$H^2$-regularity for a two-dimensional transmission problem with geometric constraint

Abstract

The $H^2$-regularity of variational solutions to a two-dimensional transmission problem with geometric constraint is investigated, in particular when part of the interface becomes part of the outer boundary of the domain due to the saturation of the geometric constraint. In such a situation, the domain includes some non-Lipschitz subdomains with cusp points, but it is shown that this feature does not lead to a regularity breakdown. Moreover, continuous dependence of the solutions with respect to the domain is established.

1 Introduction

The $H^2$-regularity of variational solutions to a two-dimensional transmission problem with geometric constraint is investigated, in particular when part of the interface becomes part of the outer boundary of the domain due to the geometric constraint, a situation in which the domain includes some non-Lipschitz subdomains with cusp points. Such a regularity is required in particular to guarantee that the variational solutions satisfy the strong formulation of the transmission problem. $H^2$-regularity is, however, not true in general and known to depend heavily on the geometry and smoothness of the domain and the interfaces. In fact, when interfaces intersect the outer boundary of the domain, regularity of variational solutions to transmission problems in non-smooth domains is a challenging issue, even for transversal intersections, see [1,2,3, 5, 10, 11, 13] and the references therein. Motivated by the mathematical study of microelectromechanical systems (MEMS), we identify herein a class of two-dimensional domains possibly featuring cusps for which $H^2$-regularity is true. We actually derive $H^2$-estimates which hold uniformly with respect to suitable perturbations of the underlying domain. We point out that such quantitative estimates are not contained in the above mentioned literature, but they turn out to be instrumental for a thorough study of MEMS models [9].

To set up the geometric framework, let $D:=(-L,L)$ be a finite interval of ${\mathbb {R}}$, $L>0$, and let $H>0$ and $d>0$ be two positive parameters. Given a function $u\in C({\bar{D}},[-H,\infty ))$ with $u(\pm L)=0$, we define the subdomain $\Omega (u)$ of $D\times (-H,\infty )$ by

$$\begin{aligned} \Omega (u) := \left\{ (x,z)\in D\times {\mathbb {R}} \,:\, -H< z < u(x)+d \right\} = \Omega _1(u)\cup \Omega _2(u)\cup \Sigma (u)\,, \end{aligned}$$

where

$$\begin{aligned} \Omega _1(u) := \left\{ (x, z)\in D\times {\mathbb {R}} \,:\, -H< z< {u}(x)\right\} \end{aligned}$$

and

$$\begin{aligned} \Omega _2(u):= \left\{ (x,z)\in D\times {\mathbb {R}}\,:\, u(x)< z < u(x)+d\right\} \end{aligned}$$

are separated by the interface

$$\begin{aligned} \Sigma (u) := \left\{ (x,z)\in D\times {\mathbb {R}}\,:\, z= u(x)>-H \right\} \,. \end{aligned}$$

Fig. 1

Geometry of $\Omega (v)$ for a state $v\in {\mathcal {S}}$ with empty coincidence set

Fig. 2

Geometry of $\Omega (w)$ for a state $w\in {\bar{{\mathcal {S}}}}$ with non-empty coincidence set

Owing to the (geometric) constraint $u\ge -H$, the lower boundary of $\Omega _2(u)$, given by the graph of the function u, cannot go beyond the lower boundary $D\times \{-H\}$ of $\Omega _1(u)$ but may coincide partly with it, along the so-called coincidence set

$$\begin{aligned} {\mathcal {C}}(u) := \{x\in D\,:\, u(x)=-H\}\,, \end{aligned}$$

(1.1)

see Figs. 1 and 2. Clearly, the geometry of $\Omega (u)$, as well as the regularity of its boundary, heavily depends on whether $\min _D\{u\}>-H$ or $\min _D\{u\}=-H$. Indeed, if $\min _D\{u\}>-H$ (i.e. the graph of u is strictly separated from $D\times \{-H\}$ as in Fig. 1), then the coincidence set ${\mathcal {C}}(u)$ is empty and $\Omega _1(u)$ is connected. In contrast, if $\min _D\{u\}=-H$ so that the graph of u intersects $D\times \{-H\}$, then ${\mathcal {C}}(u)\ne \emptyset$ and $\Omega _1(u)$ is disconnected with at least two (and possibly infinitely many) connected components, see Figs. 2 and 3.

For such a geometry, we study the regularity of variational solutions to the transmission problem

$$\begin{aligned} \mathrm {div}(\sigma \nabla \psi _u)&=0 \quad \text {in }\ \Omega (u)\,, \end{aligned}$$

(1.2a)

$$\begin{aligned} \llbracket \psi _u \rrbracket =\llbracket \sigma \nabla \psi _u \rrbracket \cdot {\mathbf {n}}_{ \Sigma (u)}&=0 \quad \text {on }\ \Sigma (u)\,, \end{aligned}$$

(1.2b)

$$\begin{aligned} \psi _u&=h_u\quad \text {on }\ \partial \Omega (u)\,, \end{aligned}$$

(1.2c)

where

$$\begin{aligned} \sigma := \sigma _1 {\mathbf {1}}_{\Omega _1(u)} + \sigma _2 {\mathbf {1}}_{\Omega _2(u)} \end{aligned}$$

for some positive constants $\sigma _1\ne \sigma _2$, and ${\mathbf {n}}_{ \Sigma (u)}$ denotes the unit normal vector field to $\Sigma (u)$ (pointing into ${\Omega }_2(u)$) given by

$$\begin{aligned} {\mathbf {n}}_{ \Sigma (u)}:=\frac{(-\partial _x u, 1)}{\sqrt{1+(\partial _x u)^2}}\,. \end{aligned}$$

In (1.2c), $h_u$ is a suitable function reflecting the boundary behavior of $\psi _u$, see Section 2 for details. In addition, $\llbracket \cdot \rrbracket$ denotes the (possible) jump across the interface $\Sigma (u)$; that is,

$$\begin{aligned} \llbracket f \rrbracket (x,u(x)) := f|_{\Omega _1(u)}(x,u(x)) - f|_{\Omega _2(u)}(x,u(x))\,, \qquad x\in D\,, \end{aligned}$$

whenever meaningful for a function $f:\Omega (u)\rightarrow {\mathbb {R}}$.

Let us already mention that there are several features of the specific geometry of $\Omega (u)$ which may hinder the $H^2$-regularity of the solution $\psi _u$ to (1.2). Indeed, on the one hand, the interface $\Sigma (u)$ always intersects with the boundary $\partial \Omega (u)$ of $\Omega (u)$ and it follows from [10] that this sole property prevents the $H^2$-regularity of $\psi _u$, unless $\sigma$ and the angles between $\Sigma (u)$ and $\partial \Omega (u)$ at the intersection points satisfy some additional conditions. On the other hand, $\Omega (u)$ and $\Omega _2(u)$ are at best Lipschitz domains, while $\Omega _1(u)$ may consist of non-Lipschitz domains with cusp points.

The particular geometry $\Omega (u)= \Omega _1(u)\cup \Omega _2(u)\cup \Sigma (u)$, in which the boundary value problem (1.2) is set, is encountered in the investigation of an idealized electrostatically actuated MEMS as already pointed out and described in detail in [6, 14]. Such a device consists of an elastic plate of thickness d which is fixed at its boundary $\{\pm L\} \times (0,d)$ and suspended above a rigid conducting ground plate located at $z=-H$. The elastic plate is made up of a dielectric material and deformed by a Coulomb force induced by holding the ground plate and the top of the elastic plate at different electrostatic potentials. In this context, u represents the vertical deflection of the bottom of the elastic plate, so that the elastic plate is given by $\Omega _2(u)$, while $\Omega _1(u)$ denotes the free space between the elastic plate and the ground plate. An important feature of the model is that the elastic plate cannot penetrate the ground plate, resulting in the geometric constraint $u\ge -H$. Still, a contact between the elastic plate and the ground plate – corresponding to a non-empty coincidence set ${\mathcal {C}}(u)$ – is explicitly allowed. The dielectric properties of $\Omega _1(u)$ and $\Omega _2(u)$ are characterized by positive constants $\sigma _1$ and $\sigma _2$, respectively. The electrostatic potential $\psi _u$ is then supposed to satisfy (1.2) and is completely determined by the deflection u. The state of the MEMS device is thus described by the deflection u, and equilibrium configurations of the device are obtained as critical points of the total energy which is the sum of the mechanical and electrostatic energies, the former being a functional of u while the latter is the Dirichlet integral of $\psi _u$. Owing to the nonlocal dependence of $\psi _u$ on u, minimizing the total energy and deriving the associated Euler-Lagrange equation demand quite precise information on the regularity of the electrostatic potential $\psi _u$ for an arbitrary, but fixed function u and its continuous dependence thereon. This first step of provisioning the required information is the main purpose of the present research. In the companion paper [9], we use the results obtained herein to analyze the minimizing problem leading to the determination of u and compute the associated Euler-Lagrange equation.

