ} {| \frac{\partial v_{ϵ}}{\partial x} |}^{2} d s & ⩽ C | b_{1} |^{2} (\int_{ϵ}^{1} {| \frac{d}{d r} r^{1 / 2} |}^{2} d r + 1) \\ + C {| b_{1}^{'} |}^{2} (\int_{ϵ}^{1} {| \frac{d}{d r^{'}} r^{' 1 / 2} |}^{2} d r^{'} + 1) + C_{M} {‖ {\tilde{φ}}^{odd} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} \\ ⩽ C_{M} {‖ {\tilde{φ}}^{odd} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} | log (ϵ) |, \end{matrix} \end{array}$ and $\begin{matrix} \int_{σ} {| v_{ϵ}^{+} |}^{2} d s ⩽ \int_{σ} {| u_{}^{odd +} |}^{2} d s + C_{M} {‖ {\tilde{φ}}^{odd} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ . \end{matrix}$ Due to (34) and the minimizing property of $u^{odd}$ , we therefore calculate $\begin{array}{l} \frac{1}{2} \int_{{\tilde{Ω}}^{+}} {| \nabla u^{odd} |}^{2} d x d y + \frac{ϵ a_{ϵ}}{2} \int_{σ} {| \frac{\partial u^{odd +}}{\partial x} |}^{2} d s + \frac{a_{ϵ}}{2 ϵ} \int_{σ} {| u^{odd +} |}^{2} d s \\ ⩽ \frac{1}{2} \int_{{\tilde{Ω}}^{+}} | \nabla v_{ϵ} |^{2} d x d y + ϵ^{2} \frac{a_{ϵ}}{2 ϵ} \int_{σ} {| \frac{\partial v_{ϵ}^{+}}{\partial x} |}^{2} d s + \frac{a_{ϵ}}{2 ϵ} \int_{σ} {| v_{ϵ}^{+} |}^{2} d s \\ ⩽ \frac{1}{2} \int_{{\tilde{Ω}}^{+}} {| \nabla u_{}^{odd} |}^{2} d x d y + \frac{a_{ϵ}}{2 ϵ} \int_{σ} {| u_{}^{odd +} |}^{2} d s \\ + C_{M} {‖ {\tilde{φ}}^{odd} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} (ϵ + ϵ^{2} | log (ϵ) | + ϵ) \\ ⩽ \frac{1}{2} \int_{{\tilde{Ω}}^{+}} {| \nabla u_{}^{odd} |}^{2} d x d y + \frac{a_{ϵ}}{2 ϵ} \int_{σ} {| u_{}^{odd +} |}^{2} d s + ϵ C_{M} {‖ {\tilde{φ}}^{odd} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} . \end{array}$ In the second to last estimate we also used that $\frac{a_{ϵ}}{ϵ} < M$ . Due to (34) and (35) it follows from above that $\begin{matrix} (37) & E_{σ, \tilde{Ω}}^{ϵ} (u^{odd}) ⩽ E_{σ, \tilde{Ω}}^{ ϵ} (u_{}^{odd}) + ϵ C_{M} {‖ {\tilde{φ}}^{odd} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} . \end{matrix}$ A combination of (36) and (37) completes the proof of Lemma 5. □

