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

Abstract

Asymptotic approximations of voltage potentials in the presence of diametrically small inhomogeneities are well studied. In particular it is known that one may construct approximations that are accurate to any order (in the diameter) uniformly in the conductivity of the inhomogeneity. The corresponding problem for thin inhomogeneities is not so well understood, in particular as concerns uniformity of the approximations. If the conductivity degenerates to 0 or goes to infinity as the width of the inhomogeneity goes to zero, the voltage potential may converge to different limiting solutions, and so the construction of uniform approximations is not straightforward. For the case of thin two dimensional inhomogeneities with closed mid-curves such approximations were constructed and rigorously verified in (Chinese Annals of Mathematics, Series B 38 (2017) 293–344). The analysis relied heavily on the regularity of the approximate solutions. In this two part paper we continue this line of research, by showing that the same approximations remain valid, even when the mid-curve is open, and the corresponding approximate solutions have singularities at the endpoints of the curve.

Keywords

Elliptic boundary-value problem thin inhomogeneity uniform approximation

1. Introduction

Consider the following somewhat non-standard elliptic boundary-value problem: $\begin{array}{l} - Δ u = f in Ω ∖ σ, \\ (1) & \frac{\partial u^{+}}{\partial n} + ϵ a_{ϵ} {(\frac{\partial}{\partial τ})}^{2} u^{+} - \frac{a_{ϵ}}{2 ϵ} (u^{+} - u^{-}) = 0 on σ, \\ \frac{\partial u^{-}}{\partial n} - ϵ a_{ϵ} {(\frac{\partial}{\partial τ})}^{2} u^{-} - \frac{a_{ϵ}}{2 ϵ} (u^{+} - u^{-}) = 0 on σ, \end{array}$ with given Dirichlet data $u = φ \in H^{1 / 2} (\partial Ω)$ . Here Ω is a bounded, smooth subdomain of $R^{2}$ , and σ is a smooth non-self-intersecting open curve, with endpoints $x_{0}$ and $x_{1}$ . $u^{+}$ denotes the trace on σ, taken from the side of $Ω ∖ σ$ that the normal field n points into, $u^{-}$ denotes the trace on σ, taken from the side of $Ω ∖ σ$ that n points away from. τ denotes the tangent field obtained by $90^{\circ}$ counterclockwise rotation of n. ϵ and $a_{ϵ}$ are positive constants. To be precise, by the solution to (1) we mean the minimizer of the energy functional $\begin{array}{rcl} E_{σ, Ω}^{ϵ} (v) & = & \frac{1}{2} \int_{Ω ∖ σ} | \nabla v |^{2} d x d y + \frac{ϵ a_{ϵ}}{2} \int_{σ} ({(\frac{\partial v^{+}}{\partial τ})}^{2} + {(\frac{\partial v^{-}}{\partial τ})}^{2}) d s \\ (2) & + \frac{a_{ϵ}}{4 ϵ} \int_{σ} {(v^{+} - v^{-})}^{2} d s - \int_{Ω} f v d x d y, \end{array}$ in the set ${v \in H^{1} (Ω ∖ σ) : v^{\pm} \in H^{1} (σ), v = φ on \partial Ω}$ . We assume that f is in $L^{2} (Ω)$ and vanishes in a neighborhood of σ.

Our main goal in this two part paper is to show that the solution to (1) represents a uniform approximation to the solution of the “thin inhomogeneity” boundary value problem $\begin{matrix} (3) & - \nabla \cdot (γ_{ϵ} \nabla u_{ϵ}) = f in Ω, u = φ on \partial Ω . \end{matrix}$ Here the coefficient $γ_{ϵ}$ is given by $\begin{matrix} (4) & γ_{ϵ} (x, y) = \{\begin{matrix} 1 & if (x, y) is in Ω ∖ ω_{ϵ}, \\ a_{ϵ} & if (x, y) is in ω_{ϵ}, \end{matrix} \end{matrix}$ with $\begin{matrix} ω_{ϵ} = {(x, y) : dist ((x, y), σ) < ϵ} . \end{matrix}$ We note that $u_{ϵ}$ is the minimizer of the energy functional $\begin{matrix} (5) & E_{Ω}^{ϵ} (v) = \frac{1}{2} \int_{Ω} γ_{ϵ} | \nabla v |^{2} d x d y - \int_{Ω} f v d x d y, \end{matrix}$ in the set ${v \in H^{1} (Ω) : v = φ on \partial Ω}$ . By uniform approximation we mean that $u - u_{ϵ}$ converges to 0 as $ϵ \to 0$ , uniformly in $a_{ϵ} > 0$ , $‖ f ‖_{L^{2}} ⩽ 1$ and $‖ φ ‖_{H^{1 / 2}} ⩽ 1$ . This convergence is in $H^{k} (Ω ∖ ω)$ , for any k and any fixed neighborhood ω of σ.

To understand that the uniformity of the convergence is not obvious, we notice that if $a_{ϵ} = a > 0$ is fixed and ϵ tends to zero then $u_{ϵ}$ , the solution to (3), tends to U, the solution to $\begin{matrix} - Δ U = f in Ω, U = φ on \partial Ω . \end{matrix}$ On the other hand, if $a_{ϵ}$ goes to infinity sufficiently fast as $ϵ \to 0$ (to be precise, if $ϵ a_{ϵ} \to \infty$ as $ϵ \to 0$ ) then $u_{ϵ}$ tends to $u_{\infty}$ , the solution to $\begin{matrix} - Δ u_{\infty} = f in Ω ∖ σ, u_{\infty} = const on σ, u_{\infty} = φ on \partial Ω, \end{matrix}$ where the exact value of the constant on σ is determined by the requirement that $\int_{σ} (\frac{\partial u_{\infty}^{+}}{\partial n} - \frac{\partial u_{\infty}^{-}}{\partial n}) d s = 0$ . If $a_{ϵ}$ goes to zero sufficiently fast as $ϵ \to 0$ (to be precise, if $\frac{a_{ϵ}}{ϵ} \to 0$ as $ϵ \to 0$ ) then $u_{ϵ}$ tends to $u_{0}$ , the solution to $\begin{matrix} - Δ u_{0} = f in Ω ∖ σ, \frac{\partial u_{0}}{\partial n} = 0 on σ, u_{0} = φ on \partial Ω . \end{matrix}$ In general, there are (after extraction of a subsequence) five limiting scenarios: (i) $ϵ a_{ϵ} \to \infty$ , (ii) $ϵ a_{ϵ} \to a$ , $0 < a < \infty$ , (iii) $ϵ a_{ϵ} \to 0$ , $\frac{a_{ϵ}}{ϵ} \to \infty$ , (iv) $\frac{a_{ϵ}}{ϵ} \to b$ , $0 < b < \infty$ , and (v) $\frac{a_{ϵ}}{ϵ} \to 0$ . Each scenario gives rise to a different type of limiting boundary condition on σ – the system (1) “catches” all of these in one ϵ-dependent boundary condition, while ensuring uniform approximability. It is worth pointing out that the energy expression (2) (or the boundary value problem (1)) is not the only one that leads to such uniform approximability. We refer to Section 4 in [15], and Remark 9 at the end of Section 6 in [15], concerning alternate expressions for the case of a closed mid-curve.

The limiting behaviour associated with “thin” inhomogeneities stands in sharp contrast to that associated with diametrically small inhomogeneities (of the form $ϵ B$ , for a fixed subdomain B). For diametrically small inhomogeneities, the “background” solution U from above is the limit of $u_{ϵ}$ , uniformly with respect to the conductivity inside $ϵ B$ , cf. [20]. Certain approximate cloaking schemes based on transformation optics techniques (cf. [18]) owe their success to this fact.

The work in this paper is a direct extension of that in [15], where the same type of model was derived and analyzed in the case of a closed mid-curve σ. The variational technique, and in particular the energy convergence approach described in Section 2, is identical to that of [15]. The major novelty in this paper stems from the fact that the curve is open, and as a consequence, solutions to (1) generally have singularities associated with the endpoints $x_{0}$ and $x_{1}$ . At first sight, it might appear that these singularities could contribute at highest order to the energy and thus change the character of the “approximating” boundary value problem. However, the analysis in this paper somewhat surprisingly shows that this is not the case, and that the same problem is “approximating” for an open curve. See the remark at the end of Section 2.1 in [13] for more details.

The techniques developed in this paper could be applied to u, the solution to the three dimensional boundary value problem $\begin{array}{l} - Δ u = f in Ω ∖ σ, u = φ on \partial Ω, \\ (6) & \frac{\partial u^{+}}{\partial n} + ϵ a_{ϵ} Δ_{τ} u^{+} - \frac{a_{ϵ}}{2 ϵ} (u^{+} - u^{-}) = 0 on σ, \\ \frac{\partial u^{-}}{\partial n} - ϵ a_{ϵ} Δ_{τ} u^{-} - \frac{a_{ϵ}}{2 ϵ} (u^{+} - u^{-}) = 0 on σ, \end{array}$ where σ is a smooth non-selfintersecting two dimensional closed, or non-closed surface inside the smooth bounded domain $Ω \subset R^{3}$ , and $Δ_{τ}$ denotes the Laplace–Beltrami operator on σ. It remains an open problem to establish uniform decompositions of u into a singular part and a $C^{1, γ / 2}$ remainder, similar to those derived in two dimensions in Appendices A and B, but if this could be done, then the analysis developed here would immediately carry over to show that u is indeed a uniform approximation to the solution to the three dimensional analogue of the “thin inhomogeneity” boundary value problem (3).

Asymptotic expansions for thin inhomogeneities with fixed material properties are well-studied in the literature. We refer for instance to [3,4,9,10]. In all of these works the lowest order term is always the homogeneous background solution U, and the focus is on (one or more) higher order terms. There are fewer studies where the conductivity of the inhomogeneity has a particular variable dependence on the thickness; one such example is the analysis found in [21], where the authors derive asymptotic expansions for the case of a closed curve, when $\frac{a_{ϵ}}{ϵ} \to b$ , $0 ⩽ b < \infty$ (the cases (iv) and (v) above).