Since the regularity of the variational solution $\psi _u$ to (1.2) is intimately connected with the regularity of the boundaries of $\Omega (u)$, $\Omega _1(u)$, and $\Omega _2(u)$, let us first mention that $\Omega (u)$ and $\Omega _2(u)$ are always Lipschitz domains and that the measures of the angles at their vertices do not exceed $\pi$, a feature which complies with the $H^2$-regularity of $\psi _u$ away from the interface $\Sigma (u)$ [4]. This property is shared by $\Omega _1(u)$ when the coincidence set ${\mathcal {C}}(u)$ is empty, see Fig. 1, so that it is expected that $\psi |_{\Omega _i(u)}$ belongs to $H^2(\Omega _i(u))$, $i=1,2$, in that case. However, when ${\mathcal {C}}(u)$ is non-empty, the open set $\Omega _1(u)$ is no longer connected and the boundary of its connected components is no longer Lipschitz, but features cusp points. Moreover, there is an interplay between the transmission conditions (1.2b) and the boundary condition (1.2c) when ${\mathcal {C}}(u)\ne \emptyset$. Whether $\psi |_{\Omega _i(u)}$ still belongs to $H^2(\Omega _i(u))$, $i=1,2$, in this situation is thus an interesting question, that we answer positively in our first result. For the precise statement, we introduce the functional setting we shall work with in the sequel. Specifically, we set

$$\begin{aligned} {\bar{{\mathcal {S}}}} := \{v\in H^2(D) \cap H_0^1(D)\,:\, v\ge -H \text { in } D \;\text { and }\; \pm \llbracket \sigma \rrbracket \partial _x v(\pm L) \le 0 \} \end{aligned}$$

and

$$\begin{aligned} {\mathcal {S}}:=\{v\in H^2(D) \cap H_0^1(D)\,:\, v> -H \text { in } D \;\text { and }\; \pm \llbracket \sigma \rrbracket \partial _x v(\pm L) \le 0 \}\,. \end{aligned}$$

Clearly, the coincidence set ${\mathcal {C}}(u)$ is empty if and only if $u\in {\mathcal {S}}$. In addition, the situation already alluded to, where ${\mathcal {C}}(u)$ is non-empty and $\Omega _1(u)$ is a disconnected open set in $\mathbb {R}^2$ with a non-Lipschitz boundary, corresponds to functions $u\in {\bar{\mathcal {S}}}{\setminus } {\mathcal {S}}$. Also, we include the constraint $\pm \llbracket \sigma \rrbracket \partial _x u(\pm L) \le 0$ in the definition of ${\mathcal {S}}$ and ${\bar{\mathcal {S}}}$ to guarantee that the way $\Sigma (u)$ and $\partial \Omega (u)$ intersect does not prevent the $H^2$-regularity of $\psi _u$ in smooth situations (i.e. $u\in {\mathcal {S}}\cap W_\infty ^2(D)$), see [10].

Theorem 1.1

Suppose (2.1) below.

• (a) For each $u\in {\bar{\mathcal {S}}}$, there is a unique variational solution $\psi _u \in h_{u}+H_{0}^1(\Omega (u))$ to (1.2). Moreover, $\psi _{u,1}:= \psi _{u}|_{\Omega _1(u)} \in H^2(\Omega _1(u))$ and $\psi _{u,2} := \psi _{u}|_{\Omega _2(u)} \in H^2(\Omega _2(u))$, and $\psi _{u}$ is a strong solution to the transmission problem (1.2).

• (b) Given $\kappa >0$, there is $c(\kappa )>0$ such that, for every $u\in {\bar{\mathcal {S}}}$ satisfying $\Vert u\Vert _{H^2(D)}\le \kappa$,

  $$\begin{aligned} \Vert \psi _u\Vert _{H^1(\Omega (u))} + \Vert \psi _{u,1}\Vert _{H^2(\Omega _1(u))} + \Vert \psi _{u,2}\Vert _{H^2(\Omega _2(u))} \le c(\kappa )\,. \end{aligned}$$

It is worth emphasizing that, for $i\in \{1,2\}$, the restriction of $\psi _u$ to $\Omega _i(u)$ belongs to $H^2(\Omega _i(u))$ for all $u\in {\bar{\mathcal {S}}}$. In particular, there is no regularity breakdown when the coincidence set ${\mathcal {C}}(u)$ is non-empty. Moreover, the $H^2$-regularity of $\psi _u$ is uniformly valid when u ranges in a bounded subset of ${\bar{S}}$. A similar observation is made in [7] for a different geometric setting when one of the two subsets does not depend on the function u. Identifying other non-smooth geometries for which $H^2$-regularity of the variational solution to a transmission problem depends in a somewhat uniform way on some specific features of the domain is an interesting issue, which is worth a forthcoming investigation.

Remark 1.2

When the upper part $\Omega _2(v)$ is clamped at its lateral boundaries in the sense that

$$\begin{aligned} u\in H_0^2(D) := \{v\in H^2(D)\cap H_0^1(D)\,:\, \partial _x v(\pm L) = 0\}\,, \end{aligned}$$

Theorem 1.1 applies whatever the values of $\sigma _1$ and $\sigma _2$.

Theorem 1.1 is an immediate consequence of Proposition 4.9 below. Its proof begins with quantitative $H^2$-estimates on $\psi _u$ depending only on $\Vert u\Vert _{H^2(D)}$ for sufficiently smooth functions in ${\mathcal {S}}$, the $H^2$-regularity of $\psi _u$ being guaranteed by [10] in that case. Since the class of functions for which these estimates are valid is dense in ${\bar{\mathcal {S}}}$, we complete the proof with a compactness argument, the main difficulty to be faced being the dependence of $\Omega (u)$ on u.

More precisely, we begin with a variational approach to (1.2) and first show in Section 3 by classical arguments that, given $u\in {\bar{\mathcal {S}}}$, the variational solution $\psi _{u}$ to (1.2) corresponds to the minimizer on $h_{u}+H_{0}^1(\Omega (u))$ of the associated Dirichlet energy

$$\begin{aligned} {\mathcal {J}}(u)[\theta ]:= \frac{1}{2}\int _{\Omega (u)} \sigma |\nabla \theta |^2\,\mathrm {d}(x,z)\,, \qquad \theta \in h_u + H_0^1(\Omega (u))\,. \end{aligned}$$

Thanks to this characterization, we use $\Gamma$-convergence tools to show the $H^1$-stability of $\psi _u$ with respect to u in Sect. 3.2. Section 4 is devoted to the study of the $H^2$-regularity of $\psi _u$ which we first establish in Sect. 4.1 for smooth functions $u\in {\mathcal {S}}\cap W_\infty ^2(D)$ (thus having an empty coincidence set), relying on the analysis performed in [10]. It is worth mentioning that the constraint involving $\llbracket \sigma \rrbracket$ in the definition of ${\mathcal {S}}$ comes into play here. For $u\in {\mathcal {S}}\cap W_\infty ^2(D)$, we next derive quantitative $H^2$-estimates on $\psi _u$ which only depend on $\Vert u\Vert _{H^2(D)}$ as stated in Theorem 1.1 (b), see Sect. 4.2. The building block is an identity in the spirit of [4, Lemma 4.3.1.2] allowing us to interchange derivatives with respect to x and z in some integrals involving second-order derivatives, its proof being provided in Appendix 1. We then combine these estimates with the already proved $H^1$-stability of variational solutions to (1.2) and use a compactness argument to extend the $H^2$-regularity of $\psi _u$ to arbitrary functions $u\in {\bar{\mathcal {S}}}$ in Sect. 4.3. In this step, special care is required to cope with the variation of the functional spaces with u. In fact, as a side product of the proof of Theorem 1.1, we obtain qualitative information on the continuous dependence of $\psi _u$ with respect to u, which we collect in the next result.