We shall now relate $u_{}^{odd}$ to a problem with even symmetry through duality. For that purpose consider the field $\nabla u_{}^{odd}$ , which is in $L^{2} ({\tilde{Ω}}^{+})$ and has divergence 0 in ${\tilde{Ω}}^{+}$ . Therefore, there exists $U_{ϵ} \in H^{1} ({\tilde{Ω}}^{+})$ , with $\begin{matrix} (38) & \nabla U_{ϵ}^{⊥} = {(- \frac{\partial U_{ϵ}}{\partial y}, \frac{\partial U_{ϵ}}{\partial x})}^{t} = \nabla u_{}^{odd}, \end{matrix}$ satisfying the additional constraint $\int_{{(\partial \tilde{Ω})}^{+}} U_{ϵ} d s = 0$ . It follows that $\begin{matrix} (39) & \int_{σ} \frac{\partial U_{ϵ}}{\partial x} \frac{\partial z}{\partial x} d x = \int_{σ} \frac{\partial u_{}^{odd}}{\partial y} \frac{\partial z}{\partial x}, \end{matrix}$ for any function $z \in H^{1} (σ)$ . Here we note that, according to the earlier representation formula for $u_{}^{odd}$ , $\frac{\partial u_{}^{odd}}{\partial y}$ (and thus $\frac{\partial U_{ϵ}}{\partial x}$ ) is in $C^{γ / 2} (σ)$ , for any $γ < 1$ . Due to the boundary condition of $u_{}^{odd}$ on σ we calculate $\begin{array}{rcl} \int_{σ} \frac{\partial u_{}^{odd}}{\partial y} \frac{\partial z}{\partial x} d x & = & \frac{a_{ϵ}}{ϵ} \int_{σ} u_{}^{odd} \frac{\partial z}{\partial x} d x \\ (40) & = & - \frac{a_{ϵ}}{ϵ} \int_{σ} \frac{\partial u_{}^{odd}}{\partial x} z d x, \end{array}$ for any $z \in H^{1} (σ)$ . To obtain the last identity we used that $u_{}^{odd}$ vanishes at the endpoints of σ, which follows from the fact that it is continuous on ${y = 0}$ , and vanishes outside σ (because of the odd symmetry). Also note that the last integral makes sense, because $\frac{\partial u_{}^{odd}}{\partial x}$ is in $L^{1} (σ)$ , due to the earlier representation formula, and because z is in $C^{0} (σ)$ . Due to the relationship between $\nabla u_{}^{odd}$ and $\nabla U_{ϵ}$ , and to the fact that $u_{}^{odd}$ vanishes on ${y = 0}$ outside σ, $\begin{array}{rcl} \int_{σ} \frac{\partial u_{}^{odd}}{\partial x} z d x & = & \int_{{y = 0} \cap \tilde{Ω}} \frac{\partial u_{}^{odd}}{\partial x} z d x \\ (41) & = & - \int_{{y = 0} \cap \tilde{Ω}} \frac{\partial U_{ϵ}}{\partial y} z d x, \end{array}$ for any $z \in H^{1 / 2} ({y = 0} \cap \tilde{Ω}) \cap H^{1} (σ)$ . A combination of (39), (40), and (41) yields $\begin{matrix} \int_{{y = 0} \cap \tilde{Ω}} \frac{\partial U_{ϵ}}{\partial y} z d x = \frac{ϵ}{a_{ϵ}} \int_{σ} \frac{\partial U_{ϵ}}{\partial x} \frac{\partial z}{\partial x} d x, \end{matrix}$ for all $z \in H^{1 / 2} ({y = 0} \cap \tilde{Ω}) \cap H^{1} (σ)$ . It follows that $U_{ϵ}$ is the solution to $\begin{array}{l} - Δ U_{ϵ} = 0 in {\tilde{Ω}}^{+}, \\ \frac{\partial U_{ϵ}}{\partial y} + \frac{ϵ}{a_{ϵ}} \frac{\partial^{2} U_{ϵ}}{\partial x^{2}} = 0 on σ, \\ \frac{\partial U_{ϵ}}{\partial y} = 0 on ({y = 0} \cap \tilde{Ω}) ∖ σ, \frac{\partial U_{ϵ}}{\partial n} = \frac{\partial {\tilde{φ}}^{odd}}{\partial τ} on {(\partial \tilde{Ω})}^{+}, \end{array}$ with $\int_{{(\partial \tilde{Ω})}^{+}} U_{ϵ} d s = 0$ . Here n denotes the outer normal to ${(\partial \tilde{Ω})}^{+}$ and τ is the counterclockwise tangent. Let ${\tilde{ψ}}^{ev}$ denote $\frac{\partial {\tilde{φ}}^{odd}}{\partial τ}$ , which is well defined (and an even distribution) on all of $\partial \tilde{Ω}$ , and let ${\tilde{U}}_{ϵ} \in H^{1} (\tilde{Ω}) \cap H^{1} (σ)$ denote the (variational) solution to $\begin{array}{l} - Δ {\tilde{U}}_{ϵ} = 0 in \tilde{Ω} ∖ σ, \frac{\partial {\tilde{U}}_{ϵ}}{\partial n} = {\tilde{ψ}}^{ev} on \partial \tilde{Ω}, \\ (42) & \frac{\partial {\tilde{U}}_{ϵ}^{+}}{\partial y} + \frac{ϵ}{a_{ϵ}} \frac{\partial^{2} {\tilde{U}}_{ϵ}}{\partial x^{2}} = 0 on σ, \\ \frac{\partial {\tilde{U}}_{ϵ}^{-}}{\partial y} - \frac{ϵ}{a_{ϵ}} \frac{\partial^{2} {\tilde{U}}_{ϵ}}{\partial x^{2}} = 0 on σ, \end{array}$ with the additional constraint $\int_{\partial \tilde{Ω}} {\tilde{U}}_{ϵ} d s = 0$ . ${\tilde{U}}_{ϵ}$ is the even extension of $U_{ϵ}$ to all of $\tilde{Ω}$ , and is a minimizer of the energy $\begin{matrix} E_{σ, \tilde{Ω}}^{N, ϵ} (v) = \frac{1}{2} \int_{\tilde{Ω} ∖ σ} | \nabla v |^{2} d x d y + \frac{ϵ}{a_{ϵ}} \int_{σ} {(\frac{\partial v}{\partial x})}^{2} d x - \int_{\partial \tilde{Ω}} {\tilde{ψ}}^{ev} v d s \end{matrix}$ in the set $H_{ev}^{1} (\tilde{Ω}) \cap H^{1} (σ)$ . Furthermore $\begin{matrix} (43) & E_{σ, \tilde{Ω}}^{N, ϵ} ({\tilde{U}}_{ϵ}) = 2 (\frac{1}{2} \int_{{\tilde{Ω}}^{+}} | \nabla U_{ϵ} |^{2} d x d y + \frac{ϵ}{2 a_{ϵ}} \int_{σ} {(\frac{\partial U_{ϵ}}{\partial x})}^{2} d x - \int_{{(\partial \tilde{Ω})}^{+}} {\tilde{ψ}}^{ev} U_{ϵ} d s) . \end{matrix}$ We calculate $\begin{array}{rcl} \int_{{(\partial \tilde{Ω})}^{+}} ψ^{ev} U_{ϵ} d s & = & \int_{{(\partial \tilde{Ω})}^{+}} \frac{\partial {\tilde{φ}}^{odd}}{\partial τ} U_{ϵ} d s = - \int_{{(\partial \tilde{Ω})}^{+}} {\tilde{φ}}^{odd} \frac{\partial U_{ϵ}}{\partial τ} d s \\ = & \int_{{(\partial \tilde{Ω})}^{+}} {\tilde{φ}}^{odd} \frac{\partial u_{}^{odd}}{\partial n} d s = \int_{{\tilde{Ω}}^{+}} {| \nabla u_{}^{odd} |}^{2} d x d y + \frac{a_{ϵ}}{ϵ} \int_{σ} {| u_{}^{odd} |}^{2} d x . \end{array}$ Since $\frac{\partial U_{ϵ}}{\partial x} = \frac{\partial u_{}^{odd}}{\partial y} = \frac{a_{ϵ}}{ϵ} u_{}^{odd}$ on σ, we also have $\begin{matrix} \frac{ϵ}{2 a_{ϵ}} \int_{σ} {(\frac{\partial U_{ϵ}}{\partial x})}^{2} d x = \frac{a_{ϵ}}{2 ϵ} \int_{σ} {| u_{}^{odd} |}^{2} d x, \end{matrix}$ and due to (38) $\begin{matrix} \frac{1}{2} \int_{{\tilde{Ω}}^{+}} | \nabla U_{ϵ} |^{2} d x d y = \frac{1}{2} \int_{{\tilde{Ω}}^{+}} {| \nabla u_{}^{odd} |}^{2} d x d y . \end{matrix}$ Upon insertion of these last three identities into (43), we get $\begin{matrix} (44) & E_{σ, \tilde{Ω}}^{N, ϵ} ({\tilde{U}}_{ϵ}) = - 2 (\frac{1}{2} \int_{{\tilde{Ω}}^{+}} {| \nabla u_{}^{odd} |}^{2} d x d y + \frac{a_{ϵ}}{2 ϵ} \int_{σ} {| u_{}^{odd} |}^{2} d x) = - E_{σ, \tilde{Ω}}^{* ϵ} (u_{}^{odd}) . \end{matrix}$ Let $V^{ev}$ denote the solution to the problem $\begin{matrix} (45) & - \nabla \cdot ({\tilde{γ}}_{ϵ} \nabla V^{ev}) = 0 in \tilde{Ω}, \frac{\partial V^{ev}}{\partial n} = {\tilde{ψ}}^{ev} on \partial \tilde{Ω}, \end{matrix}$ with the constraint $\int_{\partial \tilde{Ω}} V^{ev} d s = 0$ . Here the coefficient ${\tilde{γ}}_{ϵ}$ is given by $\begin{matrix} {\tilde{γ}}_{ϵ} (x, y) = \{\begin{matrix} 1 & if (x, y) is
in \tilde{Ω} ∖ ω_{ϵ}, \\ \frac{1}{a_{ϵ}} & if (x, y) is
in ω_{ϵ} . \end{matrix} \end{matrix}$ $V^{ev}$ is a minimimizer of the energy $\begin{matrix} E_{\tilde{Ω}}^{N, ϵ} (v) = \frac{1}{2} \int_{\tilde{Ω}} {\tilde{γ}}_{ϵ} | \nabla v |^{2} d x d y - \int_{\partial \tilde{Ω}} {\tilde{ψ}}^{ev} v d s \end{matrix}$ in the space $H^{1} (\tilde{Ω})$ . $V^{ev}$ is related to $u_{ϵ}^{odd}$ , the minimizer of $E_{\tilde{Ω}}^{ϵ}$ in $H^{1} (\tilde{Ω}) \cap {v = {\tilde{φ}}^{odd} on \partial \tilde{Ω}}$ , by $\begin{matrix} γ_{ϵ} \nabla u_{ϵ}^{odd} = {(- \frac{\partial V^{ev}}{\partial y}, \frac{\partial V^{ev}}{\partial x})}^{t}, \end{matrix}$ furthermore $\begin{array}{rcl} E_{\tilde{Ω}}^{N, ϵ} (V^{ev}) & = & \frac{1}{2} \int_{\tilde{Ω}} {\tilde{γ}}_{ϵ} {| \nabla V^{ev} |}^{2} d x d y - \int_{\partial \tilde{Ω}} \frac{\partial V^{ev}}{\partial n} V^{ev} d s = - \frac{1}{2} \int_{\tilde{Ω}} {\tilde{γ}}_{ϵ} {| \nabla V^{ev} |}^{2} d x d y \\ (46) & = & - \frac{1}{2} \int_{\tilde{Ω}} γ_{ϵ} {| \nabla u_{ϵ}^{odd} |}^{2} d x d y = - E_{\tilde{Ω}}^{ϵ} (u_{ϵ}^{odd}) . \end{array}$ The analysis in Section 2 was performed with Dirichlet boundary conditions on $\partial \tilde{Ω}$ , but it could as well have been performed with Neumann boundary conditions, and with $1 / a_{ϵ}$ in place of $a_{ϵ}$ . It would then apply to the case $0 < m < ϵ / a_{ϵ}$ , which (with $m = 1 / M$ ) is exactly the case $a_{ϵ} < M ϵ$ we are dealing with here. The equivalent of (3) would now estimate the energy difference between ${\tilde{U}}_{ϵ}$ , the solution to (42), and $V^{ev}$ , the solution to (45) – it asserts that $\begin{matrix} | E_{σ, \tilde{Ω}}^{N, ϵ} ({\tilde{U}}_{ϵ}) - E_{\tilde{Ω}}^{N, ϵ} (V^{ev}) | ⩽ C_{γ, M} {‖ {\tilde{ψ}}^{ev} ‖}_{H^{- 1 / 2} (\partial \tilde{Ω})}^{2} ϵ^{γ / 4} . \end{matrix}$ If we substitute (44) and (46) into this, we arrive at the estimate $\begin{matrix} (47) & | E_{σ, \tilde{Ω}}^{ ϵ} (u_{}^{odd}) - E_{\tilde{Ω}}^{ϵ} (u_{ϵ}^{odd}) | ⩽ C_{γ, M} {‖ {\tilde{ψ}}^{ev} ‖}_{H^{- 1 / 2} (\partial \tilde{Ω})}^{2} ϵ^{γ / 4} ⩽ C_{γ, M} {‖ {\tilde{φ}}^{odd} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ^{γ / 4} . \end{matrix}$ A combination of this estimate with the estimate in Lemma 5 establishes Lemma 6.
Under the assumption that* $a_{ϵ} < M ϵ$ , for some fixed constant M, the following estimate holds $\begin{matrix} | E_{σ, \tilde{Ω}}^{ϵ} (u^{odd}) - E_{\tilde{Ω}}^{ϵ} (u_{ϵ}^{odd}) | ⩽ C_{γ, M} {‖ {\tilde{φ}}^{odd} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ^{γ / 4}, \end{matrix}$ uniformly in $0 < a_{ϵ}$ and $0 < ϵ < ϵ_{0}$ , for any $γ < 1$ .

5. Local convergence – Odd symmetry, $a_{ϵ} > ϵ$

In this section, we shall prove that in the regime $a_{ϵ} > ϵ$ , $\begin{matrix} (48) & | E_{σ, \tilde{Ω}}^{ϵ} (u^{odd}) - E_{\tilde{Ω}}^{ϵ} (u_{ϵ}^{odd}) | ⩽ C_{γ} {‖ {\tilde{φ}}^{odd} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} (\frac{ϵ}{a_{ϵ}} + ϵ^{γ}) . \end{matrix}$ We prove this estimate in two steps: first we prove $\begin{matrix} (49) & | E_{σ, \tilde{Ω}}^{ϵ} (u^{odd}) - E_{\tilde{Ω}}^{0} (u_{0}^{odd}) | ⩽ C {‖ {\tilde{φ}}^{odd} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} \frac{ϵ}{a_{ϵ}}, \end{matrix}$ and then we prove $\begin{matrix} (50) & | E_{\tilde{Ω}}^{0} (u_{0}^{odd}) - E_{\tilde{Ω}}^{ϵ} (u_{ϵ}^{odd}) | ⩽ C_{γ} {‖ {\tilde{φ}}^{odd} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} (\frac{ϵ}{a_{ϵ}} + ϵ^{γ}) . \end{matrix}$ Here $u_{0}^{odd}$ denotes the solution to $\begin{matrix} - Δ u_{0}^{odd} = 0 in \tilde{Ω}, u_{0}^{odd} = {\tilde{φ}}^{odd} on \partial \tilde{Ω}, \end{matrix}$ which is obtained through the minimization $\begin{matrix} E_{\tilde{Ω}}^{0} (u_{0}^{odd}) = min_{\begin{array}{c} u \in H^{1} (\tilde{Ω}) \\ u = {\tilde{φ}}^{odd} on \partial \tilde{Ω} \end{array}} E_{\tilde{Ω}}^{0} (u) = min_{\begin{array}{c} u \in H_{odd}^{1} (\tilde{Ω}) \\ u = {\tilde{φ}}^{odd} on \partial \tilde{Ω} \end{array}} E_{\tilde{Ω}}^{0} (u), \end{matrix}$ with $\begin{matrix} E_{\tilde{Ω}}^{0} (u) = \frac{1}{2} \int_{\tilde{Ω}} | \nabla u |^{2} d x d y . \end{matrix}$ The estimate (48) follows immediately by combination of (49) and (50). For simplicity of notation we shall in the following delete the explicit appearance of $\tilde{Ω}$ in the energy expression symbols, in other words we shall use $E_{σ}^{ϵ}$ , $E^{ϵ}$ , and $E^{0}$ in place of $E_{σ, \tilde{Ω}}^{ϵ}$ , $E_{\tilde{Ω}}^{ϵ}$ and $E_{\tilde{Ω}}^{0}$ .