Partial knowledge of terms of asymptotic expansions has been used to build effective reconstruction algorithms, see e.g. [1,6,7,11]. Such knowledge has also been used to enhance the effect of approximate transformation cloaks, cf. [2].

For the remainder of the paper we shall for simplicity assume that σ is the straight line segment $σ = {(x, y) : - 1 < x < 1, y = 0}$ . To understand that this is not a very restrictive assumption, we first note that if σ was a sufficiently smooth curve with two straight segments at the ends, then a combination of the analysis performed here (applied to the singularities and the straight segments) combined with the analysis performed in [15] (applied around the curved smooth middle segment) would immediately lead to a proof of the uniform approximation property of u. If the curve σ is smooth, but not straight near the endpoints $x_{0}$ and $x_{1}$ , then it could be made so by a simple local change of coordinates – this would not materially affect the analysis of the singularities in Appendices A and B, and the rest of our analysis would be unchanged. The fact that the differential operator is the Laplacian outside σ is also not a very restrictive assumption, our analysis could be changed to accommodate any “background” operator of the form $\nabla \cdot (γ (x, y) \nabla u)$ , where $γ (x, y)$ is a smooth positive definite matrix valued function on Ω. We suppose that the “external source” f is an element of $\begin{matrix} F_{δ_{0}} = {f \in L^{2} (Ω), supp (f) \subset Ω ∖ ω_{δ_{0}}} \end{matrix}$ for some fixed $δ_{0}$ ; we expect that an assumption like this is essential for the results obtained here.

The organization of the paper is as follows. In Section 2, we formulate a lemma (adapted from [15]) which allows us to estimate the norm distance between solutions in terms of the difference in the associated energies. We proceed in Section 3 to state a uniform, local energy convergence result (Theorem 1). The proof of Theorem 1 is the focus of part II of this paper [13]; it is facilitated by the assumption that σ is a straight line segment, which in turn allows for a convenient splitting of the minimizers into their even and odd parts. Section 4 contains the statements and proofs of the two main results of this paper: Theorem 2 (a uniform, global energy convergence result) and Theorem 3 (a uniform norm convergence result). The proof of Theorem 2 consists of a combination of fairly simple energy arguments with the local convergence result asserted in Theorem 1. Theorem 3 follows from Theorem 2 by application of the energy-to-norm-estimate lemma from Section 2. Finally in Section 5 we report on a number of computational results illustrating the nature of the uniform convergence of solutions, and we discuss some of the conjectures that naturally emanate from this computational evidence. The appendices, Appendix A and Appendix B, establish regularity results for solutions to problems that represent special cases of the system (1). These regularity results play a crucial role in our proof of the uniform, local energy convergence result in part II of this paper – the fact that these two special cases suffice for our analysis is a consequence of the symmetry splitting and a duality feature which relies on the isotropy of $a_{ϵ}$ (the fact that $a_{ϵ}$ is a positive scalar).

2. The energy convergence approach

In this section we shall slightly reformulate a lemma from [15], which allows us to estimate the norm distance between energy minimizers, by the gap in the corresponding energy minima. We refer the reader to [15] for the relatively simple proof of this result. The application of the lemma in this paper is entirely similar to that in [15]. We use the notation $v |_{\partial Ω}$ ; this refers to the standard trace, which is well-defined (and in $H^{1 / 2} (\partial Ω)$ ) for any function v that is $H^{1}$ in a neighborhood of $\partial Ω$ .

Lemma 1.
Let Ω be a bounded, smooth domain in $R^{2}$ , and let V, W be two Hilbert spaces of functions defined on Ω, such that the trace operator $\begin{matrix} V ∋ v \mapsto v |_{\partial Ω} \in H^{\frac{1}{2}} (\partial Ω) \end{matrix}$ is well-defined, continuous, and has a linear continuous right inverse $φ \mapsto v_{φ}$ (similarly for W with a mapping $φ \mapsto w_{φ}$ ). Let H be another Hilbert space, with two associated bounded linear mappings $P : V \to H$ and $Q : W \to H$ . Denote by $a : V \times V \to R$ and $b : W \times W \to R$ two symmetric, continuous and coercive bilinear forms. Coercivity is only required on $V_{0} = V \cap {v |_{\partial Ω} = 0}$ and $W_{0} = W \cap {w |_{\partial Ω} = 0}$ . For any $φ \in H^{\frac{1}{2}} (\partial Ω)$ , $ℓ \in H^{'}$ , consider the minimization problems: $\begin{array}{l} min_{\begin{array}{c} v \in V \\ v = φ on \partial Ω \end{array}} E (v), E (v) = \frac{1}{2} a (v, v) - ℓ (P v), \\ min_{\begin{array}{c} w \in W \\ w = φ on \partial Ω \end{array}} F (w), F (w) = \frac{1}{2} b (w, w) - ℓ (Q w), \end{array}$ which admit unique minimizers $v^{ℓ, φ} \in V$ , $w^{ℓ, φ} \in W$ (E and F depend on ℓ, though it is not explicit from the notation). Then the following estimate holds $\begin{matrix} (7) & sup_{\begin{array}{c} ‖ ℓ ‖_{H^{'}} ⩽ 1 \\ ‖ φ ‖_{H^{1 / 2} (\partial Ω)} ⩽ 1 \end{array}} {‖ P v^{ℓ, φ} - Q w^{ℓ, φ} ‖}_{H} ⩽ 4 sup_{\begin{array}{c} ‖ ℓ ‖_{H^{'}} ⩽ 1 \\ ‖ φ ‖_{H^{1 / 2} (\partial Ω)} ⩽ 1 \end{array}} | E (v^{ℓ, φ}) - F (w^{ℓ, φ}) | . \end{matrix}$

3. The local problem and its symmetries

Let $\tilde{Ω} = ω_{δ_{0}^{'}}$ , for some fixed $δ_{0}^{'} < δ_{0}$ . $\tilde{Ω}$ is a subdomain of Ω which contains σ, is symmetric around the line segment σ, and inside which f vanishes. Let $\tilde{u}$ denote a solution to the boundary value problem (1) in $\tilde{Ω}$ with Dirichlet boundary data $\tilde{u} |_{\partial \tilde{Ω}} = \tilde{φ} \in H^{1 / 2} (\partial \tilde{Ω})$ . We may decompose $\tilde{u}$ as $\tilde{u} = u^{odd} + u^{ev}$ , where the odd component of $\tilde{u}$ is given by $\begin{matrix} u^{odd} (x, y) = \frac{1}{2} [\tilde{u} (x, y) - \tilde{u} (x, - y)], \end{matrix}$ and the even component is given by $\begin{matrix} u^{ev} (x, y) = \frac{1}{2} [\tilde{u} (x, y) + \tilde{u} (x, - y)] . \end{matrix}$ Let $\tilde{φ} = {\tilde{φ}}^{odd} + {\tilde{φ}}^{ev}$ be the decomposition $\tilde{φ}$ into its even and odd parts. Note that each of the functions ${\tilde{φ}}^{odd}$ and ${\tilde{φ}}^{ev}$ satisfies $\begin{matrix} {‖ {\tilde{φ}}^{odd} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ⩽ ‖ \tilde{φ} ‖_{H^{1 / 2} (\partial \tilde{Ω})}, and {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} ⩽ ‖ \tilde{φ} ‖_{H^{1 / 2} (\partial \tilde{Ω})} . \end{matrix}$ We notice that $\begin{matrix} E_{σ, \tilde{Ω}}^{ϵ} (v) = E_{σ, \tilde{Ω}}^{ϵ} (v^{odd}) + E_{σ, \tilde{Ω}}^{ϵ} (v^{ev}), \end{matrix}$ for any $v \in H^{1} (\tilde{Ω} ∖ σ)$ with $v^{\pm} |_{σ} \in H^{1} (σ)$ . As a consequence the function $u^{odd}$ is the minimizer of the energy $E_{σ, \tilde{Ω}}^{ϵ}$ in the set ${v \in H_{odd}^{1} (\tilde{Ω} ∖ σ), v^{\pm} |_{σ} \in H^{1} (σ), v = {\tilde{φ}}^{odd} on \partial \tilde{Ω}}$ , and a similar statement holds for $u^{ev}$ (with $H_{odd}^{1}$ replaced by $H_{ev}^{1}$ and ${\tilde{φ}}^{odd}$ replaced by ${\tilde{φ}}^{ev}$ ). The functions $u^{odd}$ and $u^{ev}$ are entirely characterized by their restrictions to ${\tilde{Ω}}^{+} = \tilde{Ω} \cap {y > 0}$ , and these satisfy (in a variational sense) the boundary value problems $\begin{array}{l} - Δ u^{odd} = 0 in {\tilde{Ω}}^{+}, \\ (8) & \frac{\partial u^{odd}}{\partial y} + ϵ a_{ϵ} \frac{\partial^{2} u^{odd}}{\partial x^{2}} - \frac{a_{ϵ}}{ϵ} u^{odd} = 0 on σ, \\ u^{odd} = 0 on {y = 0} ∖ σ, u^{odd} = {\tilde{φ}}^{odd} on {(\partial \tilde{Ω})}^{+}, \end{array}$ and $\begin{array}{l} - Δ u^{ev} = 0 in {\tilde{Ω}}^{+}, \\ (9) & \frac{\partial u^{ev}}{\partial y} + ϵ a_{ϵ} \frac{\partial^{2} u^{ev}}{\partial x^{2}} = 0 on σ, \\ \frac{\partial u^{ev}}{\partial y} = 0 on {y = 0} ∖ σ, u^{ev} = {\tilde{φ}}^{ev} on {(\partial \tilde{Ω})}^{+}, \end{array}$ respectively. Let ${\tilde{u}}_{ϵ}$ denote a solution to the boundary value problem (3)–(4) on $\tilde{Ω}$ with Dirichlet boundary data ${\tilde{u}}_{ϵ} |_{\partial \tilde{Ω}} = \tilde{φ}$ . The function ${\tilde{u}}_{ϵ}$ also has a decomposition ${\tilde{u}}_{ϵ} = u_{ϵ}^{odd} + u_{ϵ}^{ev}$ into its even and odd parts. $u_{ϵ}^{odd}$ and $u_{ϵ}^{ev}$ are entirely characterized by their restrictions to ${\tilde{Ω}}^{+}$ , and these satisfy (in a variational sense) the boundary value problems $\begin{matrix} (10) & \begin{matrix} - \nabla \cdot (γ_{ϵ} \nabla u_{ϵ}^{odd}) = 0 in {\tilde{Ω}}^{+}, \\ u_{ϵ}^{odd} = 0 on {y = 0}, u_{ϵ}^{odd} = {\tilde{φ}}^{odd} on {(\partial \tilde{Ω})}^{+}, \end{matrix} \end{matrix}$ and $\begin{matrix} (11) & \begin{matrix} - \nabla \cdot (γ_{ϵ} \nabla u_{ϵ}^{ev}) = 0 in {\tilde{Ω}}^{+}, \\ \frac{\partial u_{ϵ}^{ev}}{\partial y} = 0 on {y = 0}, u_{ϵ}^{ev} = {\tilde{φ}}^{ev} on {(\partial \tilde{Ω})}^{+}, \end{matrix} \end{matrix}$ respectively. In Appendix A and Appendix B we establish very precise (uniformly valid) regularity results for problem (9) and a simplified version of (8). Using the splittings $\tilde{u} = u^{odd} + u^{ev}$ and ${\tilde{u}}_{ϵ} = u_{ϵ}^{odd} + u_{ϵ}^{ev}$ in connection with the aformentioned regularity results, we are able to prove the following uniform, local energy convergence theorem.