Theorem 1.3

Suppose (2.1) below. Let $\kappa >0$, $u\in {\bar{\mathcal {S}}}$, and consider a sequence $(u_n)_{n\ge 1}$ in ${\bar{\mathcal {S}}}$ such that

$$\begin{aligned} \Vert u_n\Vert _{H^2(D)}\le \kappa \,, \quad n\ge 1\,, \qquad \lim _{n\rightarrow \infty } \Vert u_n-u\Vert _{H^1(D)} = 0\,. \end{aligned}$$

(1.3)

Setting $M := d + \max \left\{ \Vert u\Vert _{L_\infty (D)} \,,\, \sup _{n\ge 1}\{\Vert u_n\Vert _{L_\infty (D)}\} \right\}$,

$$\begin{aligned} \lim _{n\rightarrow \infty } \big \Vert (\psi _{u_n} - h_{u_n}) -(\psi _{u} - h_{u}) \big \Vert _{H^1(\Omega _M)} = 0\,. \end{aligned}$$

(1.4a)

In addition, if $i\in \{1,2\}$ and $U_i$ is an open subset of $\Omega _i(u)$ such that ${\bar{U}}_i$ is a compact subset of $\Omega _i(u)$, then

$$\begin{aligned} \psi _{u_n,i}\rightharpoonup \psi _{u,i} \quad \text {in}\quad H^2(U_i)\,. \end{aligned}$$

(1.4b)

Also, for any $p\in [1,\infty )$,

$$\begin{aligned} \begin{aligned} \lim _{n\rightarrow \infty } \big \Vert \nabla \psi _{u_n,2}(\cdot ,u_n) - \nabla \psi _{u,2}(\cdot ,u) \big \Vert _{L_p(D,\mathbb {R}^2)}&= 0 \,, \\ \lim _{n\rightarrow \infty } \big \Vert \nabla \psi _{u_n,2}(\cdot ,u_n+d) - \nabla \psi _{u,2}(\cdot ,u+d) \big \Vert _{L_p(D,\mathbb {R}^2)}&= 0 \,. \end{aligned} \end{aligned}$$

(1.4c)

Clearly, the quantity M introduced in Theorem 1.3 is finite due to (1.3) and the continuous embedding of $H^1(D)$ in $C({\bar{D}})$.

Remark 1.4

An interesting issue is the extension of the above results to a three-dimensional setting, where D is a bounded domain of ${\mathbb {R}}^2$ instead of an interval. There are, however, at least two difficulties to overcome, which are both of geometric nature. On the one hand, the coincidence set ${\mathcal {C}}(u)$ defined in (1.1) is no longer a countable union of open intervals when D is a two-dimensional domain and it might have a much more complicated structure. The former property plays an essential role in the proof of Proposition 4.9 (a) below. On the other hand, the $\Gamma$-convergence argument involved in the proof of Proposition 3.3 strongly makes use of the two-dimensional geometry of $\Omega (u)$. In fact, the literature on regularity of solutions to transmission problems in non-smooth three-dimensional domains when the interfaces intersect the outer boundary seems to be rather sparse and restricted to specific geometries. We refer to [1, 3, 5, 11, 13] for results in that direction.

Notation Given $v\in {\bar{\mathcal {S}}}$, $f\in L_2(\Omega (v))$, and $i\in \{1,2\}$, we denote the restriction of f to $\Omega _i(v)$ by $f_i$; that is, $f_i := f|_{\Omega _i(v)}$.

Throughout the paper, c and $(c_k)_{k\ge 1}$ denote positive constants depending only on L, H, d, $\sigma _1$, and $\sigma _2$. The dependence upon additional parameters will be indicated explicitly.

2 The boundary values

We state the precise assumptions on the function $h_v$ occurring in (1.2c). Roughly speaking, we assume that it is the trace on $\partial \Omega (v)$ of a function $h_v\in H^1(\Omega (v))$ which is such that $h|_{\Omega _i(v)}$ belongs to $H^2(\Omega _i(v))$ for $i=1,2$ and satisfies the transmission conditions (1.2b), as well as suitable boundedness and continuity properties with respect to v.

Specifically, for every $v\in {\bar{\mathcal {S}}}$, let

$$\begin{aligned} h_v: D\times (-H,\infty )\rightarrow \mathbb {R}\end{aligned}$$

be such that

$$\begin{aligned} h_v\in H^1(\Omega (v))\,,\qquad h_{v,i}:=h_v\vert _{\Omega _i(v)}\in H^2\big (\Omega _i(v)\big )\,,\quad i=1,2\,, \end{aligned}$$

(2.1a)

and suppose that $h_v$ satisfies the transmission conditions

$$\begin{aligned} \llbracket h_v\rrbracket =\llbracket \sigma \nabla h_v\rrbracket \cdot \mathbf{n}_{\Sigma (v)}=0 \ \text { on }\ \Sigma (v)\,. \end{aligned}$$

(2.1b)

For $\kappa >0$ given, there is $c(\kappa )>0$ such that, for all $v\in {\bar{\mathcal {S}}}$ satisfying $\Vert v\Vert _{H^2(D)}\le \kappa$,

$$\begin{aligned} \Vert h_{v,i}\Vert _{H^2(\Omega _i(v))}\le c(\kappa )\,, \quad i=1,2\,. \end{aligned}$$

(2.1c)

Moreover, given $v\in {\bar{\mathcal {S}}}$ and a sequence $(v_n)_{n\ge 1}$ in ${\bar{\mathcal {S}}}$ satisfying

$$\begin{aligned} \lim _{n\rightarrow \infty } \Vert v_n - v\Vert _{H^1(D)} = 0\,, \end{aligned}$$

we assume that

$$\begin{aligned} \lim _{n\rightarrow \infty } \Vert h_{v_n} - h_{v}\Vert _{H^1(D\times (-H,M))}=0 \end{aligned}$$

(2.1d)

and

$$\begin{aligned} \lim _{n\rightarrow \infty } \Vert h_{v_n}(\cdot , v_n+d) - h_{v}(\cdot , v+d)\Vert _{C({\bar{D}})} = 0\,, \end{aligned}$$

(2.1e)

where

$$\begin{aligned} M := d + \max \left\{ \Vert v\Vert _{L_\infty (D)} \,,\, \sup _{n\ge 1}\{\Vert v_n\Vert _{L_\infty (D)}\} \right\} <\infty \,. \end{aligned}$$

Observe that the convergence of $(v_n)_{n\ge 1}$, the continuous embedding of $H^1(D)$ in $C({\bar{D}})$, and (2.1d) imply that

$$\begin{aligned} \lim _{n\rightarrow \infty } \int _{\Omega (v_n)}\sigma \vert \nabla h_{v_n}\vert ^2\,\mathrm {d}(x,z)= \int _{\Omega (v)}\sigma \vert \nabla h_{v}\vert ^2\,\mathrm {d}(x,z)\,. \end{aligned}$$

(2.2)

From now on, we impose the conditions (2.1) throughout.

We finish this short section by providing an example of $h_v$ satisfying the imposed conditions (2.1).

Example 2.1

Let $\zeta \in C^2(\mathbb {R})$ be such that $\zeta |_{(-\infty ,1]}\equiv 0$ and $\zeta |_{[1+d,\infty )} \equiv V$ for some $V>0$. Given $v\in {\bar{\mathcal {S}}}$, put

$$\begin{aligned} h_v(x,z) :=\zeta (z- v(x)+1)\,, \qquad -H \le z \,, \quad x\in {\bar{D}}\,. \end{aligned}$$

(2.3)

Then (2.1a)–(2.1e) are satisfied. In addition,

$$\begin{aligned} h_v(x,-H)=0\,, \quad h_v(x,v(x)+d)=V\,,\qquad x\in D\,. \end{aligned}$$

In the context of a MEMS device alluded to in the introduction, these additional properties mean that the ground plate and the top of the elastic plate are kept at constant potential. For instance, $\zeta (r):=V\min \{1,(r-1)^2/d^2\}$ for $r>1$ and $\zeta \equiv 0$ on $(-\infty ,1]$ will do.

3 Variational solution to (1.2)

In this section we investigate the properties of the variational solution $\psi _v$ to (1.2) for $v\in {\bar{\mathcal {S}}}$ and, in particular, its $H^1$-stability.