5.1. Proof of (49)

We shall use $u^{odd}$ also to denote the restriction of $u^{odd}$ to ${\tilde{Ω}}_{+}$ . This restriction solves the boundary value problem (33), and is the minimizer of the energy $\begin{matrix} E_{σ}^{ϵ} (v) = \frac{1}{2} \int_{{\tilde{Ω}}^{+}} | \nabla v |^{2} d x d y + \frac{ϵ a_{ϵ}}{2} \int_{σ} {(\frac{d v}{d x})}^{2} d x + \frac{a_{ϵ}}{2 ϵ} \int_{σ} v^{2} d x \end{matrix}$ in the set $\begin{matrix} V_{{\tilde{φ}}^{odd}}^{+} = {v \in H^{1} ({\tilde{Ω}}^{+}) : v |_{σ} \in H^{1} (σ), v = {\tilde{φ}}^{odd} on {(\partial \tilde{Ω})}^{+}, v = 0 on ({y = 0} \cap \tilde{Ω}) ∖ σ} . \end{matrix}$ Furthermore $E_{σ}^{ϵ} (u^{odd}) = 2 E_{σ}^{ϵ} (u^{odd})$ . We now calculate $\begin{array}{rcl} \frac{1}{2} E_{σ}^{ϵ} (u^{odd}) & = & \frac{1}{2} \int_{{\tilde{Ω}}^{+}} {| \nabla u^{odd} |}^{2} d x d y + \frac{ϵ a_{ϵ}}{2} \int_{σ} {| \frac{\partial u^{odd}}{\partial x} |}^{2} d x + \frac{a_{ϵ}}{2 ϵ} \int_{σ} | u_{odd} |^{2} d x \\ ⩽ & \frac{1}{2} \int_{{\tilde{Ω}}^{+}} {| \nabla u_{0}^{odd} |}^{2} d x d y + \frac{ϵ a_{ϵ}}{2} \int_{σ} {| \frac{\partial u_{0}^{odd}}{\partial x} |}^{2} d x + \frac{a_{ϵ}}{2 ϵ} \int_{σ} {| u_{0}^{odd} |}^{2} d x \\ = & \frac{1}{2} \int_{{\tilde{Ω}}^{+}} {| \nabla u_{0}^{odd} |}^{2} d x d y = \frac{1}{2} E^{0} (u_{0}^{odd}), \end{array}$ in other words $\begin{matrix} (51) & E_{σ}^{ϵ} (u^{odd}) - E^{0} (u_{0}^{odd}) ⩽ 0 . \end{matrix}$ In order to prove a corresponding lower bound $\begin{matrix} (52) & E^{0} (u_{0}^{odd}) - E_{σ}^{ϵ} (u^{odd}) ⩽ C {‖ {\tilde{φ}}^{odd} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} \frac{ϵ}{a_{ϵ}}, \end{matrix}$ we apply a dual variational principle for $u^{odd}$ (on ${\tilde{Ω}}^{+}$ ). It is fairly simple to verify that $\begin{array}{rcl} E_{σ}^{ϵ} (u^{odd}) & = & max_{η, Σ, S \in W} [\int_{{(\partial \tilde{Ω})}^{+}} {\tilde{φ}}^{odd} η \cdot n d s - \frac{1}{2} \int_{{\tilde{Ω}}^{+}} | η |^{2} d x d y \\ (53) & - \frac{1}{2 ϵ a_{ϵ}} \int_{σ} | Σ |^{2} d x - \frac{ϵ}{2 a_{ϵ}} \int_{σ} | S |^{2} d x], \end{array}$ where the set W is defined by $\begin{matrix} W = {(η, Σ, S) \in L^{2} {({\tilde{Ω}}^{+})}^{2} \times L^{2} (σ) \times L^{2} (σ) : \nabla \cdot η = 0 in {\tilde{Ω}}^{+}, - \frac{d Σ}{d x} + S - η_{2} = 0 on σ} . \end{matrix}$ The equations in the definition of W are to be interpreted in the sense of distributions, by which we mean $\begin{matrix} \int_{{\tilde{Ω}}^{+}} η \cdot \nabla ψ d x d y + \int_{σ} Σ \frac{\partial ψ}{\partial x} d x + \int_{σ} S ψ d x = 0, \end{matrix}$ for any smooth function ψ on ${\tilde{Ω}}^{+}$ , which vanishes on ${(\partial \tilde{Ω})}^{+}$ and on $({y = 0} \cap \tilde{Ω}) ∖ σ$ . Based on (53), and taking $(η, Σ, S) = (\nabla u_{0}^{odd}, 0, \frac{\partial u_{0}^{odd}}{\partial y}) \in W$ , we calculate $\begin{array}{rcl} \frac{1}{2} E_{σ}^{ϵ} (u^{odd}) & = & E_{σ}^{ϵ} (u^{odd}) \\ ⩾ & \int_{{(\partial \tilde{Ω})}^{+}} {\tilde{φ}}^{odd} \frac{\partial u_{0}^{odd}}{\partial n} d s - \frac{1}{2} \int_{{\tilde{Ω}}^{+}} {| \nabla u_{0}^{odd} |}^{2} d x d y - \frac{ϵ}{2 a_{ϵ}} \int_{σ} {| \frac{\partial u_{0}^{odd}}{\partial y} |}^{2} d x \\ ⩾ & \int_{{(\partial \tilde{Ω})}^{+}} {\tilde{φ}}^{odd} \frac{\partial u_{0}^{odd}}{\partial n} d s - \frac{1}{2} \int_{{\tilde{Ω}}^{+}} {| \nabla u_{0}^{odd} |}^{2} d x d y - \frac{ϵ}{a_{ϵ}} C {‖ {\tilde{φ}}^{odd} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} \\ = & \frac{1}{2} E^{0} (u_{0}^{odd}) - \frac{ϵ}{a_{ϵ}} C {‖ {\tilde{φ}}^{odd} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2}, \end{array}$ thereby establishing (52). A combination of (51) and (52) immediately leads to (49).

5.2. Proof of (50)

Given $u_{0}^{odd}$ , the minimizer of $E^{0} (\cdot)$ , we first construct an approximate minimimizer for the energy $E^{ϵ} (\cdot)$ . We define $\begin{matrix} v^{test} (x, y) = \{\begin{matrix} u_{0}^{odd} (x, y) + v_{ϵ} (x, y) & if (x, y) is in \tilde{Ω} ∖ ω_{ϵ}, \\ 0 & if (x, y) is
in ω_{ϵ}, \end{matrix} \end{matrix}$ where $v_{ϵ}$ lies in $H^{1} (\tilde{Ω} ∖ ω_{ϵ})$ with $\begin{matrix} v_{ϵ} = - u_{0}^{odd} on \partial ω_{ϵ}, v_{ϵ} = 0 on \partial \tilde{Ω}, \end{matrix}$ and $\begin{matrix} (54) & ‖ v_{ϵ} ‖_{H^{1} (\tilde{Ω} ∖ ω_{ϵ})} ⩽ C {‖ {\tilde{φ}}^{odd} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{γ}, \end{matrix}$ for any $γ < 1$ . In order to construct this $v_{ϵ}$ , we note that $\begin{array}{l} | u_{0}^{odd} | ⩽ C {‖ {\tilde{φ}}^{odd} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ on \partial ω_{ϵ}, \\ | \frac{\partial}{\partial τ} u_{0}^{odd} | ⩽ C {‖ {\tilde{φ}}^{odd} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} on \partial ω_{ϵ} \cap {| x | > 1}, \end{array}$ the “ends” of $\partial ω_{ϵ}$ , and $\begin{matrix} | \frac{\partial}{\partial τ} u_{0}^{odd} | ⩽ C {‖ {\tilde{φ}}^{odd} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ on \partial ω_{ϵ} \cap {| x | < 1}, \end{matrix}$ the horizontal parts of $\partial ω_{ϵ}$ . We then construct $v_{ϵ}$ by extending $- u_{0}^{odd}$ from $\partial ω_{ϵ}$ to $\tilde{Ω} ∖ ω_{ϵ}$ constantly along normal rays, and we apply a cut-off to $v_{ϵ}$ near $\partial \tilde{Ω}$ . With this construction it is fairly easy to check (based on the observations about $u_{0}^{odd} |_{\partial ω_{ϵ}}$ above) that (54) holds. Using $v^{test}$ we calculate $\begin{array}{rcl} E^{ϵ} (u_{ϵ}^{odd}) & ⩽ & E^{ϵ} (v^{test}) = \frac{1}{2} \int_{\tilde{Ω} ∖ ω_{ϵ}} | \nabla u_{0}^{odd} + \nabla v_{ϵ}) |^{2} d x d y \\ ⩽ & \frac{1}{2} \int_{\tilde{Ω}} {| \nabla u_{0}^{odd} |}^{2} d x d y + C_{γ} {‖ {\tilde{φ}}^{odd} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ^{γ} . \end{array}$ This verifies that $\begin{matrix} (55) & E^{ϵ} (u_{ϵ}^{odd}) - E^{0} (u_{0}^{odd}) ⩽ C_{γ} {‖ {\tilde{φ}}^{odd} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ^{γ} . \end{matrix}$ We proceed to verify the corresponding lower bound $\begin{matrix} (56) & E^{0} (u_{0}^{odd}) - E^{ϵ} (u_{ϵ}^{odd}) ⩽ \frac{ϵ}{a_{ϵ}} C {‖ {\tilde{φ}}^{odd} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} . \end{matrix}$ For that purpose, we use the dual variational formulation for $u_{ϵ}^{odd}$ $\begin{matrix} E^{ϵ} (u_{ϵ}^{odd}) = max_{\nabla \cdot ξ = 0} [\int_{\partial \tilde{Ω}} {\tilde{φ}}^{odd} ξ \cdot n d s - \frac{1}{2} \int_{\tilde{Ω}} γ_{ϵ}^{- 1} | ξ |^{2} d x d y], \end{matrix}$ and select the test field $ξ^{test} = \nabla u_{0}^{odd}$ (which satisfies $\nabla \cdot ξ^{test} = 0$ in $\tilde{Ω}$ ) to get $\begin{array}{rcl} E^{ϵ} (u_{ϵ}^{odd}) & ⩾ & \int_{\partial \tilde{Ω}} {\tilde{φ}}^{odd} \frac{\partial u_{0}^{odd}}{\partial n} d s - \frac{1}{2} \int_{\tilde{Ω} ∖ ω_{ϵ}} {| \nabla u_{0}^{odd} |}^{2} d x d y - \frac{1}{2 a_{ϵ}} \int_{ω_{ϵ}} {| \nabla u_{0}^{odd} |}^{2} d x d y \\ ⩾ & \int_{\partial \tilde{Ω}} {\tilde{φ}}^{odd} \frac{\partial u_{0}^{odd}}{\partial n} d s - \frac{1}{2} \int_{\tilde{Ω} ∖ ω_{ϵ}} {| \nabla u_{0}^{odd} |}^{2} d x d y - \frac{ϵ}{a_{ϵ}} C {‖ {\tilde{φ}}^{odd} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} \\ ⩾ & \int_{\partial \tilde{Ω}} {\tilde{φ}}^{odd} \frac{\partial u_{0}^{odd}}{\partial n} d s - \frac{1}{2} \int_{\tilde{Ω}} {| \nabla u_{0}^{odd} |}^{2} d x d y - \frac{ϵ}{a_{ϵ}} C {‖ {\tilde{φ}}^{odd} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} \\ = & E^{0} (u_{0}^{odd}) - \frac{ϵ}{a_{ϵ}} C {‖ {\tilde{φ}}^{odd} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} . \end{array}$ A rearrangement of this estimate immediately gives the lower bound (56), and a combination of (55) and (56) now leads to the desired estimate (50).