Theorem 1.
Let $\tilde{u}$ and ${\tilde{u}}_{ϵ}$ denote the minimizers of the energies ( 2 ) and ( 5 ), respectively, with Ω replaced by $\tilde{Ω} = ω_{δ_{0}^{'}}$ , and 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 $ϵ \to 0$ .

The proof of this theorem is the focus of part II of this paper. Here we proceed instead to use this result to establish our general (global) convergence results.
4. Global convergence – Main results

Theorem 1 makes it possible to establish the following energy convergence result for the full domain Ω.

Theorem 2.
Given a fixed $δ_{0} > 0$ and given $φ \in H^{1 / 2} (\partial Ω)$ , $f \in F_{δ_{0}}$ , let u and $u_{ϵ}$ denote the solutions to ( 1 ) and ( 3 ), respectively. Then $\begin{matrix} sup_{\begin{array}{c} a_{ϵ} > 0 \\ (f, φ) \neq (0, 0) \end{array}} \frac{| E_{σ, Ω}^{ϵ} (u) - E_{Ω}^{ϵ} (u_{ϵ}) |}{{(‖ f ‖_{L^{2} (Ω)} + ‖ φ ‖_{H^{1 / 2} (\partial Ω)})}^{2}} \to 0, \end{matrix}$ as $ϵ \to 0$ .
Proof of Theorem 2.
As previously, define $\tilde{Ω} = ω_{δ_{0}^{'}}$ , with $δ_{0}^{'} < δ_{0}$ (so that f vanishes in $\tilde{Ω}$ ). Let $\tilde{u}$ denote the solution to (1) in $\tilde{Ω}$ (not Ω) with $\tilde{u} |_{\partial \tilde{Ω}} = u_{ϵ} |_{\partial \tilde{Ω}}$ , and let ${\tilde{u}}_{ϵ}$ denote the solution to (3) in $\tilde{Ω}$ (not Ω) with ${\tilde{u}}_{ϵ} |_{\partial \tilde{Ω}} = u |_{\partial \tilde{Ω}}$ . We notice that, due to energy estimates, $u_{ϵ} |_{\partial \tilde{Ω}}$ as well as $u |_{\partial \tilde{Ω}}$ are bounded in the $H^{1 / 2} (\partial \tilde{Ω})$ norm by $C (‖ f ‖_{L^{2} (Ω)} + ‖ φ ‖_{H^{1 / 2} (\partial Ω)})$ , independently of ϵ and $a_{ϵ}$ . Now define extensions $\overline{u}$ and ${\overline{u}}_{ϵ}$ of the two functions $\tilde{u}$ and ${\tilde{u}}_{ϵ}$ , as follows $\begin{matrix} \overline{u} = \{\begin{matrix} u_{ϵ} & in Ω ∖ \tilde{Ω}, \\ \tilde{u} & in \tilde{Ω}, \end{matrix} and {\overline{u}}_{ϵ} = \{\begin{matrix} u & in Ω ∖ \tilde{Ω}, \\ {\tilde{u}}_{ϵ} & in \tilde{Ω} . \end{matrix} \end{matrix}$ Due to Theorem 1, we have that $\begin{array}{rcl} E_{σ, \tilde{Ω}}^{ϵ} (\tilde{u}) & ⩽ & E_{\tilde{Ω}}^{ϵ} (u_{ϵ}) + o (1) ‖ u_{ϵ} |_{\partial \tilde{Ω}} ‖_{H^{1 / 2} (\partial \tilde{Ω})}^{2} \\ (12) & ⩽ & E_{\tilde{Ω}}^{ϵ} (u_{ϵ}) + o (1) {(‖ f ‖_{L^{2} (Ω)} + ‖ φ ‖_{H^{1 / 2} (\partial Ω)})}^{2}, \end{array}$ where $o (1) \to 0$ as $ϵ \to 0$ , uniformly in $a_{ϵ}$ , f and φ. Similarly, from Theorem 1 we have that $\begin{array}{rcl} E_{\tilde{Ω}}^{ϵ} ({\tilde{u}}_{ϵ}) & ⩽ & E_{σ, \tilde{Ω}}^{ϵ} (u) + o (1) ‖ u |_{\partial \tilde{Ω}} ‖_{H^{1 / 2} (\partial \tilde{Ω})}^{2} \\ (13) & ⩽ & E_{σ, \tilde{Ω}}^{ϵ} (u) + o (1) {(‖ f ‖_{L^{2} (Ω)} + ‖ φ ‖_{H^{1 / 2} (\partial Ω)})}^{2}, \end{array}$ where $o (1) \to 0$ as $ϵ \to 0$ , uniformly in $a_{ϵ}$ , f and φ. Due to the formula for $\overline{u}$ and (12), we now calculate $\begin{array}{rcl} E_{σ, Ω}^{ϵ} (\overline{u}) & = & E_{σ, \tilde{Ω}}^{ϵ} (\tilde{u}) + \frac{1}{2} \int_{Ω ∖ \tilde{Ω}} | \nabla u_{ϵ} |^{2} d x d y - \int_{Ω ∖ \tilde{Ω}} f u_{ϵ} d x d y \\ ⩽ & E_{\tilde{Ω}}^{ϵ} (u_{ϵ}) + o (1) {(‖ f ‖_{L^{2} (Ω)} + ‖ φ ‖_{H^{1 / 2} (\partial Ω)})}^{2} \\ + \frac{1}{2} \int_{Ω ∖ \tilde{Ω}} | \nabla u_{ϵ} |^{2} d x d y - \int_{Ω ∖ \tilde{Ω}} f u_{ϵ} d x d y \\ (14) & = & E_{Ω}^{ϵ} (u_{ϵ}) + o (1) {(‖ f ‖_{L^{2} (Ω)} + ‖ φ ‖_{H^{1 / 2} (\partial Ω)})}^{2} . \end{array}$ Due to the formula for ${\overline{u}}_{ϵ}$ and (13), we also calculate $\begin{array}{rcl} E_{Ω}^{ϵ} ({\overline{u}}_{ϵ}) & = & E_{\tilde{Ω}}^{ϵ} ({\tilde{u}}_{ϵ}) + \frac{1}{2} \int_{Ω ∖ \tilde{Ω}} | \nabla u |^{2} d x d y - \int_{Ω ∖ \tilde{Ω}} f u d x d y \\ ⩽ & E_{σ, \tilde{Ω}}^{ϵ} (u) + o (1) {(‖ f ‖_{L^{2} (Ω)} + ‖ φ ‖_{H^{1 / 2} (\partial Ω)})}^{2} \\ + \frac{1}{2} \int_{Ω ∖ \tilde{Ω}} | \nabla u |^{2} d x d y - \int_{Ω ∖ \tilde{Ω}} f u d x d y \\ (15) & = & E_{σ, Ω}^{ϵ} (u) + o (1) {(‖ f ‖_{L^{2} (Ω)} + ‖ φ ‖_{H^{1 / 2} (\partial Ω)})}^{2} . \end{array}$ From (14) and the energy minimizing property of u, we conclude that $\begin{matrix} E_{σ, Ω}^{ϵ} (u) ⩽ E_{σ, Ω}^{ϵ} (\overline{u}) ⩽ E_{Ω}^{ϵ} (u_{ϵ}) + o (1) {(‖ f ‖_{L^{2} (Ω)} + ‖ φ ‖_{H^{1 / 2} (\partial Ω)})}^{2} . \end{matrix}$ From (15) and the energy minimizing property of $u_{ϵ}$ , we conclude that $\begin{matrix} E_{Ω}^{ϵ} (u_{ϵ}) ⩽ E_{Ω}^{ϵ} ({\overline{u}}_{ϵ}) ⩽ E_{σ, Ω}^{ϵ} (u) + o (1) {(‖ f ‖_{L^{2} (Ω)} + ‖ φ ‖_{H^{1 / 2} (\partial Ω)})}^{2} . \end{matrix}$ By a combination of these last two estimates we obtain $\begin{matrix} | E_{σ, Ω}^{ϵ} (u) - E_{Ω}^{ϵ} (u_{ϵ}) | ⩽ o (1) {(‖ f ‖_{L^{2} (Ω)} + ‖ φ ‖_{H^{1 / 2} (\partial Ω)})}^{2}, \end{matrix}$ exactly as asserted by Theorem 2. □