3.1 A variational approach to (1.2)

Given $v\in {\bar{\mathcal {S}}}$ we introduce the set of admissible potentials

$$\begin{aligned} {\mathcal {A}}(v):=h_{v}+H_{0}^1(\Omega (v))\,, \end{aligned}$$

on which we define the functional

$$\begin{aligned} {\mathcal {J}}(v)[\theta ]:=\frac{1}{2}\int _{ \Omega (v)} \sigma \vert \nabla \theta \vert ^2\,\mathrm {d}(x,z)\,,\qquad \theta \in {\mathcal {A}}(v)\,. \end{aligned}$$

(3.1)

The variational solution $\psi _v$ to the transmission problem (1.2) is then the minimizer of the functional ${\mathcal {J}}(v)$ on the set ${\mathcal {A}}(v)$:

Lemma 3.1

For each $v\in {\bar{\mathcal {S}}}$ there is a unique minimizer $\psi _v \in {\mathcal {A}}(v)$ of ${\mathcal {J}}(v)$ on ${\mathcal {A}}(v)$; that is,

$$\begin{aligned} {\mathcal {J}}(v)[\psi _v] = \min _{\theta \in {\mathcal {A}}(v)} {\mathcal {J}}(v)[\theta ]\,. \end{aligned}$$

(3.2)

In addition,

$$\begin{aligned} \int _{\Omega (v)}\sigma \vert \nabla \psi _v\vert ^2\,\mathrm {d}(x,z)\le \int _{\Omega (v)}\sigma \vert \nabla h_v\vert ^2\,\mathrm {d}(x,z)\,. \end{aligned}$$

(3.3)

Proof

Let $v\in {\bar{\mathcal {S}}}$ and recall that $h_v\in H^1(\Omega (v))$ according to (2.1a). Thus, the existence of a minimizer $\psi _v$ of ${\mathcal {J}}(v)$ on ${\mathcal {A}}(v)$ readily follows from the direct method of calculus of variations due to the lower semicontinuity and coercivity of ${\mathcal {J}}(v)$ on ${\mathcal {A}}(v)$, the latter being ensured by the assumption $\sigma \ge \min \{\sigma _1,\sigma _2\}>0$ and Poincaré’s inequality. The uniqueness of $\psi _v$ is guaranteed by the strict convexity of ${\mathcal {J}}(v)$. Next, since obviously $h_v\in {\mathcal {A}}(v)$, the inequality (3.3) is an immediate consequence of the minimizing property (3.2) of $\psi _v$. $\square$

For further use, we report the following version of Poincaré’s inequality for functions in $H^1_0(\Omega (v))$ with a constant depending mildly on $v\in \bar{{\mathcal {S}}}$.

Lemma 3.2

Let $v\in \bar{{\mathcal {S}}}$ and $\theta \in H_0^1(\Omega (v))$. Then

$$\begin{aligned} \Vert \theta \Vert _{L_2(\Omega (v))} \le 2 \Vert H+d+v\Vert _{L_\infty (D)} \Vert \partial _z \theta \Vert _{L_2(\Omega (v))}\,. \end{aligned}$$

Proof

For $x\in D$ and $z\in (-H,v(x)+d)$,

$$\begin{aligned} \theta (x,z)^2 = 2 \int _{-H}^z \theta (x,y) \partial _z\theta (x,y)\ \mathrm {d}y\,. \end{aligned}$$

Hence, after integration with respect to (x, z) over $\Omega (v)$,

$$\begin{aligned} \Vert \theta \Vert _{L_2(\Omega (v))}^2&= \int _{\Omega (v)} \theta (x,z)^2\ \mathrm {d}(x,z) \\&\le 2 \Vert H+d+v\Vert _{L_\infty (D)} \int _{\Omega (v)} |\theta (x,y)| |\partial _z \theta (x,y)|\ \mathrm {d}(x,z) \\&\le 2 \Vert H+d+v\Vert _{L_\infty (D)} \Vert \theta \Vert _{L_2(\Omega (v))} \Vert \partial _z \theta \Vert _{L_2(\Omega (v))}\,, \end{aligned}$$

from which we deduce the stated inequality. $\square$

3.2 $H^1$-stability of $\psi _v$

The purpose of this section is to study the continuity properties of the solution $\psi _v$ to (3.2) with respect to v. More precisely, we aim at establishing the following result.

Proposition 3.3

Consider $v\in \bar{{\mathcal {S}}}$ and a sequence $(v_n)_{n\ge 1}$ in $\bar{{\mathcal {S}}}$ such that

$$\begin{aligned} v_n\rightarrow v \ \text { in }\ H_0^1(D)\,, \end{aligned}$$

(3.4)

and set

$$\begin{aligned} M := d + \max \left\{ \Vert v\Vert _{L_\infty (D)} \,,\, \sup _{n\ge 1}\{\Vert v_n\Vert _{L_\infty (D)}\} \right\} \,, \end{aligned}$$

(3.5)

which is finite by (3.4) and the continuous embedding of $H^1(D)$ in $C({\bar{D}})$. Then

$$\begin{aligned} \lim _{n\rightarrow \infty } \left\| (\psi _{v_n}-h_{v_n}) - (\psi _v - h_v) \right\| _{H_0^1(D\times (-H,M))} = 0 \end{aligned}$$

and

$$\begin{aligned} \lim _{n\rightarrow \infty } {\mathcal {J}}(v_n)[\psi _{v_n}] = {\mathcal {J}}(v)[\psi _v]\,. \end{aligned}$$

To prove Proposition 3.3, we make use of a $\Gamma$-convergence approach and argue as in [7, Section 3.2] with minor changes. We thus omit the proof here and refer to the extended version of this paper [8] for details.

4 $H^2$-regularity

In the previous section we introduced the variational solution $\psi _v\in H^1(\Omega (v))$ to (1.2) for arbitrary $v\in \bar{{\mathcal {S}}}$ and noticed its continuous dependence in $H^1(\Omega (v))$ with respect to v. We now aim at improving the $H^1$-regularity of $\psi _v|_{\Omega _i(v)}$ to $H^2(\Omega _i(v))$ for $i=1,2$. To this end we first consider the case of smooth functions $v\in {\mathcal {S}} \cap W_\infty ^2(D)$ with empty coincidence sets and provide in Sects. 4.1 and 4.2 the corresponding $H^2$-estimates that depend only on the norm of v in $H^2(D)$ (but not on its $W_\infty ^2(D)$-norm). In Sect. 4.3 we extend these estimates to the general case $v\in \bar{{\mathcal {S}}}$ by means of a compactness argument.

4.1 $H^2$-regularity for $v\in {\mathcal {S}}\cap W_\infty ^2(D)$

Assuming that v is smoother with an empty coincidence set, see Fig. 1, the existence of a strong solution $\psi _v$ to (1.2) is a consequence of the analysis performed in [10].

Proposition 4.1

If $v\in {\mathcal {S}} \cap W_\infty ^2(D)$, then the variational solution $\psi _v$ to (3.2) satisfies

$$\begin{aligned} \psi _{v,i} := \psi _v|_{\Omega _i(v)} \in H^2(\Omega _i(v))\,, \quad i=1,2\,, \end{aligned}$$

and the transmission problem

$$\begin{aligned} \mathrm {div}(\sigma \nabla \psi _v)&=0 \quad \ \text {in }\ \Omega (v)\,, \end{aligned}$$

(4.1a)

$$\begin{aligned} \llbracket \psi _v \rrbracket =\llbracket \sigma \nabla \psi _v \rrbracket \cdot {\mathbf {n}}_{ \Sigma (v)}&=0\quad \ \text {on }\ \Sigma (v)\,, \end{aligned}$$

(4.1b)

$$\begin{aligned} \psi _v&=h_v\quad \text {on }\ \partial \Omega (v)\,. \end{aligned}$$

(4.1c)

Moreover, $\partial _x \psi _v + \partial _x v \partial _z \psi _v$ and $-\sigma \partial _x v \partial _x \psi _v +\sigma \partial _z\psi _v$ both belong to $H^1(\Omega (v))$.

Besides [10], the proof of Proposition 4.1 requires the following auxiliary result.