6. Proof of Lemma 1 and Lemma 2

In this section we combine the estimates from the previous four sections to obtain proofs of Lemma 1 and Lemma 2. We start with

Proof of Lemma 1.

Suppose the statement in Lemma 1 was not true. Then there would exist sequences $ϵ_{n} \to 0$ and $a_{ϵ_{n}} > 0$ , and a constant $c > 0$ , such that $\begin{matrix} (57) & c < sup_{{\tilde{φ}}^{ev} \neq 0} \frac{| E_{σ, \tilde{Ω}}^{ϵ_{n}} (u^{ev}) - E_{\tilde{Ω}}^{ϵ_{n}} (u_{ϵ_{n}}^{ev}) |}{‖ {\tilde{φ}}^{ev} ‖_{H^{1 / 2} (\partial \tilde{Ω})}^{2}} . \end{matrix}$ There are two possibilities concerning the sequences $ϵ_{n}$ and $a_{ϵ_{n}}$ :

There exists a constant $m > 0$ so that $m < ϵ_{n} a_{ϵ_{n}}$ .

There exists a subsequence of the $ϵ_{n}$ (for simplicity also called $ϵ_{n}$ ) so that $ϵ_{n} a_{ϵ_{n}} \to 0$ .

In case (1), we have from Section 2 that $\begin{matrix} | E_{σ, \tilde{Ω}}^{ϵ_{n}} (u^{ev}) - E_{\tilde{Ω}}^{ϵ_{n}} (u_{ϵ_{n}}^{ev}) | ⩽ C_{γ, m} {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ_{n}^{γ / 4} . \end{matrix}$ This clearly represents a contradiction to (57).

In case (2), we have from Section 3 that $\begin{matrix} | E_{σ, \tilde{Ω}}^{ϵ_{n}} (u^{ev}) - E_{\tilde{Ω}}^{ϵ_{n}} (u_{ϵ_{n}}^{ev}) | ⩽ C_{γ} {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} (ϵ_{n} a_{ϵ_{n}} + ϵ_{n}^{γ}) . \end{matrix}$ However, this is clearly also a contradiction to (57), as $ϵ_{n} a_{ϵ_{n}} + ϵ_{n}^{γ} \to 0$ .

In summary we have established a contradiction to (57) in both cases (1) and (2), and so we conclude that the statement of Lemma 1 is indeed true. □

Proof of Lemma 2.

Suppose the statement in Lemma 2 was not true. Then there would exist sequences $ϵ_{n} \to 0$ and $a_{ϵ_{n}} > 0$ , and a constant $c > 0$ such $\begin{matrix} (58) & c < sup_{{\tilde{φ}}^{odd} \neq 0} \frac{| E_{σ, \tilde{Ω}}^{ϵ_{n}} (u^{odd}) - E_{\tilde{Ω}}^{ϵ_{n}} (u_{ϵ_{n}}^{odd}) |}{‖ {\tilde{φ}}^{odd} ‖_{H^{1 / 2} (\partial \tilde{Ω})}^{2}} . \end{matrix}$ There are two possibilities concerning the sequences $ϵ_{n}$ and $a_{ϵ_{n}}$ :

There exists a constant M so that $a_{ϵ_{n}} < M ϵ_{n}$ .

There exists a subsequence of the $ϵ_{n}$ (for simplicity also called $ϵ_{n}$ ) so that $\frac{a_{ϵ_{n}}}{ϵ_{n}} \to \infty$ .

In case (1′), we have from Section 4, Lemma 6, that $\begin{matrix} | E_{σ, \tilde{Ω}}^{ϵ_{n}} (u^{odd}) - E_{\tilde{Ω}}^{ϵ_{n}} (u_{ϵ_{n}}^{odd}) | ⩽ C_{γ, M} {‖ {\tilde{φ}}^{odd} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ_{n}^{γ / 4} . \end{matrix}$ This clearly represents a contradiction to (58).

In case (2′), we have from Section 5 that $\begin{matrix} | E_{σ, \tilde{Ω}}^{ϵ_{n}} (u^{odd}) - E_{\tilde{Ω}}^{ϵ_{n}} (u_{ϵ_{n}}^{odd}) | ⩽ C_{γ} {‖ {\tilde{φ}}^{odd} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} (\frac{ϵ_{n}}{a_{ϵ_{n}}} + ϵ_{n}^{γ}) . \end{matrix}$ However, this is clearly also a contradiction to (58) as $\frac{ϵ_{n}}{a_{ϵ_{n}}} + ϵ_{n}^{γ} \to 0$ .

In summary we have established a contradiction to (58) in both cases (1′) and (2′), and so we conclude that the statement of Lemma 2 is indeed true. □

Footnotes

Acknowledgements

This work was partially supported by NSF grant DMS-12-11330. Additionally MV was supported by the NSF IR/D program while serving at the National Science Foundation. Any opinion, findings, and conclusions or recommendations expressed in this paper are those of the authors, and do not necessarily reflect the views of the National Science Foundation.

Proof of Lemma 3

From the representation formula for $u^{ev}$ in Proposition 1 of Appendix A in [1], and the remark immediately following that proposition, we get that $\begin{matrix} (59) & u^{ev} = b_{1} r^{1 / 2} sin (θ / 2) ϕ (r) + b_{1}^{'} r^{' 1 / 2} sin (θ^{'} / 2) ϕ (r^{'}) + u^{*}, \end{matrix}$ where $(r, θ)$ and $(r^{'}, θ^{'})$ are polar coordinates around the two endpoints of σ, and $u^{*}$ is bounded in $C_{ev}^{1, γ / 2} (K ∖ σ)$ , up to the curve σ, by $C ‖ {\tilde{φ}}^{ev} ‖_{H^{1 / 2} (\partial \tilde{Ω})}$ ; similarly $b_{1}$ and $b_{1}^{'}$ are bounded by $C ‖ {\tilde{φ}}^{ev} ‖_{H^{1 / 2} (\partial \tilde{Ω})}$ . Here we make use of the fact that Lemma 3 only concerns the case $ϵ a_{ϵ} > m > 0$ . We decompose $z_{ϵ}$ , the solution to (9), as follows $\begin{matrix} z_{ϵ} = z_{ϵ}^{(1)} + z_{ϵ}^{(2)}, \end{matrix}$ where $z_{ϵ}^{(1)}$ is the solution to $\begin{matrix} (60) & \{\begin{matrix} - Δ z_{ϵ}^{(1)} = 0 & in \tilde{Ω} ∖ \overline{ω_{ϵ}}, \\ z_{ϵ}^{(1)} = 0 & on \partial \tilde{Ω}, \\ z_{ϵ}^{(1)} = - b_{1} r^{1 / 2} sin (θ / 2) ϕ (r) - b_{1}^{'} r^{' 1 / 2} sin (θ^{'} / 2) ϕ (r^{'}) & on \partial ω_{ϵ}, \end{matrix} \end{matrix}$ and $z_{ϵ}^{(2)}$ is the solution to $\begin{matrix} (61) & \{\begin{matrix} - Δ z_{ϵ}^{(2)} = 0 & in \tilde{Ω} ∖ \overline{ω_{ϵ}}, \\ z_{ϵ}^{(2)} = 0 & on \partial \tilde{Ω}, \\ z_{ϵ}^{(2)} = u^{*} \circ p_{σ} - u^{*} & on \partial ω_{ϵ} . \end{matrix} . \end{matrix}$ Notice that $(b_{1} r^{1 / 2} sin (θ / 2) ϕ (r) + b_{1}^{'} r^{' 1 / 2} sin (θ^{'} / 2) ϕ (r^{'})) \circ p_{σ} = 0$ .