We may now apply the result from Lemma 1 in Section 2 to obtain the following norm convergence result. Theorem 3.
Given a fixed $δ_{0} > 0$ and given $φ \in H^{1 / 2} (\partial Ω)$ , $f \in F_{δ_{0}}$ , let u and $u_{ϵ}$ denote the solutions to ( 1 ) and ( 3 ), respectively. For any $s ⩾ 0$ and any $δ > 0$ $\begin{matrix} sup_{\begin{array}{c} a_{ϵ} > 0 \\ (f, φ) \neq (0, 0) \end{array}} \frac{‖ u - u_{ϵ} ‖_{H^{s} (Ω ∖ ω_{δ})}}{‖ f ‖_{L^{2} (Ω)} + ‖ φ ‖_{H^{1 / 2} (\partial Ω)}} \to 0, \end{matrix}$ as $ϵ \to 0$ .
Proof of Theorem 3.
The uniform convergence in $L^{2} (Ω ∖ ω_{δ_{0}})$ follows immediately from a combination of Theorem 2 and Lemma 1. In this particular application of Proposition 1 we take $V = H^{1} (Ω)$ , with the standard $H^{1}$ norm, and $W = {w \in H^{1} (Ω ∖ σ) : w^{+} |_{σ}, w^{-} |_{σ} \in H^{1} (σ)}$ with the norm $‖ w ‖_{H^{1} (Ω ∖ σ)} + ‖ w^{+} |_{σ} ‖_{H^{1} (σ)} + ‖ w^{-} |_{σ} ‖_{H^{1} (σ)}$ . For H we take $F_{δ_{0}} = {f \in L^{2} (Ω) : supp (f) \subset Ω ∖ ω_{δ_{0}}}$ . The operators P and Q are both multiplication by the indicator function for $Ω ∖ ω_{δ_{0}}$ : $\begin{matrix} P : V ∋ v \to 1_{Ω ∖ ω_{δ_{0}}} v \in H and Q : W ∋ w \to 1_{Ω ∖ ω_{δ_{0}}} w \in H . \end{matrix}$ As bilinear forms we choose the two families $\begin{matrix} a^{ϵ} (v, v) = \int_{Ω} γ_{ϵ} | \nabla v |^{2} d x d y, \end{matrix}$ where $γ_{ϵ}$ is as defined earlier, and $\begin{matrix} b^{ϵ} (w, w) = \int_{Ω ∖ σ} | \nabla v |^{2} d x d y + ϵ a_{ϵ} \int_{σ} ({(\frac{\partial w^{+}}{\partial x})}^{2} + {(\frac{\partial w^{-}}{\partial x})}^{2}) d x + \frac{a_{ϵ}}{2 ϵ} \int_{σ} {(w^{+} - w^{-})}^{2} d x . \end{matrix}$ As $F_{δ_{0}} \subset F_{δ}$ for any $δ ⩽ δ_{0}$ , it actually follows from this argument that we have uniform convergence in $L^{2} (Ω ∖ ω_{δ})$ , for any fixed $δ > 0$ . The uniform convergence in the higher Sobolev norms on $Ω ∖ ω_{δ}$ , for any fixed $δ > 0$ , now follows by interior elliptic regularity using the facts that $Δ (u - u_{ϵ})$ vanishes in $Ω ∖ \overline{ω_{ϵ}}$ and that $u - u_{ϵ}$ vanishes on $\partial Ω$ . □
Remark.
For our analysis we have assumed that the conductivity inside the inhomogeneity $ω_{ϵ}$ is isotropic and constant, and that the inhomogeneity has constant thickness $2 ϵ$ . These assumptions were used to derive the energy formulas and the asymptotic estimates. Interesting physical problems arise when one or more of these are dropped. For an inhomogeneity of variable conductivity (but constant thickness), a process very similar to the one introduced in this two part paper can be used to derive an approximate energy for the reduced problem on σ. However, questions arise as to how simple and explicit this energy can be made, while still allowing for uniform approximability. Furthermore, given an explicit candidate for this energy, a major difficulty related to a rigorous proof of uniform approximability is the possible presence of different asymptotic ranges along the curve σ. There may indeed be a need for some assumptions on how “irregularly” the conductivity can behave in order to get uniform approximation estimates. Variable-thickness inhomogeneities are another area of interest; the analysis presented in this two part paper immediately extends to the case when the inhomogeneity is symmetric around the “mid-curve” σ, has minimal and maximal thickness of the same order of magnitude, and is of constant, isotropic conductivity (cf. [12]). The problem of a general variable-thickness inhomogeneity is still open, and the difficulties of proof are similar to those encountered in the case of variable conductivities.

Even the simplest, constant anisotropic conductivity makes the situation much more complicated. Suppose σ is the line segment from $(- 1, 0)$ to $(1, 0)$ , and suppose the conductivity $a_{ϵ} : = {{(a_{ϵ})}_{i j}}$ is diagonal. The fact that ${(a_{ϵ})}_{11}$ and ${(a_{ϵ})}_{22}$ may be of distinctly different orders complicates the construction of an approximate energy (i.e., complicates the “inner minimization” leading to (2)) and it also significantly complicates any rigorous proof for odd symmetry, due to the fact that it will not be possible to separate the roles played by the different parts of the approximate energy. For a general anisotropic conductivity it is even worse; here the splitting according to even and odd symmetry is not very useful, due to the “coupling” of the resulting equations.

5. Computational results and related comments

In this section, we present some numerical results to illustrate the accuracy with which the solution u to (1) approximates the solution $u_{ϵ}$ to (3), uniformly in $a_{ϵ}$ , as $ϵ \to 0$ . We take the domain Ω to be $B_{2} (0)$ , the ball of radius 2 centered at the origin, and we take the curve σ to be the straight line segment from $(- 1, 0)$ to $(1, 0)$ . The function f is taken to be identically zero in Ω (instead of in some smaller subset $ω_{δ_{0}}$ ) and the specified Dirichlet boundary data corresponds to the function $φ (x, y) = (1 - x) + x^{2} y - (x + 1) y^{2} + 3 x y^{3}$ . The computations were done on the half-domain $Ω^{+}$ , making full use of the symmetry properties discussed in Section 3. This means the actual calculations are done with $\begin{matrix} φ^{ev} (x, y) = 1 - x - (x + 1) y^{2}, and φ^{odd} (x, y) = x^{2} y + 3 x y^{3} . \end{matrix}$ All of the results were computed using the FEniCS finite element software package, with piecewise linear elements. The meshes were taken fine enough to resolve the thin inhomogeneity $ω_{ϵ}$ .

5.1. Even symmetry

The first problem we discuss is the one with even symmetry, i.e., Dirichlet data $φ^{ev}$ from above. In this case, the problem in $Ω^{+}$ is given by (9). As the only changing parameter in the problem is $ϵ a_{ϵ}$ , the behaviour of the solution as $ϵ \to 0$ should only depend on whether $ϵ a_{ϵ}$ converges to 0, is bounded and strictly positive in the limit, or goes to infinity. In the first graphics we compare the solutions along a horizontal segment at $y = 0.5$ , a fixed distance away from the curve σ. The plots in Fig. 1 show both the solution $u^{ev}$ and $u_{ϵ}^{ev}$ , and in the case where $ϵ a_{ϵ} \to 0$ , the solution $u_{0}^{ev}$ as well.

Fig. 1.

Numerical results for even symmetry along the line $y = 0.5$ . In each plot, the solid line with circles represents the solution $u_{ϵ}^{ev}$ , and the dashed line with squares represents the solution $u^{ev}$ . In the last row, the dot-dash line with triangles shows the solution $u_{0}^{ev}$ .

Fig. 2.

Numerical results for even symmetry along the line $y = 0$ . In each plot, the solid line with circles represents the solution $u_{ϵ}^{ev}$ , and the dashed line with squares represents the solution $u^{ev}$ . In the last row, the dot-dash line with triangles shows the solution $u_{0}^{ev}$ .

As expected from the results proved in this paper, we have convergence of the two solutions as $ϵ \to 0$ in all cases, due to the fact that we stay away from the curve σ. For a second pass over the results, in Fig. 2 we look at the values of the different solutions on a horizontal segment at $y = 0$ , which includes σ.

In particular, these images show the behaviour that we expect to see at the endpoints of σ, due to Proposition 1 of Appendix A. For both $ϵ a_{ϵ} \to \infty$ and $ϵ a_{ϵ}$ bounded and away from zero, there is a distinct $r^{1 / 2}$ behaviour near these endpoints. However, in the case where $ϵ a_{ϵ} \to 0$ , our numerics indicate that the singular behaviour disappears, consistent with the fact that $u_{0}^{ev}$ has no singularity. We also observe that the solutions seem to converge uniformly in $a_{ϵ}$ even on σ, something our rigorous analysis does not show.

5.2. Odd symmetry

In the case of odd symmetry, the approximating problem takes the form of (8), which now has two changing parameters: $ϵ a_{ϵ}$ and $\frac{a_{ϵ}}{ϵ}$ . However, the numerical results seem to indicate that the qualitative behaviour of the solutions in terms of how well they converge only depends on whether or not $\frac{a_{ϵ}}{ϵ}$ goes to infinity. The Dirichlet boundary data is $φ^{odd}$ , from above. In Fig. 3, we first look at the solutions along the line $y = 0.5$ , including the solution $u_{0}^{odd}$ when it is an appropriate approximation. As expected from the results in this paper, we see convergence of the solution at $y = 0.5$ , uniformly in $a_{ϵ}$ .

Fig. 3.

Numerical results for odd symmetry along the line $y = 0.5$ . In each plot, the solid line with circles represents the solution $u_{ϵ}^{odd}$ and the dashed line with squares represents the solution $u^{odd}$ . In the first row, the dot-dash line with triangles represents the solution $u_{0}^{odd}$ , while in the second row, it is the solution $u_{*}^{odd}$ (see Appendix B).

Fig. 4.

Numerical results for odd symmetry along the line $y = 0$ . In each plot, the solid line with circles represents the solution $u_{ϵ}^{odd}$ , the dashed line with squares represents the solution $u^{odd}$ , and the dot-dash line with triangles represents the solution $u_{*}^{odd}$ .