Lemma 4.2

Let $v\in \bar{{\mathcal {S}}}$ and consider $\phi \in L_2(\Omega (v))$ such that

$$\begin{aligned} \phi _i := \phi |_{\Omega _i(v)} \in H^1(\Omega _i(v))\,, \quad i=1,2\,, \end{aligned}$$

and $\llbracket \phi \rrbracket =0$ on $\Sigma (v)$. Then $\phi \in H^1(\Omega (v))$ and

$$\begin{aligned} \Vert \phi \Vert _{H^1(\Omega (v))} \le \Vert \phi _1\Vert _{H^1(\Omega _1(v))} + \Vert \phi _2\Vert _{H^1(\Omega _2(v))} \,. \end{aligned}$$

(4.2)

Proof

We set $e_x=(1,0)$ and $e_z=(0,1)$. Given $\theta \in C_c^\infty \big (\Omega (v)\big )$ and $j\in \{x,z\}$ we note that

$$\begin{aligned} \begin{aligned} \int _{\Omega (v)}\phi \partial _j\theta \,\mathrm {d}(x,z)&= \int _{\Omega (v)} \mathrm {div}(\phi \theta e_j)\, \mathrm {d}(x,z) - \sum _{i=1}^2 \int _{\Omega _i(v)} \theta \partial _j \phi _i\,\mathrm {d}(x,z)\\&=\int _{\Sigma (v)} \llbracket \phi \rrbracket \,\theta e_j\cdot {\mathbf {n}}_{ \Sigma (v)}\,\mathrm {d}\sigma _{\Sigma (v)} - \sum _{i=1}^2 \int _{\Omega _i(v)} \theta \partial _j \phi _i\,\mathrm {d}(x,z)\,, \end{aligned} \end{aligned}$$

due to Gauß’ theorem. Thus, since $\llbracket \phi \rrbracket =0$ on $\Sigma (v)$,

$$\begin{aligned} \left| \int _{\Omega (v)}\phi \partial _j\theta \,\mathrm {d}(x,z)\right| \le \big (\Vert \phi _1\Vert _{H^1(\Omega _1(v))}+\Vert \phi _2\Vert _{H^1(\Omega _2(v))}\big )\,\Vert \theta \Vert _{L_2(\Omega (v))}\,, \end{aligned}$$

for $j=x,z$ and $\theta \in C_c^\infty \big (\Omega (v)\big )$. Consequently, $\phi \in H^1(\Omega (v))$. $\square$

Proof of Proposition 4.1

We check that the transmission problem (4.1) fits into the framework of [10]. Since $v\in {\mathcal {S}} \cap W_\infty ^2(D)$ and $v(\pm L) = 0$, the boundaries of $\Omega _1(v)$ and $\Omega _2(v)$ are $W_\infty ^2$-smooth curvilinear polygons and the interface $\Sigma (v)$ meets the boundary $\partial \Omega (v)$ of $\Omega (v)$ at the vertices $A_\pm := (\pm L,0)$. Moreover, at the vertex $A_\pm$, the measures $\omega _{\pm ,1}$ and $\omega _{\pm ,2}$ of the angles between $-e_z$ and $(1,\mp \partial _x v(\pm L))$ and between $(1,\mp \partial _x v(\pm L))$ and $e_z$, respectively, satisfy $\omega _{\pm ,1}+\omega _{\pm ,2} = \pi$, as well as

$$\begin{aligned} \begin{aligned} \omega _{\pm ,2}&\ge \frac{\pi }{2} \;\text { if }\; \llbracket \sigma \rrbracket < 0\,, \\ \omega _{\pm ,2}&\le \frac{\pi }{2} \;\text { if }\; \llbracket \sigma \rrbracket > 0\,, \end{aligned} \end{aligned}$$

by definition of ${\mathcal {S}}$. According to the analysis performed in [10], these conditions guarantee that the variational solution $\psi _v$ to (3.2) provided by Lemma 3.1 satisfies $\psi _{v,i} = \psi _v|_{\Omega _i(v)} \in H^2(\Omega _i(v))$ for $i=1,2$ and solves the transmission problem (1.2) in a strong sense.

Next, owing to the just established $H^2$-regularity of $\psi _{v,1}$ and $\psi _{v,2}$, we may differentiate with respect to x the transmission condition $\llbracket \psi _v \rrbracket (x,v(x))=0$, $x\in D$, and find that

$$\begin{aligned} \llbracket \partial _x \psi _v + \partial _x v \partial _z \psi _v \rrbracket = 0 \quad \ \text { on }\ \Sigma (v)\,. \end{aligned}$$

The stated $H^1$-regularity of $\partial _x \psi _v + \partial _x v \partial _z \psi _v$ then follows from Lemma 4.2 and the boundedness of $\partial _x v$ and $\partial _x^2 v$. In the same vein, due to (1.2b), the regularity of v, and the identity

$$\begin{aligned} \frac{\llbracket - \sigma \partial _x v \partial _x \psi _v + \sigma \partial _z \psi _v \rrbracket }{\sqrt{1+(\partial _x v)^2}} = \llbracket \sigma \nabla \psi _v \rrbracket \cdot {\mathbf {n}}_{ \Sigma (v)} = 0\,, \end{aligned}$$

the claimed $H^1$-regularity of $- \sigma \partial _x v \partial _x \psi _v + \sigma \partial _z \psi _v$ is again a consequence of Lemma 4.2 and the boundedness of $\partial _x v$ and $\partial _x^2 v$. $\square$

4.2 $H^2$-Estimates on $\psi _v$ for $v\in {\mathcal {S}} \cap W_\infty ^2(D)$

The $H^2$-regularity of $\psi _v$ being guaranteed by Proposition 4.1 for $v\in {\mathcal {S}}\cap W_\infty ^2(D)$, the next step is to show that this property extends to any $v\in \bar{{\mathcal {S}}}$. To this end, we shall now derive quantitative $H^2$-estimates on $\psi _v$, paying special attention to their dependence upon the regularity of v. As in [7], it turns out to be more convenient to study a non-homogeneous transmission problem with homogeneous Dirichlet boundary conditions instead of (4.1). Specifically, for $v\in {\mathcal {S}}\cap W_\infty ^2(D)$, we define

$$\begin{aligned} \chi =\chi _v:=\psi _v-h_v\in H_{0}^1(\Omega (v))\,, \end{aligned}$$

(4.3)

where $\psi _v\in H^1(\Omega (v))$ is the unique solution to (4.1) provided by Proposition 4.1. Since $\psi _{v,i}=\psi _v|_{\Omega _i(v)}$ belongs to $H^2(\Omega _i(v))$ for $i=1, 2$, we readily infer from (2.1a) and (4.3) that

$$\begin{aligned} \chi _{i} := \chi _{v}|_{\Omega _i(v)}\in H^2(\Omega _i(v))\,, \quad i=1,2\,. \end{aligned}$$

(4.4)

We omit in the following the dependence of $\chi$ on v for ease of notation.

According to (2.1a), (2.1b), and Proposition 4.1, $\chi$ solves the transmission problem

$$\begin{aligned} \mathrm {div}(\sigma \nabla \chi )&= - \mathrm {div}(\sigma \nabla h_v) \quad \text {in }\ \Omega (v)\,, \end{aligned}$$

(4.5a)

$$\begin{aligned} \llbracket \chi \rrbracket = \llbracket \sigma \nabla \chi \rrbracket \cdot \mathbf{n}_{\Sigma (v)}&=0\quad \text {on }\ \Sigma (v)\,, \end{aligned}$$

(4.5b)

$$\begin{aligned} \chi&=0\quad \text {on }\ \partial \Omega (v)\,, \end{aligned}$$

(4.5c)

and it follows from (2.1a) that it is equivalent to derive $H^2$-estimates on $(\psi _{v,1},\psi _{v,2})$ or $(\chi _1,\chi _2)$.