First, we estimate $z_{ϵ}^{(2)}$ . For that purpose, we divide $\partial ω_{ϵ}$ into two parts $\begin{matrix} \partial ω_{ϵ}^{int} = {(x, \pm ϵ) : - 1 < x < 1}, \end{matrix}$ and $\begin{matrix} \partial ω_{ϵ}^{ends} = \partial ω_{ϵ} ∖ \partial ω_{ϵ}^{int} . \end{matrix}$ On $\partial ω_{ϵ}^{int}$ it follows immediately from the fact that $u^{*}$ is $C^{1, γ / 2}$ (up to σ, but not across σ) that $\begin{array}{l} {‖ u^{*} \circ p_{σ} - u^{*} ‖}_{L^{\infty} (\partial ω_{ϵ}^{int})} + {‖ \frac{\partial}{\partial x} (u^{*} \circ p_{σ} - u^{*}) ‖}_{L^{\infty} (\partial ω_{ϵ}^{int})} \\ (62) & ⩽ C ϵ^{γ / 2} {‖ u^{*} ‖}_{C^{1, γ / 2} (K ∖ σ)} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{γ / 2} . \end{array}$ On $\partial ω_{ϵ}^{ends}$ the function $u^{*} \circ p_{σ}$ is identically equal to $u^{*} (- 1, 0)$ (on the left component of $\partial ω_{ϵ}^{ends}$ ) or $u^{*} (1, 0)$ (on the right component of $\partial ω_{ϵ}^{ends}$ ) and $\nabla (u^{*} \circ p_{σ})$ is identically equal to 0. From Lemma 2 in Appendix A of [1] we have that $\frac{\partial}{\partial x} u^{*} (\pm 1, 0) = 0$ , and from the even symmetry of $u^{*}$ we also have that $\frac{\partial}{\partial y} u^{*} (\pm 1, 0) = 0$ ; in summary $\begin{matrix} (63) & \nabla u^{*} (\pm 1, 0) = 0 . \end{matrix}$ Since $u^{*}$ is $C^{1, γ / 2}$ (up to σ) it now immediately follows that $\begin{matrix} {‖ u^{*} \circ p_{σ} - u^{*} ‖}_{L^{\infty} (\partial ω_{ϵ}^{ends})} ⩽ C ϵ^{1 + γ / 2} {‖ u^{*} ‖}_{C^{1, γ / 2} (K ∖ σ)}, \end{matrix}$ and $\begin{matrix} {‖ \nabla (u^{*} \circ p_{σ} - u^{*}) ‖}_{L^{\infty} (\partial ω_{ϵ}^{ends})} = {‖ \nabla u^{*} ‖}_{L^{\infty} (\partial ω_{ϵ}^{ends})} ⩽ C ϵ^{γ / 2} {‖ u^{*} ‖}_{C^{1, γ / 2} (K ∖ σ)} . \end{matrix}$ Together these yield $\begin{array}{l} {‖ u^{*} \circ p_{σ} - u^{*} ‖}_{L^{\infty} (\partial ω_{ϵ}^{ends})} + {‖ \nabla (u^{*} \circ p_{σ} - u^{*}) ‖}_{L^{\infty} (\partial ω_{ϵ}^{ends})} \\ (64) & ⩽ C ϵ^{γ / 2} {‖ u^{*} ‖}_{C^{1, γ / 2} (K ∖ σ)} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{γ / 2} . \end{array}$

By a combination of (62) and (64) it follows that $\begin{matrix} {‖ z_{ϵ}^{(2)} ‖}_{W^{1, \infty} (\partial ω_{ϵ})} = {‖ u^{*} \circ p_{σ} - u^{*} ‖}_{W^{1, \infty} (\partial ω_{ϵ})} ⩽ {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{γ / 2} . \end{matrix}$ The function $z_{ϵ}^{(2)}$ is indeed $C^{1}$ on $\partial ω_{ϵ}$ , but we don’t use that fact here. By a simple construction, we may extend the trace $z_{ϵ}^{(2)} |_{\partial ω_{ϵ}}$ to a function $Z_{ϵ}^{(2)}$ defined on the whole domain $\tilde{Ω} ∖ \overline{ω_{ϵ}}$ with $Z_{ϵ}^{(2)} = 0$ on $\partial \tilde{Ω}$ and $\begin{matrix} {‖ Z_{ϵ}^{(2)} ‖}_{W^{1, \infty} (\tilde{Ω} ∖ \overline{ω_{ϵ}})} ⩽ C {‖ z_{ϵ}^{(2)} ‖}_{W^{1, \infty} (\partial ω_{ϵ})} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{γ / 2} . \end{matrix}$ One such extension can be obtained by extending constantly on normal rays emanating from $ω_{ϵ}$ , and then multiplying by a smooth cut-off function to get value zero at $\partial \tilde{Ω}$ . The standard variational formulation of (61) asserts that $\begin{matrix} \int_{\tilde{Ω} ∖ \overline{ω_{ϵ}}} \nabla (z_{ϵ}^{(2)} - Z_{ϵ}^{(2)}) \nabla z_{ϵ}^{(2)} d x = 0; \end{matrix}$ in other words, $\begin{matrix} \int_{\tilde{Ω} ∖ \overline{ω_{ϵ}}} {| \nabla z_{ϵ}^{(2)} |}^{2} d x = \int_{\tilde{Ω} ∖ \overline{ω_{ϵ}}} \nabla Z_{ϵ}^{(2)} \nabla z_{ϵ}^{(2)} d x, \end{matrix}$ and so $\begin{matrix} (65) & {‖ \nabla z_{ϵ}^{(2)} ‖}_{L^{2} (\tilde{Ω} ∖ \overline{ω_{ϵ})}} ⩽ {‖ \nabla Z_{ϵ}^{(2)} ‖}_{L^{2} (\tilde{Ω} ∖ \overline{ω_{ϵ})}} ⩽ C {‖ Z_{ε}^{(2)} ‖}_{W^{1, \infty} (\tilde{Ω} ∖ \overline{ω_{ϵ})}} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{γ / 2}, \end{matrix}$ which is the desired estimate for $z_{ϵ}^{(2)}$ . It remains to estimate the function $z_{ϵ}^{(1)}$ , defined by (60). To that end we define an extension of $z_{ϵ}^{(1)} |_{\partial ω_{ϵ}}$ to a function $Z_{ϵ}^{(1)}$ , defined on the whole domain $\tilde{Ω} ∖ \overline{ω_{ϵ}}$ , with $Z_{ϵ}^{(1)} = 0$ on $\partial \tilde{Ω}$ and $\begin{matrix} {‖ Z_{ϵ}^{(1)} ‖}_{H^{1} (\tilde{Ω} ∖ \overline{ω_{ϵ}})} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{1 / 2} \sqrt{| log ϵ |} . \end{matrix}$ We start by defining $Z_{ϵ}^{(1)}$ in $(\tilde{Ω} ∖ \overline{ω_{ϵ}}) \cap {x < - 1}$ , the region denoted A in Fig. 1. On the part of $\partial ω_{ϵ}$ which lies inside ${x < - 1}$ , $z_{ϵ}^{(1)} |_{\partial ω_{ϵ}}$ is given by $- b_{1} ϵ^{1 / 2} sin (θ / 2)$ . We now define $\begin{matrix} Z_{ϵ}^{(1)} = - b_{1} ϵ^{1 / 2} sin (θ / 2) ψ (r) in (\tilde{Ω} ∖ \overline{ω_{ϵ}}) \cap {x < - 1}, \end{matrix}$ where $ψ (r)$ is a smooth cut-off function (with $ψ \equiv 1$ for $r < r_{0} / 2$ and $ψ \equiv 0$ for $r > r_{0}$ ) which ensures that $Z_{ϵ}^{(1)}$ vanishes on $\partial \tilde{Ω} \cap {x < - 1}$ . It is clear from this definition that $\begin{matrix} (66) & {‖ Z_{ϵ}^{(1)} ‖}_{H^{1} ((\tilde{Ω} ∖ \overline{ω_{ϵ}}) \cap {x < - 1})} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{1 / 2} \sqrt{| log ϵ |} . \end{matrix}$ We now continue to define $Z_{ϵ}^{(1)}$ on $(Ω ∖ \overline{ω_{ϵ}}) \cap {- 1 < x < 0}$ , the region denoted $B \cup B^{'}$ in Fig. 1, such that $\begin{array}{l} Z_{ϵ}^{(1)} = - b_{1} ϵ^{1 / 2} \frac{\sqrt{2}}{2} ψ (y) on (\tilde{Ω} ∖ \overline{ω_{ϵ}}) \cap {x = - 1}, \\ (67) & Z_{ϵ}^{(1)} = - b_{1} r^{1 / 2} sin (θ / 2) ϕ (r) on {- 1 < x < 0, y = \pm ϵ}, \\ Z_{ϵ}^{(1)} vanishes
on (\tilde{Ω} ∖ \overline{ω_{ϵ}}) \cap {x = 0} and
on \partial \tilde{Ω} \cap {- 1 < x < 0}, \end{array}$ and most importantly $\begin{matrix} {‖ Z_{ϵ}^{(1)} ‖}_{H^{1} ((\tilde{Ω} ∖ \overline{ω_{ϵ}}) \cap {- 1 < x < 0})} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{1 / 2} \sqrt{| log ϵ |} . \end{matrix}$ In order to verify the existence of this $H^{1}$ extension it suffices to prove that the prescribed set of boundary values (67) are bounded by $C ‖ {\tilde{φ}}^{ev} ‖_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{1 / 2} \sqrt{| log ϵ |}$ in the $H^{1 / 2}$ norm. It is clear that $\begin{array}{rcl} {‖ Z_{ϵ}^{(1)} ‖}_{H^{1 / 2} ((\tilde{Ω} ∖ \overline{ω_{ϵ}}) \cap {x = - 1})} & = & {‖ b_{1} ϵ^{1 / 2} \frac{\sqrt{2}}{2} ψ (y) ‖}_{H^{1 / 2} ((\tilde{Ω} ∖ \overline{ω_{ϵ}}) \cap {x = - 1})} \\ (68) & ⩽ & C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{1 / 2} . \end{array}$ Concerning $z = b_{1} r^{1 / 2} sin (θ / 2) ϕ (r)$ , we calculate $\begin{array}{l} \begin{matrix} z (x, y) & = b_{1} r^{1 / 2} \frac{1}{\sqrt{2}} {(1 - cos θ)}^{1 / 2} ϕ (r) = \frac{b_{1}}{\sqrt{2}} {(\sqrt{{(x + 1)}^{2} + ϵ^{2}} - (x + 1))}^{1 / 2} ϕ (r) \\ = \frac{b_{1}}{\sqrt{2}} \frac{ϵ}{{(\sqrt{{(x + 1)}^{2} + ϵ^{2}} + (x + 1))}^{1 / 2}} ϕ (r), \end{matrix} \\ \begin{matrix} \frac{\partial}{\partial x} z (x, y) & = - \frac{b_{1}}{2 \sqrt{2}} \frac{ϵ}{{(\sqrt{{(x + 1)}^{2} + ϵ^{2}} + (x + 1))}^{3 / 2}} (\frac{2 (x + 1)}{2 \sqrt{{(x + 1)}^{2} + ϵ^{2}}} + 1) ϕ (r) \\ + z (x, ϵ) \frac{\partial}{\partial x} ϕ (r), \end{matrix} \end{array}$ on ${- 1 < x < 0, y = ϵ}$ , from which we conclude that $\begin{array}{l} ‖ z ‖_{L^{2} ({- 1 < x < 0, y = ϵ})} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ \sqrt{| log ϵ |}, and \\ ‖ z ‖_{H^{1} ({- 1 < x < 0, y = ϵ})} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}, \end{array}$ and so by logarithmic convexity of the norms $\begin{array}{rcl} {‖ Z_{ϵ}^{(1)} ‖}_{H^{1 / 2} ({- 1 < x < 0, y = ϵ})} & = & ‖ z ‖_{H^{1 / 2}} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{1 / 2} | log ϵ |^{1 / 4} \\ (69) & ⩽ & C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{1 / 2} | log ϵ |^{1 / 2} . \end{array}$ Finally, consider the integral $\begin{array}{l} \int_{0}^{r_{0} / 2} \frac{| Z_{ϵ}^{(1)} (- 1, ϵ + s) - Z_{ϵ}^{(1)} (- 1 + s, ϵ) |^{2}}{s} d s \\ = \frac{| b_{1} |^{2}}{2} \int_{0}^{r_{0} / 2} {(ϵ^{1 / 2} - \frac{ϵ}{{(\sqrt{{(s)}^{2} + ϵ^{2}} + s)}^{1 / 2}})}^{2} / s d s \\ = ϵ \frac{| b_{1} |^{2}}{2} \int_{0}^{r_{0} / 2 ϵ} \frac{{[{(\sqrt{t^{2} + 1} + t)}^{1 / 2} - 1]}^{2}}{t (\sqrt{t^{2} + 1} + t)} d t \\ (70) & ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ | log ϵ | . \end{array}$ By combining (68), (69) and (70) with the fact that $Z_{ϵ}^{(1)}$ vanishes on $(\tilde{Ω} ∖ \overline{ω_{ϵ}}) \cap {x = 0}$ and on $\partial \tilde{Ω} \cap {- 1 < x < 0}$ , we now conclude that the prescribed values of $Z_{ϵ}^{(1)}$ on the boundary of $\tilde{Ω} \cap {- 1 < x < 0, ϵ < y}$ (the region denoted B in Fig. 1) are bounded by $C ‖ {\tilde{φ}}^{ev} ‖_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{1 / 2} \sqrt{| log ϵ |}$ in the $H^{1 / 2}$ norm.1