Fig. 5.

Numerical results for odd symmetry along the line $x = 0$ . In each plot, the solid line with circles represents the solution $u_{ϵ}^{odd}$ and the dashed line with squares represents the solution $u^{odd}$ . In the first row, the dot-dash line with triangles represents the solution $u_{0}^{odd}$ , while in the second row, it is the solution $u_{*}^{odd}$ .

For the second set of images (Fig. 4) we choose to focus on a segment of the line $y = 0$ , which includes the curve σ. In Section 4 of part II of this paper [13] it is shown that the solutions $u^{odd}$ and $u_{*}^{odd}$ are energy close when $\frac{a_{ϵ}}{ϵ} < M$ , but as the numerical results indicate, this is true more generally. For a definition of $u_{*}^{odd}$ , see Appendix B. Notice that due to its continuity across ${y = 0}$ and its odd symmetry, $u_{ϵ}^{odd}$ always vanishes at ${y = 0}$ . However, only when $a_{ϵ} / ϵ \to \infty$ will $u^{odd}$ (and $u_{*}^{odd}$ ) approach zero on all of ${y = 0}$ (otherwise they exhibit “limiting” jumps across σ).

In Fig. 5 we look at the solutions along the vertical line $x = 0$ . We observe that the solutions match up away from the curve σ, but if $\frac{a_{ϵ}}{ϵ}$ remains bounded there appears to be a boundary layer forming near the curve.

5.3. Some natural conjectures

There are two main types of results that these numerical computations seem to suggest, but that we have not yet proven. One of these concerns the behaviour on σ. For all values of $a_{ϵ}$ in the even symmetry case, we see that the solution $u^{ev}$ seems to approach the solution $u_{ϵ}^{ev}$ , even on the curve σ, while the only results we have been able to prove thus far involve convergence in the far-field. For odd symmetry, the situation is more precarious: when $\frac{a_{ϵ}}{ϵ}$ goes to infinity, the solution $u^{odd}$ appears to converge to 0 on σ (which is the value of $u_{ϵ}^{odd}$ ) except for either a boundary layer or numerical anomaly at the two endpoints. However, if $\frac{a_{ϵ}}{ϵ}$ remains bounded, we can not expect to see convergence to $u_{ϵ}^{odd}$ on σ, because, heuristically speaking, if the solution to (8) was to converge to 0 on σ, its Cauchy data would appear to converge to 0 on this curve, and this contradicts the fact that $u^{odd}$ does not “limit” to zero everywhere.

The second type of result concerns rate of convergence. Since the local convergence results of this paper are based on supremum arguments, where in each case only one of the two estimates yields a strict control on the rate of the error, the final result does not have this kind of bound. To guess what might happen with the error as $ϵ \to 0$ , we performed calculations of the $L^{2}$ error of the solution on $ω_{0.8} ∖ ω_{0.5}$ . The plots below show the error as a function of ϵ for several different asymptotics of $a_{ϵ}$ and for the two different boundary data (even and odd).

Fig. 6.

Error plot (log-log scale) for Dirichlet boundary data $1 - x - (x + 1) y^{2}$ . The value in the legend for each data set represents the power of ϵ in $a_{ϵ} = 3 ϵ^{α}$ .

Fig. 7.

Error plot (log-log scale) for Dirichlet boundary data $x^{2} y + 3 x y^{3}$ . The value in the legend for each data set represents the power of ϵ in $a_{ϵ} = 3 ϵ^{α}$ .

From the results in Section 2 and Section 4 of part II of this paper [13], we expect convergence at a rate of (at least) $ϵ^{γ / 4}$ , $γ < 1$ , in the case of even symmetry, as long as $ϵ a_{ϵ} > m > 0$ , and in the case of odd symmetry, as long as $a_{ϵ} < M ϵ$ . The numerics, however, seems to suggest that a rate estimate of ϵ may well hold in all cases. The slopes of the best-fit lines to the data are presented in Table 1.

Table 1

Experimental rates of convergence for the approximate problem to the full problem as $ϵ \to 0$ for a variety of conductivities $a_{ϵ}$

$a_{ϵ}$	Even symmetry	Odd symmetry
$3 ϵ^{- 2}$	1.016	1.157
$3 ϵ^{- 1}$	1.016	1.144
3	0.813	1.071
$3 ϵ$	1.006	1.014
$3 ϵ^{2}$	1.041	0.979

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.

Some regularity considerations

For our local energy convergence arguments in part II of this paper, it is essential to know the precise behaviour of $u^{ev}$ (and $u^{odd}$ ) near the curve σ. These functions are smooth from above and below near interior points of σ, with bounds of the form $C ‖ {\tilde{φ}}^{ev} ‖_{H^{1 / 2} (\partial Ω)}$ (and $C ‖ {\tilde{φ}}^{odd} ‖_{H^{1 / 2} (\partial Ω)}$ ) as already shown in [15]. We now proceed to analyse the (singular) behaviour of $u^{ev}$ near the endpoints of σ, using a square root transformation technique similar to that employed in [16] for a “classical” crack. We refer the reader to [14,17] for related studies of the regularity of solutions to elliptic problems near corner points, and points of change in boundary conditions. It suffices to consider only the left endpoint $(- 1, 0)$ . Let $B_{r_{0}}$ denote the ball of radius $r_{0}$ centered at $(- 1, 0)$ , and select $r_{0} < 1$ sufficiently small that $B_{r_{0}} \subset \subset \tilde{Ω}$ . The function $u^{ev}$ satisfies $\begin{array}{l} - Δ u^{ev} = 0 in B_{r_{0}}^{+}, \\ \frac{\partial u^{ev}}{\partial y} + ϵ a_{ϵ} \frac{\partial^{2} u^{ev}}{\partial x^{2}} = 0 on {- 1 < x < - 1 + r_{0}, y = 0}, \\ \frac{\partial u^{ev}}{\partial y} = 0 on {- 1 - r_{0} < x < - 1, y = 0}, \end{array}$ with $\begin{matrix} {‖ u^{ev} ‖}_{H^{1} (B_{r_{0}}^{+})} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} . \end{matrix}$ The constant C depends on $r_{0}$ , but we have omitted that dependence from the notation; it is, as always, independent of ϵ and $a_{ϵ}$ . Let W be the function which, in polar coordinates around $(- 1, 0)$ , on the first quadrant $Q = {r < \sqrt{r_{0}}, 0 < θ < π / 2}$ is given by $W (r, θ) = u^{ev} (r^{2}, 2 θ)$ , and extend W (using the same notation) as an even function across $x = - 1$ to all of $B_{\sqrt{r_{0}}}^{+}$ . This function W now satisfies $\begin{array}{l} - Δ W = 0 in B_{\sqrt{r_{0}}}^{+}, \\ \frac{\partial W}{\partial y} = - ϵ a_{ϵ} \frac{\partial}{\partial x} (\frac{1}{2 | x + 1 |} \frac{\partial}{\partial x} W (x, 0)) on {y = 0} . \end{array}$ It is easy to see that the fact that $u^{ev}$ is in $H^{1} (B_{r_{0}}^{+})$ (and bounded independently of ϵ and $a_{ϵ}$ ) implies that W is in $H^{1} (B_{\sqrt{r_{0}}}^{+})$ . To be precise $\begin{matrix} ‖ W ‖_{H^{1} (B_{\sqrt{r_{0}}}^{+})} ⩽ C {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} . \end{matrix}$ Let $I_{r_{0}^{'}}$ denote the interval $(- r_{0}^{'} - 1, - 1 + r_{0}^{'})$ for any $r_{0}^{'} ⩽ \sqrt{r_{0}}$ . The weakly defined trace $\frac{\partial}{\partial y} W |_{y = 0}$ , restricted to $I_{r_{0}^{'}}$ , is in $H_{00}^{- 1 / 2} (I_{r_{0}^{'}})$ , the dual of $H_{00}^{1 / 2} (I_{r_{0}^{'}})$ 1

$H_{00}^{1 / 2} (I_{r_{0}^{'}})$ denotes those elements of $H^{1 / 2} (I_{r_{0}^{'}})$ which are in $H^{1 / 2} (R)$ , when extended by zero outside $I_{r_{0}^{'}}$ , cf. [19].

– it thus lies in

H^{- s} (I_{r_{0}^{'}})

(the dual of

H_{0}^{s} (I_{r_{0}^{'}})

) for any

s > 1 / 2

. By “integration” (inversion of the operator

d / d x

) it now follows that

\frac{1}{2 | x + 1 |} \frac{\partial}{\partial x} W (x, 0)

is in

H^{1 - s} (I_{r_{0}^{'}})

, with

\begin{matrix} (16) & {‖ \frac{1}{2 | x + 1 |} \frac{\partial}{\partial x} W (x, 0) ‖}_{H^{1 - s} (I_{r_{0}^{'}})} ⩽ \frac{C_{s}}{ϵ a_{ϵ}} {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} . \end{matrix}

The constant

C_{s}

is independent of ϵ and

a_{ϵ}

. Here we used the fact that the function

\frac{1}{2 | x + 1 |} \frac{\partial}{\partial x} W (x, 0)

is odd in

x = - 1

, and we also used the fact that the differentiation operator

\frac{d}{d x}

maps

H^{1 - s} (I_{r_{0}^{'}}) \cap {v : \int_{I_{r_{0}^{'}}} v = 0}

continuously and one-to-one onto

H^{- s} (I_{r_{0}^{'}})

for any

1 / 2 < s < 1

This latter fact is most easily proven by noticing that $- d / d x$ maps $H_{0}^{s} (I_{r_{0}^{'}})$ continuously and one-to-one onto $H^{s - 1} (I_{r_{0}^{'}}) \cap {v : \int_{I_{r_{0}^{'}}} v = 0}$ , for any $1 / 2 < s < 1$ , and then using that the adjoint of this map has the desired property and equals $d / d x$ .