For that purpose, we transform (4.5) to a transmission problem on the rectangle ${\mathcal {R}}:=D\times (0,1+d)$. More precisely, we introduce the transformation

$$\begin{aligned} T_1(x,z):=\left( x,\frac{z+H}{v(x)+H}\right) \,,\qquad (x,z)\in \Omega _1(v)\,, \end{aligned}$$

(4.6)

mapping $\Omega _1(v)$ onto the rectangle ${\mathcal {R}}_1:=D\times (0,1)$, and the transformation

$$\begin{aligned} T_2(x,z):=\left( x,z-v(x)+1\right) \,,\qquad (x,z)\in \Omega _2(v)\,, \end{aligned}$$

(4.7)

mapping $\Omega _2(v)$ onto the rectangle ${\mathcal {R}}_2:=D\times (1,1+d)$. The interface separating ${\mathcal {R}}_1$ and ${\mathcal {R}}_2$ is

$$\begin{aligned} \Sigma _0:=D\times \{1\}\,, \end{aligned}$$

so that

$$\begin{aligned} {\mathcal {R}}=D\times (0,1+d)={\mathcal {R}}_1\cup {\mathcal {R}}_2\cup \Sigma _0\,. \end{aligned}$$

It is worth pointing out here that $T_1$ is well-defined due to $v\in {\mathcal {S}}$. Let $(x,\eta )$ denote the new variables in ${\mathcal {R}}$; that is, $(x,\eta )=T_1(x,z)$ for $(x,z)\in {\mathcal {R}}_1$ and $(x,\eta )=T_2(x,z)$ for $(x,z)\in {\mathcal {R}}_2$. Then, (4.4) implies

$$\begin{aligned} \Phi :=\Phi _1 {\mathbf {1}}_{{\mathcal {R}}_1} + \Phi _2 {\mathbf {1}}_{{\mathcal {R}}_2}\in H_0^1({\mathcal {R}})\,,\qquad \Phi _i:= \chi _i\circ T_i^{-1}\in H^2({\mathcal {R}}_i)\,,\quad i=1,2\,.\qquad \end{aligned}$$

(4.8)

For further use, we also introduce

$$\begin{aligned} {{\hat{\sigma }}}(x,\eta ):=\left\{ \begin{array}{cl} \dfrac{\sigma _1}{v(x)+H}\,, &{} (x,\eta )\in {\mathcal {R}}_1\,,\\ \sigma _2 \,, &{} (x,\eta )\in {\mathcal {R}}_2\,, \end{array}\right. \end{aligned}$$

and derive the following fundamental identity for $\Phi$, which provides a connection between some integrals involving products of second-order derivatives of $\Phi$ and is in the spirit of [4, Lemma 4.3.1.2], [7, Lemma 3.4], and [10, Lemme II.2.2].

Lemma 4.3

Given $v\in {\mathcal {S}} \cap W_\infty ^2(D)$, the function $\Phi$ defined in (4.8) satisfies

Proof

We adapt the proof of [7, Lemma 3.4] and [10, Lemme II.2.2]. Note that (4.5b), (4.6), (4.7), and (4.8) imply $\llbracket \Phi \rrbracket =0$ on $\Sigma _0$, so that

$$\begin{aligned} \llbracket \partial _x\Phi \rrbracket =0\quad \text { on }\ \Sigma _0\,. \end{aligned}$$

(4.9)

Consequently, since $(\partial _x\Phi _1,\partial _x\Phi _2)$ lies in $H^1({\mathcal {R}}_1)\times H^1({\mathcal {R}}_2)$ by (4.8), we may argue as in the proof of Lemma 4.2 and deduce from (4.9) that

$$\begin{aligned} F:=\partial _x\Phi \in H^1({\mathcal {R}})\,. \end{aligned}$$

Moreover, by (4.8),

$$\begin{aligned} F(x,0) = F(x,1+d) = 0\,, \qquad x\in D\,. \end{aligned}$$

(4.10)

Similarly, setting

$$\begin{aligned} G:=-\sigma \frac{\partial _x v}{1+(\partial _x v)^2}\partial _x\Phi +{\hat{\sigma } \partial _\eta \Phi }, \end{aligned}$$

we derive from (4.8) that $G_i := G|_{{\mathcal {R}}_i}\in H^1({\mathcal {R}}_i)$ for $i=1,2$, while (4.5b), (4.6), (4.7), and (4.8) imply that, for $x\in D$,

$$\begin{aligned} G_1(x,1)&= \frac{\sigma _1}{\sqrt{1+(\partial _x v(x))^2}} \left[ - \partial _x v(x) \partial _x \chi _1(x,v(x)) + \partial _z \chi _1(x,v(x)) \right] \\&= \frac{\sigma _2}{\sqrt{1+(\partial _x v(x))^2}} \left[ - \partial _x v(x) \partial _x \chi _2(x,v(x)) + \partial _z \chi _2(x,v(x)) \right] = G_2(x,1)\,; \end{aligned}$$

that is, $\llbracket G\rrbracket =0$ on $\Sigma _0$, and we argue as in the proof of Lemma 4.2 to conclude that

$$\begin{aligned} G \in H^1({\mathcal {R}})\,. \end{aligned}$$

In addition, by (4.8),

$$\begin{aligned} G(\pm L, \eta )&= - \sigma (\pm L,\eta ) \left( \frac{\partial _x v}{1 + (\partial _x v)^2} \right) (\pm L) \partial _x \Phi (\pm L,\eta ) + {\hat{\sigma }}(\pm L,\eta ) \partial _\eta \Phi (\pm L,\eta ) \\&= - \sigma (\pm L,\eta ) \left( \frac{\partial _x v}{1 + (\partial _x v)^2} \right) (\pm L) \partial _x \Phi (\pm L,\eta ) \end{aligned}$$

for $\eta \in (0,1+d)$. Hence,

$$\begin{aligned} G(\pm L, \eta ) + \sigma (\pm L,\eta ) \left( \frac{\partial _x v}{1 + (\partial _x v)^2} \right) (\pm L) F(\pm L,\eta )= 0\,, \qquad \eta \in (0,1+d)\,. \qquad \end{aligned}$$

(4.11)

Owing to (4.10), (4.11), and the $H^1$-regularity of F and G, we are in a position to apply Lemma A.1 (see Appendix 1) with

$$\begin{aligned} (V,W)=(F,G) \;\;\text { and }\;\; \tau ^{\pm } = \sigma \left( \frac{\partial _x v}{1+(\partial _x v)^2} \right) (\pm L)\,, \end{aligned}$$

to obtain the identity

$$\begin{aligned} \int _{\mathcal {R}}\partial _x F\partial _\eta G\,\mathrm {d}(x,\eta )=\int _{\mathcal {R}}\partial _{\eta }F\partial _x G\,\mathrm {d}(x,\eta )\,. \end{aligned}$$

(4.12)

Using the definitions of F and G, the identity (4.12) reads

$$\begin{aligned} \begin{aligned}&\sum _{i=1}^2\int _{{\mathcal {R}}_i} \partial _x^2\Phi _i \left( -\sigma \frac{\partial _x v}{1+(\partial _x v)^2} \partial _{x}\partial _{\eta }\Phi _i + {\hat{\sigma }} \partial _\eta ^2 \Phi _i \right) \,\mathrm {d}(x,\eta )\\ {}&= \sum _{i=1}^2\int _{{\mathcal {R}}_i}\partial _{x}\partial _{\eta }\Phi _i \left( -\sigma \frac{\partial _x v}{1+(\partial _x v)^2} \partial _{x}^2\Phi _i - \sigma \frac{\partial _x^2 v [1 -(\partial _x v)^2]}{[1+(\partial _x v)^2]^2} \partial _x\Phi _i \right) \, \mathrm {d}(x,\eta ) \\ {}&\qquad + \sum _{i=1}^2\int _{{\mathcal {R}}_i}\partial _{x}\partial _{\eta }\Phi _i \Big ( \partial _x{{\hat{\sigma } }} \partial _\eta \Phi _i +{\hat{\sigma } \partial }_{x}\partial _{\eta }\Phi _i \Big ) \,\mathrm {d}(x,\eta )\,. \end{aligned} \end{aligned}$$

Noticing that the first terms on both sides of the above identity are the same and that

$$\begin{aligned} \partial _x\Phi _i\partial _{x}\partial _{\eta }\Phi _i=\frac{1}{2} \partial _\eta \big ((\partial _x\Phi _i)^2\big ) \end{aligned}$$

implies that

the assertion follows, recalling that $\partial _x {\hat{\sigma }}=0$ in ${\mathcal {R}}_2$. $\square$

Remark 4.4

If $\partial _x v(\pm L)=0$, then (4.11) reduces to $G(\pm L,\eta ) = 0$ for $\eta \in (0,1+d)$ and the crucial identity (4.12) used in the proof of Lemma 4.3 directly follows from [4, Lemma 4.3.1.2]. For the general case $v\in {\mathcal {S}}$, we require the extension given in Lemma A.1.

We now translate the outcome of Lemma 4.3 in terms of the solution $\chi$ to (4.5).