Here we use the well known fact that a function v, which lies in $H^{1 / 2} (- 1, 0)$ and in $H^{1 / 2} (0, 1)$ and for which $\int_{0}^{1 / 2} | v (s) - v (- s) |^{2} / s d s$ is finite, indeed lies in $H^{1 / 2}$ on the entire interval $(- 1, 1)$ . The sum of the $H^{1 / 2} (- 1, 0)$ norm, the $H^{1 / 2} (0, 1)$ norm and square root of this integral is equivalent to the $H^{1 / 2} (- 1, 1)$ norm, [3]. It is not essential that the two segments meet at an angle of π, they could form a corner, and the same norm characterization would still be valid.

It follows immediately that we may extend

Z_{ϵ}^{(1)}

to all of

\tilde{Ω} \cap {- 1 < x < 0, ϵ < y}

, with

\begin{matrix} {‖ Z_{ϵ}^{(1)} ‖}_{H^{1} (\tilde{Ω} \cap {- 1 < x < 0, ϵ < y})} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{1 / 2} \sqrt{| log ϵ |} .^{2} \end{matrix}

The domain $\tilde{Ω} \cap {- 1 < x < 0, ϵ < y}$ does depend on ϵ, but the fact that it is uniformly diffeomorphic to a fixed domain ensures that the constant C is independent of ϵ.

A similar construction works for

B^{'} = \tilde{Ω} \cap {- 1 < x < 0, y < - ϵ}

, so that altogether we have constructed

Z_{ϵ}^{(1)}

with the desired boundary values and with

\begin{matrix} (71) & {‖ Z_{ϵ}^{(1)} ‖}_{H^{1} ((\tilde{Ω} ∖ \overline{ω_{ϵ}}) \cap {- 1 < x < 0})} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{1 / 2} \sqrt{| log ϵ |} . \end{matrix}

By joining together the two constructions of

Z_{ϵ}^{(1)}

in the disjoint regions

A = (\tilde{Ω} ∖ \overline{ω_{ϵ}}) \cap {x < - 1}

and

B \cup B^{'} = (\tilde{Ω} ∖ \overline{ω_{ϵ}}) \cap {- 1 < x < 0}

, we obtain (due to the continuity across the line

x = - 1

) a function

Z_{ϵ}^{(1)}

in all of

A \cup B \cup B^{'} = (\tilde{Ω} ∖ \overline{ω_{ϵ}}) \cap {x < 0}

which satisfies

\begin{array}{l} Z_{ϵ}^{(1)} = 0 on \partial \tilde{Ω} \cap {x < 0} and Z_{ϵ}^{(1)} = 0 on (\tilde{Ω} ∖ ω_{ϵ}) \cap {x = 0}, \\ Z_{ϵ}^{(1)} = - b_{1} r^{1 / 2} sin (θ / 2) ϕ (r) - b_{1}^{'} r^{' 1 / 2} sin (θ^{'} / 2) ϕ (r^{'}) on \partial ω_{ϵ} \cap {x < 0}, \end{array}

together with the estimate

\begin{matrix} {‖ Z_{ϵ}^{(1)} ‖}_{H^{1} ((\tilde{Ω} ∖ \overline{ω_{ϵ}}) \cap {x < 0})} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{1 / 2} \sqrt{| log ϵ |} . \end{matrix}

This last estimate is obtained by a combination of (66) and (71). An entirely similar construction works on the domain

(\tilde{Ω} ∖ ω_{ϵ}) \cap {0 < x}

, and by merging these two constructions (using the fact that both constructions vanish on

x = 0

) we have constructed the desired function,

Z_{ϵ}^{(1)} \in H^{1} (\tilde{Ω} ∖ ω_{ϵ})

, with

\begin{array}{l} Z_{ϵ}^{(1)} = - b_{1} r^{1 / 2} sin (θ / 2) ϕ (r) - b_{1}^{'} r^{' 1 / 2} sin (θ^{'} / 2) ϕ (r^{'}) on \partial ω_{ϵ}, \\ (72) & Z_{ϵ}^{(1)} = 0 on \partial \tilde{Ω}, and \\ {‖ Z_{ϵ}^{(1)} ‖}_{H^{1} (\tilde{Ω} ∖ \overline{ω_{ϵ}})} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{1 / 2} \sqrt{| log ϵ |} . \end{array}

We may now use this function

Z_{ϵ}^{(1)}

, the same way we used the function

Z_{ϵ}^{(2)}

, to show that

z_{ϵ}^{(1)}

, defined by (60), satisfies

\begin{matrix} {‖ \nabla z_{ϵ}^{(1)} ‖}_{L^{2} (\tilde{Ω} ∖ \overline{ω_{ϵ}})} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{1 / 2} \sqrt{| log ϵ |}, \end{matrix}

\begin{matrix} (73) & {‖ \nabla z_{ϵ}^{(1)} ‖}_{L^{2} (\tilde{Ω} ∖ \overline{ω_{ϵ}})} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{γ / 2}, \end{matrix}

for any

γ < 1

. Due to (65) and (73) we now get that the corrector function

z_{ϵ}

, defined by (9), satisfies

\begin{matrix} ‖ \nabla z_{ϵ} ‖_{L^{2} (\tilde{Ω} ∖ \overline{ω_{ϵ}})} ⩽ {‖ \nabla z_{ϵ}^{(1)} ‖}_{L^{2} (\tilde{Ω} ∖ \overline{ω_{ϵ}})} + {‖ \nabla z_{ϵ}^{(2)} ‖}_{L^{2} (\tilde{Ω} ∖ \overline{ω_{ϵ}})} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{γ / 2}, \end{matrix}

for any

γ < 1

, as asserted in Lemma 3.