From (16) we conclude that

\frac{\partial}{\partial x} W (x, 0)

is in

H^{1 - s} (I_{r_{0}^{'}})

(with a similar bound) and so

W (x, 0)

is in

H^{2 - s} (I_{r_{0}^{'}})

with

\begin{matrix} {‖ W (x, 0) ‖}_{H^{2 - s} (I_{r_{0}^{'}})} ⩽ C_{s} (\frac{1}{ϵ a_{ϵ}} + 1) {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}, \end{matrix}

for any

s > 1 / 2

. A standard elliptic “local” (or “interior”) regularity result now implies that

\begin{matrix} W lies in H^{2 - δ} (B_{r_{0}^{'}}^{+}), \end{matrix}

for any fixed

r_{0}^{'} < \sqrt{r_{0}}

and

δ > 0

, with the associated bound

\begin{matrix} ‖ W ‖_{H^{2 - δ} (B_{r_{0}^{'}}^{+})} ⩽ C_{δ} (\frac{1}{ϵ a_{ϵ}} + 1) ‖ \tilde{φ} ‖_{H^{1 / 2} (\partial \tilde{Ω})} . \end{matrix}

The trace

\frac{\partial}{\partial y} W |_{y = 0}

, restricted to

I_{r_{0}^{'}}

, is thus in

H^{1 / 2 - δ} (I_{r_{0}^{'}})

, and so

\frac{1}{2 | x + 1 |} \frac{\partial}{\partial x} W (x, 0)

is in

H^{3 / 2 - δ} (I_{r_{0}^{'}})

, with the bound

\begin{matrix} (17) & {‖ \frac{1}{2 | x + 1 |} \frac{\partial}{\partial x} W (x, 0) ‖}_{H^{3 / 2 - δ} (I_{r_{0}^{'}})} ⩽ \frac{C_{δ}}{ϵ a_{ϵ}} (\frac{1}{ϵ a_{ϵ}} + 1) {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} . \end{matrix}

Here we again used that the function

\frac{1}{2 | x + 1 |} \frac{\partial}{\partial x} W (x, 0)

is odd in

x = - 1

. Due to this odd symmetry it vanishes at

x = - 1

– in other words

\frac{1}{2 | x + 1 |} \frac{\partial}{\partial x} W (x, 0) \in H^{3 / 2 - δ} (I_{r_{0}^{'}}) \cap {V : V (- 1) = 0}

. It is easy to see that the mapping

V (\cdot) \to | \cdot + 1 | V (\cdot)

continuously

\begin{matrix} maps H^{1} (I_{r_{0}^{'}}) \cap {V : V (- 1) = 0} into H^{1} (I_{r_{0}^{'}}) \cap {V : V (- 1) = 0}, \end{matrix}

and since

\begin{matrix} (18) & \frac{d}{d x} (| x + 1 | V (x)) = \{\begin{matrix} V (x) & for x > - 1 \\ - V (x) & for x < - 1 \end{matrix} + | x + 1 | \frac{d}{d x} V (x) \end{matrix}

this same map also continuously

\begin{matrix} maps H^{2} (I_{r_{0}^{'}}) \cap {V : V (- 1) = 0} into H^{2} (I_{r_{0}^{'}}) \cap {V : V (- 1) = 0} . \end{matrix}

By interpolation (cf. [5]) it follows that the mapping

V (\cdot) \to | \cdot + 1 | V (\cdot)

continuously

\begin{matrix} maps H^{3 / 2 - δ} (I_{r_{0}^{'}}) \cap {V : V (- 1) = 0} into H^{3 / 2 - δ} (I_{r_{0}^{'}}) \cap {V : V (- 1) = 0} . \end{matrix}

Using this result with

V (x) = \frac{1}{2 | x + 1 |} \frac{\partial}{\partial x} W (x, 0)

we conclude from (17) that

\frac{\partial}{\partial x} W (x, 0)

lies in

H^{3 / 2 - δ} (I_{r_{0}^{'}})

, so that

W (x, 0)

lies in

H^{5 / 2 - δ} (I_{r_{0}^{'}})

and thus (by local elliptic regularity) it follows that

\begin{matrix} W lies in H^{3 - δ} (B_{r_{0}^{'}}^{+}), \end{matrix}

for any fixed

r_{0}^{'} < \sqrt{r_{0}}

, with the associated estimate

\begin{matrix} ‖ W ‖_{H^{3 - δ} (B_{r_{0}^{'}}^{+})} ⩽ C_{δ} (\frac{1}{{(ϵ a_{ϵ})}^{2}} + 1) {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} . \end{matrix}

Continuing this line of reasoning we get that the trace $\frac{\partial}{\partial y} W |_{y = 0}$ is in $H^{3 / 2 - δ} (I_{r_{0}^{'}})$ , and so $\frac{1}{2 | x + 1 |} \frac{\partial}{\partial x} W (x, 0)$ is in $H^{5 / 2 - δ} (I_{r_{0}^{'}}) \cap {V : V (- 1) = 0}$ with the bound $\begin{matrix} (19) & {‖ \frac{1}{2 | x + 1 |} \frac{\partial}{\partial x} W (x, 0) ‖}_{H^{5 / 2 - δ} (I_{r_{0}^{'}})} ⩽ \frac{C_{δ}}{ϵ a_{ϵ}} (\frac{1}{{(ϵ a_{ϵ})}^{2}} + 1) {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} . \end{matrix}$ By taking a further derivative of (18) we obtain $\begin{array}{rcl} (20) & \frac{d^{2}}{d x^{2}} (| x + 1 | V (x)) = \{\begin{matrix} 2 V^{'} (x) & for x > - 1 \\ - 2 V^{'} (x) & for x < - 1 \end{matrix} + | x + 1 | \frac{d^{2}}{d x^{2}} V (x), \end{array}$ for any $V \in H^{2} (I_{r_{0}^{'}}) \cap {V : V (- 1) = 0}$ . Based on (18) and (20) we conclude that $\begin{matrix} | \cdot + 1 | V (\cdot) \in H^{5 / 2 - δ} (I_{r_{0}^{'}}) \cap {V : V (- 1) = 0} \end{matrix}$ for any $V \in H^{5 / 2 - δ} (I_{r_{0}^{'}}) \cap {V : V (- 1) = 0}$ . By application of this result with $V (x) = \frac{1}{2 | x + 1 |} \frac{\partial}{\partial x} W (x, 0)$ we get from (19) that $\begin{matrix} \frac{\partial}{\partial x} W (x, 0) lies in H^{5 / 2 - δ} (I_{r_{0}^{'}}), \end{matrix}$ for any $δ > 0$ . This immediately implies that $W (x, 0)$ lies in $H^{7 / 2 - δ} (I_{r_{0}^{'}})$ , and W lies in $H^{4 - δ} (B_{r_{0}^{'}}^{+})$ with the bound $\begin{matrix} ‖ W ‖_{H^{4 - δ} (B_{r_{0}^{'}}^{+})} ⩽ C_{δ} (\frac{1}{{(ϵ a_{ϵ})}^{3}} + 1) {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})}, \end{matrix}$ for any $r_{0}^{'} < \sqrt{r_{0}}$ and $δ > 0$ . Let U denote the harmonic conjugate of W in $B_{r_{0}^{'}}^{+}$ , selected so that $U (- 1, 0) = 0$ (note: with this choice, U is odd in $x = - 1$ ). Since $\begin{matrix} \nabla U = \nabla W^{⊥} \end{matrix}$ it follows that U lies in $H^{4 - δ} (B_{r_{0}^{'}}^{+})$ , for any $δ > 0$ (with a bound similar to that of W). The function W, given by $\begin{matrix} x + i y = z \to W (z) : = W (x, y) + i U (x, y), \end{matrix}$ is holomorphic in a $C^{+} = C \cap {ℑ z > 0}$ neigborhood of $- 1$ ( $B_{r_{0}^{'}}^{+}$ ), and $C^{2, γ}$ (for any $γ < 1$ ) on the closure of this neighborhood. It thus has a Taylor approximation of degree 2, in the sense that $\begin{array}{rcl} W (z) & = & W (- 1) + B_{1} (z + 1) + \frac{1}{2} B_{2} {(z + 1)}^{2} + 0 (| z + 1 |^{2 + γ}) \\ = & B_{0} + B_{1} (z + 1) + \frac{1}{2} B_{2} {(z + 1)}^{2} + E (z), \end{array}$ with the constants $B_{0}$ , $B_{1}$ , and $B_{2}$ bounded by $C ‖ {\tilde{φ}}^{ev} ‖_{H^{1 / 2} (\partial \tilde{Ω})} (\frac{1}{{(ϵ a_{ϵ})}^{3}} + 1)$ , and with $\begin{array}{l} | E (z) | ⩽ C_{γ} {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} (\frac{1}{{(ϵ a_{ϵ})}^{3}} + 1) | z + 1 |^{2 + γ}, \\ | \frac{d}{d z} E (z) | ⩽ C_{γ} {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} (\frac{1}{{(ϵ a_{ϵ})}^{3}} + 1) | z + 1 |^{1 + γ}, and \\ | \frac{d^{2}}{d z^{2}} E (z) | ⩽ C_{γ} {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} (\frac{1}{{(ϵ a_{ϵ})}^{3}} + 1) | z + 1 |^{γ}, \end{array}$ in $B_{r_{0}^{'}}^{+}$ (for any fixed $γ < 1$ ). Since the real part of W (i.e., W) is even in $x = - 1$ it follows that $B_{1} = - i b_{1}$ and $B_{2} = 2 b_{2}$ , where $b_{1}$ and $b_{2}$ are real, and since the imaginary part, U, has $U (- 1, 0) = 0$ , it follows that $B_{0} = b_{0}$ is also real. The coefficients $b_{0}$ , $b_{1}$ and $b_{2}$ are bounded by $C_{γ} ‖ {\tilde{φ}}^{ev} ‖_{H^{1 / 2} (\partial \tilde{Ω})} (\frac{1}{{(ϵ a_{ϵ})}^{3}} + 1)$ . Let $\sqrt{z + 1}$ be the square root with a “cut” along the half axis $x > - 1$ (in other words: $\sqrt{r e^{i θ}} = \sqrt{r} e^{i θ / 2}$ in polar coordinates around $x = - 1$ ) – we now define $\begin{matrix} w (z) = W (\sqrt{z + 1} - 1) for z \in B_{r_{0}^{″}}^{+} . \end{matrix}$ With this notation it follows that $\begin{matrix} w (z) = b_{0} - i b_{1} \sqrt{z + 1} + b_{2} (z + 1) + e (z), \end{matrix}$ with $b_{0}$ , $b_{1}$ , and $b_{2} \in R$ (bounded by $C_{γ} ‖ {\tilde{φ}}^{ev} ‖_{H^{1 / 2} (\partial \tilde{Ω})} (\frac{1}{{(ϵ a_{ϵ})}^{3}} + 1)$ ), and $\begin{array}{l} | e (z) | ⩽ C_{γ} {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} (\frac{1}{{(ϵ a_{ϵ})}^{3}} + 1) | z + 1 |^{1 + \frac{γ}{2}}, \\ | \frac{d}{d z} e (z) | ⩽ C_{γ} {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} (\frac{1}{{(ϵ a_{ϵ})}^{3}} + 1) | z + 1 |^{\frac{γ}{2}}, and \\ | \frac{d^{2}}{d z^{2}} e (z) | ⩽ C_{γ} {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} (\frac{1}{{(ϵ a_{ϵ})}^{3}} + 1) | z + 1 |^{- 1 + \frac{γ}{2}}, \end{array}$ in $B_{r_{0}^{″}}^{+}$ (for any fixed $γ < 1$ ). We also note that $\begin{matrix} u^{ev} (x, y) = ℜ w (x + i y), (x, y) \in B_{r_{0}^{″}}^{+}, \end{matrix}$ with ℜ denoting the real part. We thus get $\begin{matrix} (21) & u^{ev} (x, y) = b_{0} + b_{1} r^{1 / 2} sin (θ / 2) + b_{2} r cos θ + e (x, y), \end{matrix}$ in $B_{r_{0}^{″}}^{+}$ , with $\begin{array}{l} | e (x, y) | ⩽ C_{γ} {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} (\frac{1}{{(ϵ a_{ϵ})}^{3}} + 1) r^{1 + \frac{γ}{2}}, \\ | \nabla e (x, y) | ⩽ C_{γ} {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} (\frac{1}{{(ϵ a_{ϵ})}^{3}} + 1) r^{\frac{γ}{2}}, \\ | D^{2} e (x, y) | ⩽ C_{γ} {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} (\frac{1}{{(ϵ a_{ϵ})}^{3}} + 1) r^{- 1 + \frac{γ}{2}}, \end{array}$ in $B_{r_{0}^{″}}^{+}$ . $(r, θ)$ denotes polar coordinates around $(- 1, 0)$ , and $r_{0}^{″} < r_{0}$ . By defining $u^{*} = b_{0} + b_{2} r cos θ + e (x, y)$ the decomposition (21) becomes $\begin{matrix} u^{ev} (x, y) = b_{1} r^{1 / 2} sin (θ / 2) + u^{*} (x, y), \end{matrix}$ with $\begin{array}{l} {‖ u^{*} ‖}_{C^{1, γ / 2} (\overline{B_{r_{0}^{″}}^{+}})} ⩽ C_{γ} {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} (\frac{1}{{(ϵ a_{ϵ})}^{3}} + 1), \\ (22) & | D^{2} u^{*} (x, y) | ⩽ C_{γ} {‖ {\tilde{φ}}^{ev} ‖}_{H^{1 / 2} (\partial \tilde{Ω})} (\frac{1}{{(ϵ a_{ϵ})}^{3}} + 1) r^{- 1 + \frac{γ}{2}} in B_{r_{0}^{″}}^{+}, and \\ u^{*} (- 1, 0) = b_{0}, \frac{\partial u^{*}}{\partial x} (- 1, 0) = b_{2} . \end{array}$ We may perform a similar analysis around the right endpoint of σ, $(1, 0)$ . Combining these two results with the fact that $u^{ev}$ is $C^{1 . γ / 2}$ in $\overline{{\tilde{Ω}}^{+}}$ (with a bound of $C ‖ {\tilde{φ}}^{ev} ‖_{H^{1 / 2} (\partial \tilde{Ω})}$ ) outside neighborhoods of $(- 1, 0)$ and $(1, 0)$ , we obtain the following result.