Lemma 4.5

Let $v\in {\mathcal {S}} \cap W_\infty ^2(D)$. The solution $\chi =\psi _v-h_v$ to (4.5) satisfies

Proof

Let us first recall the regularity of $\Phi$ stated in (4.8) which validates the subsequent computations. Using the transformations $T_1$ and $T_2$ introduced in (4.6) and (4.7), respectively, we obtain

$$\begin{aligned} \sum _{i=1}^2 \int _{\Omega _i(v)}\sigma \,&\partial _x^2\chi _i\,\partial _z^2\chi _i\,\mathrm {d}(x,z)\\ =&\int _{{\mathcal {R}}_1}\frac{\sigma _1}{v+H} \bigg [\partial _x^2\Phi _1 + \eta \Big ( 2\Big (\frac{\partial _x v}{v+H}\Big )^2 - \frac{\partial _x^2 v}{v+H}\Big ) \, \partial _\eta \Phi _1 - 2 \eta \frac{\partial _x v}{v+H}\partial _{x}\partial _{\eta }\Phi _1\\& +\eta ^2\Big (\frac{\partial _x v}{v+H}\Big )^2\partial _\eta ^2\Phi _1\bigg ] \,\partial _\eta ^2\Phi _1\,\mathrm {d}(x,\eta )\\&+\int _{{\mathcal {R}}_2}\sigma _2\Big [\partial _x^2\Phi _2-2\partial _xv\partial _{x}\partial _{\eta }\Phi _2-\partial _x^2 v\partial _\eta \Phi _2 +(\partial _x v)^2 \partial _\eta ^2\Phi _2 \Big ] \, \partial _\eta ^2\Phi _2\,\mathrm {d}(x,\eta ) \\ =&\sum _{i=1}^2 \int _{{\mathcal {R}}_i} {\hat{\sigma }} \partial _x^2\Phi _i \,\partial _\eta ^2\Phi _i\,\mathrm {d}(x,\eta )\\&+\int _{{\mathcal {R}}_1}\frac{\sigma _1}{v+H} \bigg [ \eta \Big ( 2 \Big (\frac{\partial _x v}{v+H}\Big )^2 - \frac{\partial _x^2 v}{v+H} \Big ) \, \partial _\eta \Phi _1 - 2\eta \frac{\partial _x v}{v+H} \partial _{x}\partial _{\eta }\Phi _1\\& +\eta ^2\Big (\frac{\partial _x v}{v+H}\Big )^2\partial _\eta ^2\Phi _1\bigg ] \,\partial _\eta ^2\Phi _1\,\mathrm {d}(x,\eta )\\&+\int _{{\mathcal {R}}_2}\sigma _2\Big [-2\partial _xv\partial _{x}\partial _{\eta }\Phi _2-\partial _x^2 v\partial _\eta \Phi _2 +(\partial _x v)^2 \partial _\eta ^2\Phi _2 \Big ] \, \partial _\eta ^2\Phi _2\,\mathrm {d}(x,\eta )\,. \end{aligned}$$

We use Lemma 4.3 to express the first integral on the right-hand side and get

(4.13)

We then compute separately the integrals over ${\mathcal {R}}_i$, $i=1,2$, and begin with the contribution of ${\mathcal {R}}_1$. We complete the square to get

$$\begin{aligned} I_1&:= \int _{{\mathcal {R}}_1} \frac{\sigma _1}{v+H} \bigg [ \vert \partial _{x}\partial _{\eta }\Phi _1\vert ^2 -\frac{\partial _x v}{v+H} \partial _{\eta }\Phi _1\partial _{x}\partial _{\eta }\Phi _1 - 2 \eta \frac{\partial _x v}{v+H} \partial _{x}\partial _{\eta }\Phi _1 \partial _\eta ^2\Phi _1 \\&\quad +\eta ^2 \Big (\frac{\partial _x v}{v+H}\Big )^2 \big \vert \partial _\eta ^2\Phi _1\big \vert ^2 + 2\eta \Big (\frac{\partial _x v}{v+H}\Big )^2 \,\partial _\eta \Phi _1 \partial _\eta ^2\Phi _1 \\&\quad - \eta \frac{\partial _x^2 v}{v+H}\,\partial _\eta \Phi _1 \partial _\eta ^2\Phi _1 \bigg ] \, \,\mathrm {d}(x,\eta ) \\&= \int _{{\mathcal {R}}_1} \frac{\sigma _1}{v+H}\bigg [ \big | \partial _{x}\partial _{\eta }\Phi _1 \big |^2 + \Big (\frac{\partial _x v}{v+H}\Big )^2 \big | \partial _\eta \Phi _1 \big |^2 + \eta ^2\Big (\frac{\partial _x v}{v+H}\Big )^2 \big | \partial _\eta ^2\Phi _1\big |^2 \\&\quad -2\eta \frac{\partial _x v}{v+H} \partial _{x}\partial _{\eta }\Phi _1 \partial _\eta ^2\Phi _1 + 2\eta \Big (\frac{\partial _x v}{v+H}\Big )^2\,\partial _\eta \Phi _1 \partial _\eta ^2\Phi _1 \\&\quad - 2 \frac{\partial _x v}{v+H} \partial _{\eta }\Phi _1 \partial _{x}\partial _{\eta }\Phi _1 \bigg ]\ \mathrm {d}(x,\eta )\\&\quad + \int _{{\mathcal {R}}_1} \frac{\sigma _1}{v+H} \bigg [ - \Big (\frac{\partial _x v}{v+H}\Big )^2 \big | \partial _\eta \Phi _1 \big |^2 + \frac{\partial _x v}{v+H} \partial _{\eta }\Phi _1 \partial _{x}\partial _{\eta }\Phi _1 \\&\quad - \eta \frac{\partial _x^2 v}{v+H} \partial _\eta \Phi _1 \partial _\eta ^2 \Phi _1\bigg ]\ \mathrm {d}(x,\eta ) \\&= \int _{{\mathcal {R}}_1} \sigma _1 (v+H) \bigg [ \frac{\partial _{x}\partial _{\eta }\Phi _1}{v+H} -\frac{\partial _x v}{(v+H)^2} \partial _{\eta }\Phi _1-\eta \frac{\partial _x v}{(v+H)^2}\, \partial _\eta ^2\Phi _1\bigg ]^2\,\mathrm {d}(x,\eta )\\&\quad +\int _{{\mathcal {R}}_1} \sigma _1 \partial _x v\,\bigg [\frac{1}{(v+H)^2}\partial _{\eta }\Phi _1\partial _{x}\partial _{\eta }\Phi _1 -\frac{\partial _x v}{(v+H)^3}\big (\partial _\eta \Phi _1\big )^2\bigg ] \,\mathrm {d}(x,\eta )\\&\quad - \int _{{\mathcal {R}}_1} \sigma _1 \frac{\partial _x^2 v}{(v+H)^2}\,\eta \,\partial _\eta \Phi _1 \partial _\eta ^2\Phi _1 \,\mathrm {d}(x,\eta )\,. \end{aligned}$$

Thanks to the identities

$$\begin{aligned} \frac{1}{(v+H)^2}\partial _\eta \Phi _1\partial _{x}\partial _{\eta }\Phi _1 - \frac{\partial _x v}{(v+H)^3} \big (\partial _\eta \Phi _1\big )^2= & {} \frac{1}{2} \partial _x \left( \left( \frac{\partial _\eta \Phi _1}{v+H} \right) ^2\right) \,, \\ \partial _\eta \Phi _1\partial _\eta ^2\Phi _1= & {} \frac{1}{2} \partial _\eta \big (\partial _\eta \Phi _1\big )^2\,, \end{aligned}$$

and the property $\partial _\eta \Phi _1(\pm L,\eta )= 0$ for $\eta \in (0,1)$ stemming from (4.8), we may perform integration by parts in the last two integrals on the right-hand side of the previous identity and obtain

$$\begin{aligned} I_1&=\int _{{\mathcal {R}}_1} \sigma _1 (v+H)\bigg [\frac{\partial _{x}\partial _{\eta }\Phi _1 }{v+H} -\frac{\partial _x v}{(v+H)^2} \partial _{\eta }\Phi _1-\eta \frac{\partial _x v}{(v+H)^2}\, \partial _\eta ^2\Phi _1\bigg ]^2\,\mathrm {d}(x,\eta )\\&\qquad -\frac{\sigma _1}{2}\int _{D}\frac{\partial _x^2 v}{(v+H)^2}\,\big (\partial _\eta \Phi _1(x,1)\big )^2\,\mathrm {d}x\,. \end{aligned}$$