Proof of Lemma 4

We notice that Lemma 4 only concerns the case $ϵ a_{ϵ} > m > 0$ . A simple computation shows that $\begin{array}{l} - \int_{\partial ω_{ϵ}^{ends}} \frac{\partial}{\partial n} u^{ev} - \int_{- 1}^{1} \frac{\partial}{\partial y} u^{ev} (x, ϵ) + \int_{- 1}^{1} \frac{\partial}{\partial y} u^{ev} (x, - ϵ) \\ + \int_{- 1}^{1} \frac{\partial}{\partial y} u^{ev} (x, 0_{+}) - \int_{- 1}^{1} \frac{\partial}{\partial y} u^{ev} (x, 0_{-}) = 0 . \end{array}$ Here we use the fact that $Δ u^{ev} = 0$ in $ω_{ϵ} ∖ (- 1, 1)$ . Since $\begin{matrix} \int_{- 1}^{1} D U_{\pm} (x) d x = \int_{- 1}^{1} \frac{\partial}{\partial y} u^{ev} (x, 0_{\pm}) d x, \end{matrix}$ it immediately follows that the assigned values for $ξ_{ϵ} \cdot n |_{\partial ω_{ϵ}}$ from (22) satisfy $\begin{matrix} \int_{\partial ω_{ϵ}} ξ_{ϵ} \cdot n = 0 . \end{matrix}$ Given these assigned values for $ξ_{ϵ} \cdot n$ , we define $\begin{matrix} V (s) = \int_{0}^{s} ξ_{ϵ} \cdot n d σ, \end{matrix}$ where s denotes (counterclockwise) arclength along $\partial ω_{ϵ}$ , and the integral is taken along $\partial ω_{ϵ}$ , from arclength 0 to arclength s. We define arclength 0 to correspond to the point $(- 1, ϵ)$ on $\partial ω_{ϵ}$ (the point between components A and B in Fig. 2). The representation formula in Proposition 1 of Appendix A in [1] implies that $\begin{matrix} \frac{\partial}{\partial n} u^{ev} = \frac{b_{1}}{2} ϵ^{- 1 / 2} sin (θ / 2) + \frac{\partial}{\partial r} u^{*}, \end{matrix}$ on $\partial ω_{ϵ} \cap {x < - 1}$ . Due to the $C^{1, γ / 2}$ nature of $u^{*}$ , and the fact that $\nabla u^{*} (- 1, 0) = 0$ (see (63) of Appendix A in this paper), we have that $\begin{matrix} | \frac{\partial}{\partial r} u^{*} | ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{γ / 2} on \partial ω_{ϵ} \cap {x < - 1} . \end{matrix}$ By counterclockwise integration the function V has the representation $\begin{array}{rcl} V (ϵ, θ) & = & - ϵ \int_{π / 2}^{θ} \frac{b_{1}}{2} ϵ^{- 1 / 2} sin (\tilde{θ} / 2) d \tilde{θ} - ϵ \int_{π / 2}^{θ} \frac{\partial}{\partial r} u^{*} |_{r = ϵ} d \tilde{θ} \\ = & b_{1} ϵ^{1 / 2} cos (θ / 2) - \frac{b_{1}}{\sqrt{2}} ϵ^{1 / 2} - ϵ \int_{π / 2}^{θ} \frac{\partial}{\partial r} u^{*} |_{r = ϵ} d \tilde{θ} \end{array}$ in polar coordinates on $\partial ω_{ϵ} \cap {x < - 1}$ . By extending V as a constant function along straight rays emanating from $\partial ω_{ϵ} \cap {x < - 1}$ , i.e., by defining $\begin{matrix} V (r, θ) = V (ϵ, θ) for
any r > ϵ, \end{matrix}$ we obtain (as in the case of $Z_{ϵ}^{(1)}$ and $Z_{ϵ}^{(2)}$ ) an $H^{1}$ function V on $(\tilde{Ω} ∖ ω_{ϵ}) \cap {x < - 1}$ (the component A in Fig. 2), which vanishes on ${x = - 1, y > ϵ}$ , and satisfies $\begin{matrix} (74) & ‖ V ‖_{H^{1} ((\tilde{Ω} ∖ ω_{ϵ}) \cap {x_{1} < - 1})} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{γ / 2}, \end{matrix}$ for any $γ < 1$ . On ${- 1 < x < - 1 / 2, y = ϵ}$ (the lower boundary of the component B in Fig. 2) the function V has the formula $\begin{array}{rcl} V (x, ϵ) & = & - \int_{- 1}^{x} {(ξ_{ϵ})}_{2} (s, ϵ) d s = \int_{- 1}^{x} (\frac{\partial}{\partial y} u^{ev} (s, ϵ) - D U_{+} (s)) \\ = & \int_{0}^{x + 1} (\frac{b_{1}}{\sqrt{2}} \frac{\partial}{\partial y} \sqrt{{(s^{2} + y^{2})}^{1 / 2} - s} |_{y = ϵ} - \frac{b_{1}}{2} max {s, ϵ}^{- 1 / 2} - d_{ϵ}) d s \\ + \int_{- 1}^{x} (\frac{\partial u^{*}}{\partial y} (s, ϵ) - \frac{\partial u^{*}}{\partial y} (s, 0_{+})) d s \\ = & \frac{b_{1}}{2} \int_{0}^{x + 1} (\frac{\sqrt{{(s^{2} + ϵ^{2})}^{1 / 2} + s}}{\sqrt{2 (s^{2} + ϵ^{2})}} - max {s, ϵ}^{- 1 / 2}) d s - d_{ϵ} (x + 1) \\ (75) & + \int_{- 1}^{x} (\frac{\partial u^{*}}{\partial y} (s, ϵ) - \frac{\partial u^{*}}{\partial y} (s, 0_{+})) d s . \end{array}$ Let $g (t)$ denote the function $\begin{matrix} g (t) = \int_{0}^{t} (\frac{\sqrt{{(s^{2} + ϵ^{2})}^{1 / 2} + s}}{\sqrt{2 (s^{2} + ϵ^{2})}} - max {s, ϵ}^{- 1 / 2}) d s . \end{matrix}$ A change of variable yields $\begin{matrix} (76) & g (t) = \sqrt{ϵ} \int_{0}^{t / ϵ} (\frac{\sqrt{{(s^{2} + 1)}^{1 / 2} + s}}{\sqrt{2 (s^{2} + 1)}} - max {s, 1}^{- 1 / 2}) d s . \end{matrix}$ We have that $\begin{array}{rcl} \frac{\sqrt{{(s^{2} + 1)}^{1 / 2} + s}}{\sqrt{2 (s^{2} + 1)}} - max {s, 1}^{- 1 / 2} & = & \frac{\sqrt{s} \sqrt{{(s^{2} + 1)}^{1 / 2} + s} - \sqrt{2 (s^{2} + 1)}}{\sqrt{s} \sqrt{2 (s^{2} + 1)}} \\ = & \frac{s \sqrt{{(1 + s^{- 2})}^{1 / 2} + 1} - s \sqrt{2 + 2 s^{- 2}}}{s^{3 / 2} \sqrt{2 + 2 s^{- 2}}} \\ = & O (s^{- 5 / 2}), as s \to \infty, \end{array}$ and $\begin{matrix} \frac{\sqrt{{(s^{2} + 1)}^{1 / 2} + s}}{\sqrt{2 (s^{2} + 1)}} - max {s, 1}^{- 1 / 2} = O (1), as s \to 0 . \end{matrix}$ It immediately follow that $\begin{matrix} | g (t) | ⩽ C \sqrt{ϵ}, \end{matrix}$ with $\begin{matrix} C = \int_{0}^{\infty} | \frac{\sqrt{{(s^{2} + 1)}^{1 / 2} + s}}{\sqrt{2 (s^{2} + 1)}} - max {s, 1}^{- 1 / 2} | d s < \infty . \end{matrix}$ Since $| d_{ϵ} | ⩽ C ‖ {\tilde{φ}}^{ev} ‖_{H^{1 / 2} (\partial \tilde{Ω})} \sqrt{ϵ}$ , it follows that $\begin{matrix} | d_{ϵ} (x + 1) | ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} \sqrt{ϵ}, \end{matrix}$ and since $\frac{\partial u^{*}}{\partial y}$ is in $C^{γ / 2} (K^{+})$ we have that $\begin{matrix} | \int_{- 1}^{x} (\frac{\partial u^{*}}{\partial y} (s, ϵ) - \frac{\partial u^{*}}{\partial y} (s, 0_{+})) d s | ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{γ / 2}, \end{matrix}$ for $- 1 < x < - 1 / 2$ . By combination of these last three estimates with the representation (75) we conclude that $\begin{matrix} (77) & {‖ V (\cdot, ϵ) ‖}_{L^{2} (- 1, - 1 / 2)} ⩽ {‖ V (\cdot, ϵ) ‖}_{L^{\infty} (- 1, - 1 / 2)} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{γ / 2} . \end{matrix}$ On ${- 1 < x < - 1 / 2}$ , the first derivative of $V (\cdot, ϵ)$ has the representation $\begin{array}{rcl} \frac{\partial}{\partial x} V (x, ϵ) & = & \frac{b_{1}}{2} (\frac{\sqrt{{({(x + 1)}^{2} + ϵ^{2})}^{1 / 2} + x + 1}}{\sqrt{2 ({(x + 1)}^{2} + ϵ^{2})}} - max {x + 1, ϵ}^{- 1 / 2}) \\ (78) & - d_{ϵ} + \frac{\partial u^{*}}{\partial y} (x, ϵ) - \frac{\partial u^{*}}{\partial y} (x, 0_{+}) . \end{array}$ We calculate, $\begin{array}{l} \int_{- 1}^{- 1 / 2} {| \frac{\sqrt{{({(x + 1)}^{2} + ϵ^{2})}^{1 / 2} + x + 1}}{\sqrt{2 ({(x + 1)}^{2} + ϵ^{2})}} - max {x + 1, ϵ}^{- 1 / 2} |}^{2} d x \\ = \int_{0}^{1 / 2} {| \frac{\sqrt{{(s^{2} + ϵ^{2})}^{1 / 2} + s}}{\sqrt{2 (s^{2} + ϵ^{2})}} - max {s, ϵ}^{- 1 / 2} |}^{2} d s \\ = \int_{0}^{1 / 2 ϵ} {| \frac{\sqrt{{(s^{2} + 1)}^{1 / 2} + s}}{\sqrt{2 (s^{2} + 1)}} - max {s, 1}^{- 1 / 2} |}^{2} d s \\ (79) & ⩽ C, \end{array}$ with $\begin{matrix} C = \int_{0}^{\infty} {| \frac{\sqrt{{(s^{2} + 1)}^{1 / 2} + s}}{\sqrt{2 (s^{2} + 1)}} - max {s, 1}^{- 1 / 2} |}^{2} d s < \infty . \end{matrix}$ Due to the bound on $| d_{ϵ} |$ and the $C^{1, γ / 2} (K^{+})$ nature of $u^{*}$ , we also have $\begin{matrix} (80) & {‖ - d_{ϵ} + \frac{\partial u^{*}}{\partial y} (\cdot, ϵ) - \frac{\partial u^{*}}{\partial y} (\cdot, 0_{+}) ‖}_{L^{2} (- 1, - 1 / 2)} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{γ / 2} . \end{matrix}$ A combination of (78), (79), and (80) now gives $\begin{matrix} {‖ \frac{\partial}{\partial x} V (\cdot, ϵ) ‖}_{L^{2} (- 1, - 1 / 2)} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}, \end{matrix}$ so that, due to (77), we may conclude $\begin{matrix} {‖ V (\cdot, ϵ) ‖}_{H^{1} (- 1, - 1 / 2)} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} . \end{matrix}$ This estimate in combination with (77) and the logarithmic convexity of the Sobolev norms yields $\begin{matrix} (81) & {‖ V (\cdot, ϵ) ‖}_{H^{1 / 2} (- 1, - 1 / 2)} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{γ / 4}, \end{matrix}$ for any $γ < 1$ . We also calculate the integral $\begin{array}{rcl} \int_{- 1}^{- 1 / 2} \frac{| V (x, ϵ) |^{2}}{| x + 1 |} d x & ⩽ & \frac{| b_{1} |^{2}}{2} \int_{- 1}^{- 1 / 2} \frac{| g (x + 1) |^{2}}{| x + 1 |} d x \\ (82) & + 2 \int_{- 1}^{- 1 / 2} \frac{| \int_{- 1}^{x} (- d_{ϵ} + \frac{\partial u^{*} (s, ϵ)}{\partial y} - \frac{\partial u^{*} (s, 0_{+})}{\partial y}) d s |^{2}}{| x + 1 |} d x . \end{array}$ By application of Hölder’s inequality to the formula (76) for g we get $\begin{array}{rcl} | g (t) | & ⩽ & \sqrt{ϵ} {(\int_{0}^{t / ϵ} {| \frac{\sqrt{{(s^{2} + 1)}^{1 / 2} + s}}{\sqrt{2 (s^{2} + 1)}} - max {s, 1}^{- 1 / 2} |}^{p} d s)}^{1 / p} \times {(\int_{0}^{t / ϵ} 1 d s)}^{1 / q} \\ ⩽ & ϵ^{\frac{1}{2} - \frac{1}{q}} t^{1 / q} {(\int_{0}^{\infty} {| \frac{\sqrt{{(s^{2} + 1)}^{1 / 2} + s}}{\sqrt{2 (s^{2} + 1)}} - max {s, 1}^{- 1 / 2} |}^{p} d s)}^{1 / p} \\ ⩽ & C ϵ^{\frac{1}{2} - \frac{1}{q}} t^{1 / q}, \end{array}$ for any $1 < p < 2$ and $\frac{1}{q} = 1 - \frac{1}{p} (< \frac{1}{2})$ . Based on this estimate we obtain $\begin{matrix} (83) & \int_{- 1}^{- 1 / 2} \frac{| g (x + 1) |^{2}}{| x + 1 |} d x ⩽ C ϵ^{1 - \frac{2}{q}} \int_{0}^{1 / 2} s^{\frac{2}{q} - 1} d s ⩽ C ϵ^{1 - \frac{2}{q}}, \end{matrix}$ for any $q > 2$ . We also have $\begin{matrix} | - d_{ϵ} + \frac{\partial u^{*} (s, ϵ)}{\partial y} - \frac{\partial u^{*} (s, 0_{+})}{\partial y} | ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{γ / 2} for s \in (- 1, - 1 / 2), \end{matrix}$ and so $\begin{matrix} | \int_{- 1}^{x} (- d_{ϵ} + \frac{\partial u^{*} (s, ϵ)}{\partial y} - \frac{\partial u^{*} (s, 0_{+})}{\partial y}) d s | ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{γ / 2} | x + 1 |, \end{matrix}$ which gives $\begin{matrix} (84) & \int_{- 1}^{- 1 / 2} {| \int_{- 1}^{x} (- d_{ϵ} + \frac{\partial u^{*} (s, ϵ)}{\partial y} - \frac{\partial u^{*} (s, 0_{+})}{\partial y}) d s |}^{2} / | x + 1 | d x ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ^{γ} . \end{matrix}$ Insertion of (83) and (84) into (82) leads to the estimate $\begin{matrix} (85) & \int_{- 1}^{- 1 / 2} \frac{| V (x, ϵ) |^{2}}{| x + 1 |} d x ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}^{2} ϵ^{γ}, \end{matrix}$ for any $γ < 1$ . In light of the estimates (81) and (85), we conclude that the boundary data, consisting of the function 0 on ${x = - 1, ϵ < y} \cap \tilde{Ω}$ and the function $- \int_{- 1}^{x} {(ξ_{ϵ})}_{2} (s, ϵ) d s$ on ${- 1 < x < - 1 / 2, y = ϵ}$ , is in $H^{1 / 2}$ with a norm bounded by $C ‖ {\tilde{φ}}^{ev} ‖_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{γ / 4}$ , and thus this combined datum permits an $H^{1}$ extension V to the region $B = \tilde{Ω} \cap {- 1 < x < - 1 / 2, ϵ < y}$ in Fig. 2, with $\begin{matrix} ‖ V ‖_{H^{1} (\tilde{Ω} \cap {- 1 < x < - 1 / 2, ϵ < y})} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{γ / 4} . \end{matrix}$ The same construction works on $\tilde{Ω} \cap {- 1 < x < - 1 / 2, y < - ϵ}$ , the region denoted $B^{'}$ in Fig. 2, given constant boundary data $\int_{0}^{ϵ π} ξ_{ϵ} \cdot n$ on ${x = - 1, y < - ϵ}$ and boundary data $\int_{0}^{s} ξ_{ϵ} \cdot n$ , $ϵ π < s < ϵ π + 1 / 2$ on ${- 1 < x < - 1 / 2, y = - ϵ}$ (note that arclength = $ϵ π$ corresponds to the point $(- 1, - ϵ)$ on $\partial ω_{ϵ}$ ). By combining the (three) constructions carried out so far, we obtain an $H^{1}$ function V on $A \cup B \cup B^{'} = (\tilde{Ω} ∖ ω_{ϵ}) \cap {x < - 1 / 2}$ , with $\begin{matrix} V |_{\partial ω_{ϵ}} (s) = \int_{0}^{s} (ξ_{ϵ}) \cdot n on \partial ω_{ϵ}, and ‖ V ‖_{H^{1} ((\tilde{Ω} ∖ ω_{ϵ}) \cap {x_{1} < - 1 / 2})} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{γ / 4} . \end{matrix}$ A similar construction for $x > 1 / 2$ would lead to a function $V \in H^{1} ((\tilde{Ω} ∖ ω_{ϵ}) \cap {1 / 2 < x})$ , with $\begin{matrix} V |_{\partial ω_{ϵ}} (s) = \int_{0}^{s} ξ_{ϵ} \cdot n on \partial ω_{ϵ}, and ‖ V ‖_{H^{1} ((Ω ∖ ω_{ϵ}) \cap {1 / 2 < x})} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{γ / 4} . \end{matrix}$ On the “interior” segments ${- 3 / 4 < x < 3 / 4, y = \pm ϵ}$ of $\partial ω_{ϵ}$ it is clear that the function $\int_{0}^{s} (ξ_{ϵ}) \cdot n$ has $H^{1 / 2}$ -norms bounded by $C ‖ {\tilde{φ}}^{ev} ‖_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{γ / 2}$ (actually its $W^{1, \infty}$ -norms will be bounded by $C ‖ {\tilde{φ}}^{ev} ‖_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{γ / 2}$ ) and so we may also easily construct an $H^{1}$ function V on $C \cup C^{'} = (\tilde{Ω} ∖ ω_{ϵ}) \cap {- 3 / 4 < x < 3 / 4}$ with $\begin{matrix} V |_{\partial ω_{ϵ}} (s) = \int_{0}^{s} ξ_{ϵ} \cdot n on \partial ω_{ϵ} and ‖ V ‖_{H^{1} ((\tilde{Ω} ∖ ω_{ϵ}) \cap {- 3 / 4 < x < 3 / 4})} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{γ / 2} . \end{matrix}$ By pasting these constructs for $x < - 1 / 2$ , $x > 1 / 2$ , and $- 3 / 4 < x < 3 / 4$ together using a partition of unity (in x) we finally obtain a function $V \in H^{1} (\tilde{Ω} ∖ ω_{ϵ})$ with $\begin{matrix} V |_{\partial ω_{ϵ}} (s) = \int_{0}^{s} ξ_{ϵ} \cdot n on \partial ω_{ϵ}, and ‖ V ‖_{H^{1} (\tilde{Ω} ∖ ω_{ϵ})} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{γ / 4} . \end{matrix}$ By multiplication with a smooth cutoff function (vanishing near $\partial \tilde{Ω}$ and equal to 1 in a fixed, ϵ independent subdomain containing $ω_{ϵ}$ ) we may assume that V vanishes in a neighborhood of $\partial \tilde{Ω}$ . Having constructed this extension V in all of $\tilde{Ω} ∖ ω_{ϵ}$ , we now define $\begin{matrix} ξ_{ϵ} = - \nabla V^{⊥} = {(\frac{\partial V}{\partial y}, - \frac{\partial V}{\partial x})}^{t} \in L^{2} {(\tilde{Ω} ∖ ω_{ϵ})}^{2} . \end{matrix}$ This $ξ_{ϵ}$ is a solution to (22) (since $ξ_{ϵ} \cdot n = \frac{\partial V}{\partial s}$ assumes the required boundary data on $\partial ω_{ϵ}$ and $\partial \tilde{Ω}$ ), with $\begin{matrix} ‖ ξ_{ϵ} ‖_{L^{2} (\tilde{Ω} ∖ ω_{ϵ})} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ϵ^{γ / 4}, \end{matrix}$ as asserted in Lemma 4.