We shall need one additional property of $u^{*}$ , as asserted by the next lemma. Lemma 2.

Let $u^{*}$ be the remainder from Proposition 1 . This function satisfies $\begin{matrix} \frac{\partial}{\partial x} u^{*} (- 1, 0) = \frac{\partial}{\partial x} u^{*} (1, 0) = 0 . \end{matrix}$ In other words, its tangential derivatives vanish at the endpoints of the segment $σ = {- 1 < x < 1, y = 0}$ .

Proof of Lemma 2.

Let T denote a bounded linear operator $H^{1 / 2} (σ) \to H^{1} ({\tilde{Ω}}^{+}) \cap {u = 0 on {(\partial \tilde{Ω})}^{+}}$ , with the property that $T v |_{σ} = v$ . The variational formulation of the boundary value problem (9), i.e., the vanishing of the first variation of the energy functional (2) on $H_{ev}^{1} (\tilde{Ω}) \cap H^{1} (σ)$ , asserts that $\begin{array}{rcl} \int_{{\tilde{Ω}}^{+}} \nabla u^{ev} \nabla T v d x & = & - ϵ a_{ϵ} \int_{σ} \frac{\partial u^{ev}}{\partial x} \frac{\partial T v}{\partial x} d s \\ = & - ϵ a_{ϵ} \int_{σ} \frac{\partial u^{ev}}{\partial x} \frac{\partial v}{\partial x} d s \\ = & - ϵ a_{ϵ} \int_{σ} \frac{\partial u^{*}}{\partial x} \frac{\partial v}{\partial x} d s, \end{array}$ for any $v \in H^{1} (σ)$ . Here we used that $T v |_{σ} = v$ , and that $u^{ev} = u^{*}$ on σ. Reading from bottom to top and using that T is a continuous operator from $H^{1 / 2} (σ)$ to $H^{1} ({\tilde{Ω}}^{+})$ , we conclude that $\begin{matrix} (23) & the linear functional v \to \int_{σ} \frac{\partial u^{*}}{\partial x} \frac{\partial v}{\partial x} d s is continuously extendable to H^{1 / 2} (σ) . \end{matrix}$ Since $u^{*} |_{σ}$ is $C^{2}$ in the interior of σ and $C^{1, γ / 2}$ on the closure of σ and satisfies (22), we have that $\begin{matrix} (24) & \int_{σ} \frac{\partial u^{*}}{\partial x} \frac{\partial v}{\partial x} d s = - \int_{σ} \frac{\partial^{2} u^{*}}{\partial x^{2}} v d s + \frac{\partial u^{*}}{\partial x} v |_{x = - 1}^{x = 1}, \end{matrix}$ for any $v \in H^{1} (σ)$ . Concerning the first term in the right hand side we have, for any $0 < s < 1 / 2$ , $\begin{array}{rcl} | \int_{σ} \frac{\partial^{2} u^{*}}{\partial x^{2}} v d s | & ⩽ & \int_{σ} | \frac{\partial^{2} u^{*}}{\partial x^{2}} | | v | d s \\ ⩽ & C {(\int_{σ} dist {(x, \partial σ)}^{2 (- 1 + \frac{γ}{2} + s)})}^{1 / 2} {(\int_{σ} \frac{| v |^{2}}{dist {(x, \partial σ)}^{2 s}})}^{1 / 2} \\ ⩽ & C {(\int_{σ} dist {(x, \partial σ)}^{2 (- 1 + \frac{γ}{2} + s)})}^{1 / 2} ‖ v ‖_{H^{s} (σ)} . \end{array}$ Here we have used the fact that $\begin{matrix} | \frac{\partial^{2} u^{*}}{\partial x^{2}} (x, 0) | ⩽ C dist {((x, 0), \partial σ)}^{- 1 + \frac{γ}{2}}, - 1 < x < 1 \end{matrix}$ (see (22)), and the fact that $\begin{matrix} \int_{σ} \frac{| v (x) |^{2}}{dist {((x, 0), \partial σ)}^{2 s}} d x < C_{s} ‖ v ‖_{H^{s} (σ)}, 0 < s < 1 / 2 . \end{matrix}$ The constant in the first inequality depends on γ, ϵ, $a_{ϵ}$ and ${\tilde{φ}}^{ev}$ . The last inequality follows from the classical Hardy’s inequality combined with interpolation. Since γ can be taken arbitrarily close to 1, we may, for any given $s > 0$ , achieve that $\begin{matrix} \int_{σ} dist {(x, \partial σ)}^{2 (- 1 + \frac{γ}{2} + s)} < \infty . \end{matrix}$ It now follows that $\begin{matrix} | \int_{σ} \frac{\partial^{2} u^{*}}{\partial x^{2}} v d s | ⩽ C_{s} ‖ v ‖_{H^{s} (σ)}, 0 < s, \end{matrix}$ in other words: the first term in the right hand side of (24) is continuously extendable to $H^{s}$ for any $s > 0$ . Thus in particular: this term is continuously extendable to $H^{1 / 2} (σ)$ . According to (23), the left hand side of (24) is also continuously extendable to $H^{1 / 2} (σ)$ , and so it follows that $\begin{array}{l} the linear functional v \to \frac{\partial u^{*}}{\partial x} (1, 0) v (1) - \frac{\partial u^{*}}{\partial x} (- 1, 0) v (- 1) \\ is continuously extendable to H^{1 / 2} (σ) . \end{array}$ However, it is well-known that none of the trace operators $v \to v (\pm 1)$ are continuously extendable to $H^{1 / 2} (σ)$ , and so this latter statement can only hold provided $\begin{matrix} \frac{\partial u^{*}}{\partial x} (1, 0) = \frac{\partial u^{*}}{\partial x} (- 1, 0) = 0, \end{matrix}$ exactly as desired. □