References

Charnley and

M.S.

Vogelius, A uniformly valid model for the limiting behaviour of voltage potentials in the presence of thin inhomogeneities I. The case of an open mid-curve, Asymptotic Analysis (2019).

Dapogny and

M.S.

Vogelius, Uniform asymptotic expansion of the voltage potential in the presence of thin inhomogeneities with arbitrary conductivity, Chinese Annals of Mathematics, Series B 38 (2017), 293–344. doi:10.1007/s11401-016-1072-3.

J.-L.

Lions and

Magenes, Non-Homogeneous Boundary Value Problems and Applications I, Grundlehren der Mathematischen Wissenschaften, Vol. 181, Springer-Verlag, Berlin/Heidelberg/New York, 1972.

Morgenstern and

Szabo, Vorlesungen über Theoretische Mechanik, Die Grundlehren der Mathematischen Wissenschaften in Einzeldarstellungen, Vol. 112, Springer-Verlag, Berlin–Göttingen–Heidelberg, 1961.

A uniformly valid model for the limiting behaviour of voltage potentials in the presence of thin inhomogeneities II. A local energy approximation result

Abstract

Keywords

1. Introduction

2.1. Proof of (6)

3.1. Proof of (26)

3.2. Proof of (27)

4. Local convergence – Odd symmetry, a ϵ < M ϵ

5.1. Proof of (49)

5.2. Proof of (50)

6. Proof of Lemma 1 and Lemma 2

Footnotes

Acknowledgements

Proof of Lemma 3

Proof of Lemma 4

References

4. Local convergence – Odd symmetry, $a_{ϵ} < M ϵ$