Further regularity considerations

An analysis very similar to that from the previous section may be applied to the solution to $\begin{array}{l} - Δ u_{*}^{odd} = 0 in {\tilde{Ω}}^{+}, \\ (25) & \frac{\partial u_{*}^{odd}}{\partial y} = \frac{a_{ϵ}}{ϵ} u_{*}^{odd} on σ, \\ u_{*}^{odd} = 0 on {y = 0} ∖ σ, \end{array}$ with $\begin{matrix} u_{*}^{odd} = {\tilde{φ}}^{odd} on {(\partial \tilde{Ω})}^{+}, \end{matrix}$ where ${\tilde{φ}}^{odd} \in H^{1 / 2} (\partial \tilde{Ω})$ is defined on all of $\partial \tilde{Ω}$ , and odd in y. We note that $u_{*}^{odd}$ is a good approximation to $u^{odd}$ in energy (see Section 4 of part II of this paper [13], for the case $a_{ϵ} < M ϵ$ ). The regularity result associated with (25) is

We briefly sketch the analysis, but due to the similarities, we refer the reader back to the previous section for complete details. It suffices to analyze the behaviour near the left endpoint $(- 1, 0)$ of σ. Let W be the function which, in polar coordinates around $(- 1, 0)$ on the first “quadrant” $Q = {r < \sqrt{r_{0}}, 0 < θ < π / 2}$ , is given by $W (r, θ) = u_{*}^{odd} (r^{2}, 2 θ)$ , and extend W (using the same notation) as an odd function across $x = - 1$ to all of $B_{\sqrt{r_{0}}}^{+}$ . This function W now satisfies $\begin{matrix} (26) & \begin{matrix} - Δ W = 0 in B_{\sqrt{r_{0}}}^{+}, \\ \frac{\partial W}{\partial y} = \frac{a_{ϵ}}{ϵ} 2 | x + 1 | W (x, 0) on {y = 0} . \end{matrix} \end{matrix}$ It is easy to see that the fact that $u_{*}^{odd}$ is in $H^{1} (B_{r_{0}}^{+})$ (and bounded by $C ‖ {\tilde{φ}}^{odd} ‖_{H^{1 / 2} (\partial \tilde{Ω})}$ independently of ϵ and $a_{ϵ}$ ), implies that W is in $H^{1} (B_{\sqrt{r_{0}}}^{+})$ with a similar bound. The trace of W on $I_{\sqrt{r_{0}}} = (- \sqrt{r_{0}} - 1, - 1 + \sqrt{r_{0}})$ is thus in $H^{1 / 2} (I_{\sqrt{r_{0}}})$ , and so is $| x + 1 | W (x, 0)$ (bounded independently of ϵ and $a_{ϵ}$ ); a standard elliptic regularity result now immediately implies that $\begin{matrix} W lies in H^{2} (B_{r_{0}^{'}}^{+}), \end{matrix}$ for any $r_{0}^{'} < \sqrt{r_{0}}$ , with a bound of $C ‖ {\tilde{φ}}^{odd} ‖_{H^{1 / 2} (\partial \tilde{Ω})} (\frac{a_{ϵ}}{ϵ} + 1)$ . The trace of W on $I_{r_{0}^{'}}$ is thus in $H^{3 / 2} (I_{r_{0}^{'}})$ (and $W (- 1, 0) = 0$ , since W is odd around $x = - 1$ ). The derivative of $| x + 1 | W (x, 0)$ is given by $\begin{matrix} \frac{d}{d x} (| x + 1 | W (x, 0)) = \{\begin{matrix} (x + 1) \frac{d}{d x} W (x, 0) + W (x, 0) & for x > - 1, \\ - (x + 1) \frac{d}{d x} W (x, 0) - W (x, 0) & for x < - 1 . \end{matrix} \end{matrix}$ The function $(x + 1) \frac{d}{d x} W (x, 0) + W (x, 0)$ is in $H^{1 / 2} (- 1, - 1 + r_{0}^{'})$ as well as $H^{1 / 2} (- r_{0}^{'} - 1, - 1)$ , and since $W (- 1, 0) = 0$ the integrals $\begin{array}{l} \int_{- 1}^{- 1 + r_{0}^{'}} {[(x + 1) \frac{d}{d x} W (x, 0) + W (x, 0)]}^{2} / (x + 1) d x, and \\ \int_{- r_{0}^{'} - 1}^{- 1} {[(x + 1) \frac{d}{d x} W (x, 0) + W (x, 0)]}^{2} / (x + 1) d x \end{array}$ are both finite (and bounded by $C ‖ {\tilde{φ}}^{odd} ‖_{H^{1 / 2} (\partial \tilde{Ω})} (\frac{a_{ϵ}}{ϵ} + 1)$ ); it now follows that the derivative of $| x + 1 | W (x, 0)$ is in $H^{1 / 2} (I_{r_{0}^{'}})$ (cf. [19]). As a consequence $\begin{matrix} | x + 1 | W (x, 0) lies in H^{3 / 2} (I_{r_{0}^{'}}), \end{matrix}$ and is bounded by $C ‖ {\tilde{φ}}^{odd} ‖_{H^{1 / 2} (\partial \tilde{Ω})} (\frac{a_{ϵ}}{ϵ} + 1)$ . By one more iteration of the earlier local elliptic regularity argument it follows that $\begin{matrix} W lies in H^{3} (B_{r_{0}^{'}}^{+}), \end{matrix}$ for any $r_{0}^{'} < \sqrt{r_{0}}$ . Therefore the trace of W on $I_{r_{0}^{'}}$ lies in $H^{5 / 2} (I_{r_{0}^{'}})$ , and is bounded by $C ‖ {\tilde{φ}}^{odd} ‖_{H^{1 / 2} (\partial \tilde{Ω})} ({(\frac{a_{ϵ}}{ϵ})}^{2} + 1)$ . The second derivative of $| x + 1 | W (x, 0)$ is given by $\begin{matrix} \frac{d^{2}}{d x^{2}} (| x + 1 | W (x, 0)) = \{\begin{matrix} (x + 1) \frac{d^{2}}{d x^{2}} W (x, 0) + 2 \frac{d}{d x} W (x, 0) & for x > - 1, \\ - (x + 1) \frac{d^{2}}{d x^{2}} W (x, 0) - 2 \frac{d}{d x} W (x, 0) & for x < - 1 . \end{matrix} \end{matrix}$ The function $(x + 1) \frac{d^{2}}{d x^{2}} W (x, 0) + 2 \frac{d}{d x} W (x, 0)$ is in $H^{1 / 2} (- 1, - 1 + r_{0}^{'})$ as well as $H^{1 / 2} (- r_{0}^{'} - 1, - 1)$ , and the integrals $\begin{array}{l} \int_{- 1}^{- 1 + r_{0}^{'}} {[(x + 1) \frac{d^{2}}{d x^{2}} W (x, 0) + 2 \frac{d}{d x} W (x, 0)]}^{2} / | x + 1 |^{θ} d x and \\ \int_{- r_{0}^{'} - 1}^{- 1} {[(x + 1) \frac{d^{2}}{d x^{2}} W (x, 0) + 2 \frac{d}{d x} W (x, 0)]}^{2} / | x + 1 |^{θ} d x \end{array}$ are both finite for any $θ < 1$ , (and bounded by $C_{θ} ‖ {\tilde{φ}}^{odd} ‖_{H^{1 / 2} (\partial \tilde{Ω})} ({(\frac{a_{ϵ}}{ϵ})}^{2} + 1)$ ); it now follows that the second derivative of $| x + 1 | W (x, 0)$ is in $H^{θ / 2} (I_{r_{0}^{'}})$ for any $θ < 1$ . As a consequence $\begin{matrix} | x + 1 | W (x, 0) lies in H^{5 / 2 - δ} (I_{r_{0}^{'}}), \end{matrix}$ for any $δ > 0$ , and is bounded by $C_{δ} ‖ {\tilde{φ}}^{odd} ‖_{H^{1 / 2} (\partial \tilde{Ω})} ({(\frac{a_{ϵ}}{ϵ})}^{2} + 1)$ . By local elliptic regularity arguments it follows that $\begin{matrix} W lies in H^{4 - δ} (B_{r_{0}^{'}}^{+}), \end{matrix}$ for any $δ > 0$ , and is bounded by $C_{δ} ‖ {\tilde{φ}}^{odd} ‖_{H^{1 / 2} (\partial \tilde{Ω})} ({(\frac{a_{ϵ}}{ϵ})}^{3} + 1)$ . Let U denote the harmonic conjugate of W in $B_{r_{0}^{'}}$ , also selected so that $U (- 1, 0) = 0$ . Since $\begin{matrix} \nabla U = \nabla W^{⊥} \end{matrix}$ it follows that U lies in $H^{4 - δ} (B_{r_{0}^{'}}^{+})$ , for any $δ > 0$ (bounded by $C_{δ} ‖ {\tilde{φ}}^{odd} ‖_{H^{1 / 2} (\partial \tilde{Ω})} ({(\frac{a_{ϵ}}{ϵ})}^{3} + 1)$ ). The function W, given by $\begin{matrix} x + i y = z \to W (z) : = W (x, y) + i U (x, y), \end{matrix}$ is holomorphic in a $C_{+}$ neigborhood of $- 1$ , ( $B_{r_{0}^{'}}^{+}$ ), and $C^{2, γ}$ (for any $γ < 1$ ) in the closure of this neighborhood. It thus has a Taylor approximation of degree 2, in the sense that $\begin{matrix} W (z) = W (0) + B_{1} (z + 1) + \frac{1}{2} B_{2} {(z + 1)}^{2} + 0 (| z + 1 |^{2 + γ}) = B_{1} (z + 1) + \frac{1}{2} B_{2} {(z + 1)}^{2} + E (z), \end{matrix}$ and an analysis, identical to that in the proof of Proposition 1 of Appendix A, now completes the proof of Proposition 2.

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