Transforming the above identity back to $\Omega _1(v)$ yields

$$\begin{aligned} I_1=\int _{\Omega _1(v)}\sigma _1\big \vert \partial _{x}\partial _{z}\chi _1\big \vert ^2\,\mathrm {d}(x,z) - \frac{\sigma _1}{2} \int _{D} \partial _x^2 v(x)\,\big ( \partial _z\chi _1 (x,v(x))\big )^2\,\mathrm {d}x\,. \end{aligned}$$

(4.14)

Next, arguing in a similar way,

$$\begin{aligned} I_2 :=&\, \sigma _2 \int _{{\mathcal {R}}_2}\Big [ \big \vert \partial _x\partial _\eta \Phi _2\big |^2 -2\partial _x v \partial _{x}\partial _{\eta }\Phi _2 \partial _\eta ^2\Phi _2-\partial _x^2 v\partial _\eta \Phi _2\partial _\eta ^2\Phi _2 +(\partial _x v)^2\big \vert \partial _\eta ^2\Phi _2\big \vert ^2\Big ]\,\mathrm {d}(x,\eta ) \\ =&\, \sigma _2 \int _{{\mathcal {R}}_2} \Big [\big | \partial _{x}\partial _{\eta }\Phi _2 \big |^2 - 2 \partial _x v\partial _{x}\partial _{\eta }\Phi _2 \partial _\eta ^2\Phi _2+(\partial _x v)^2\big \vert \partial _\eta ^2\Phi _2\big \vert ^2\Big ]\,\mathrm {d}(x,\eta ) \\&\qquad - \frac{\sigma _2}{2}\int _{{\mathcal {R}}_2}\partial _x^2 v \partial _\eta \left( \partial _\eta \Phi _2 \right) ^2 \,\mathrm {d}(x,\eta ) \\ =&\, \sigma _2 \int _{{\mathcal {R}}_2} \Big [\partial _{x}\partial _{\eta }\Phi _2 - \partial _x v \partial _\eta ^2\Phi _2\Big ]^2\,\mathrm {d}(x,\eta ) - \frac{\sigma _2}{2}\int _{D} \partial _x^2 v(x)\,\big ( \partial _\eta \Phi _2 (x,1+d)\big )^2\,\mathrm {d}x\\&\qquad +\frac{\sigma _2 }{2}\int _{D} \partial _x^2 v(x)\,\big ( \partial _\eta \Phi _2 (x,1)\big )^2\,\mathrm {d}x\,. \end{aligned}$$

Transforming this formula back to $\Omega _2(v)$ yields

$$\begin{aligned} \begin{aligned} I_2&= \sigma _2 \int _{\Omega _2(v)} \big |\partial _{x}\partial _{z}\chi _2\big |^2\,\mathrm {d}(x,z) - \frac{\sigma _2}{2} \int _{D} \partial _x^2 v(x)\,\big ( \partial _z\chi _2 (x,v(x)+d)\big )^2\,\mathrm {d}x\\&\qquad + \frac{\sigma _2}{2}\int _{D} \partial _x^2 v(x)\,\big ( \partial _z\chi _2 (x,v(x))\big )^2\,\mathrm {d}x \,. \end{aligned} \end{aligned}$$

(4.15)

Finally,

and we deduce from (4.13), (4.14), (4.15), and the above identity that

(4.16)

It remains to simplify the last two integrals on the right-hand side of (4.16). To this end, we first recall that the regularity of $\chi$ allows us to differentiate with respect to x the transmission condition $\llbracket \chi \rrbracket =0$ on $\Sigma (v)$ to deduce that

$$\begin{aligned} \llbracket \partial _x \chi + \partial _x v \partial _z \chi \rrbracket = 0 \;\text { on }\; \Sigma (v)\,, \end{aligned}$$

(4.17)

while the second transmission condition in (4.5b) reads

$$\begin{aligned} \llbracket \sigma \big ( \partial _x v \partial _x \chi - \partial _z \chi \big ) \rrbracket = 0 \;\text { on }\; \Sigma (v)\,. \end{aligned}$$

(4.18)

In particular, (4.17) and (4.18) imply that, on $\Sigma (v)$,

$$\begin{aligned} \llbracket \sigma \big ( \partial _x v \partial _x \chi - \partial _z \chi \big ) \big ( \partial _x \chi + \partial _x v \partial _z \chi \big ) \rrbracket&= \big ( \partial _x \chi _1 + \partial _x v \partial _z \chi _1 \big ) \llbracket \sigma \big ( \partial _x v \partial _x \chi - \partial _z \chi \big ) \rrbracket \\&\qquad + \sigma _2 \big ( \partial _x v \partial _x \chi _2 - \partial _z \chi _2 \big ) \llbracket \big ( \partial _x \chi + \partial _x v \partial _z \chi \big ) \rrbracket \\&= 0\,. \end{aligned}$$

Therefore,

Hence,

(4.19)

Consequently, (4.16) and (4.19) entail

as claimed. $\square$

In order to estimate the boundary and the transmission terms in Lemma 4.5, we first report the following trace estimates.

Lemma 4.6

Given $\kappa >0$ and $\alpha \in (0,1/2]$, there is $c(\alpha ,\kappa )>0$ such that, for any $v\in \bar{{\mathcal {S}}}$ satisfying $\Vert v\Vert _{H^2(D)}\le \kappa$ and $\theta \in H^1(\Omega _2(v))$,

$$\begin{aligned} \Vert \theta (\cdot ,v)\Vert _{H^\alpha (D)}+ \Vert \theta (\cdot ,v+d)\Vert _{H^\alpha (D)}\le c(\alpha ,\kappa )\, \Vert \theta \Vert _{L_2(\Omega _2(v))}^{(1-2\alpha )/2}\, \Vert \theta \Vert _{H^1(\Omega _2(v))}^{(2\alpha +1)/2}\,. \end{aligned}$$

Proof

Let $\theta \in H^1(\Omega _2(v))$. Using the transformation $T_2$ defined in (4.7) which maps $\Omega _2(v)$ onto the rectangle ${\mathcal {R}}_2 =D\times (1,1+d)$, we note that $\phi :=\theta \circ T_2^{-1}$ belongs to $H^1({\mathcal {R}}_2)$ with

$$\begin{aligned} \Vert \phi \Vert _{L_2({\mathcal {R}}_2)} = \Vert \theta \Vert _{L_2(\Omega _2(v))} \end{aligned}$$

(4.20)

and

$$\begin{aligned} \Vert \nabla \phi \Vert _{L_2({\mathcal {R}}_2)}^2 = \Vert \partial _x \theta + \partial _x v \partial _z \theta \Vert _{L_2(\Omega _2(v))}^2 + \Vert \partial _z \theta \Vert _{L_2(\Omega _2(v))}^2\,, \end{aligned}$$

so that the continuous embedding of $H^2(D)$ in $W_\infty ^1(D)$ and the assumed bound on v readily imply that

$$\begin{aligned} \Vert \phi \Vert _{H^1({\mathcal {R}}_2)} \le c(\kappa ) \Vert \theta \Vert _{H^1(\Omega _2(v))} \,. \end{aligned}$$

(4.21)

By complex interpolation,

$$\begin{aligned}{}[L_2({\mathcal {R}}_2 ),H^1({\mathcal {R}}_2 )]_{\alpha +1/2}\doteq H^{\alpha +1/2} ({\mathcal {R}}_2 )\,, \end{aligned}$$

from which we deduce that

$$\begin{aligned} \Vert \phi \Vert _{H^{\alpha +1/2}({\mathcal {R}}_2 )}\le c(\alpha ) \Vert \phi \Vert _{L_2({\mathcal {R}}_2 )}^{(1-2\alpha )/2} \Vert \phi \Vert _{H^1({\mathcal {R}}_2 )}^{(2\alpha +1)/2}\,. \end{aligned}$$

Since $\alpha >0$, the trace maps $H^{\alpha +1/2}({\mathcal {R}}_2 )$ continuously on $H^{\alpha }(D\times \{1\})$, and we thus infer from (4.20) and (4.21) that

$$\begin{aligned} \begin{aligned} \Vert \theta (\cdot ,v)\Vert _{H^{\alpha }(D)}&=\Vert \phi (\cdot , 1) \Vert _{H^{\alpha }(D)}\le c(\alpha ) \Vert \phi \Vert _{H^{\alpha +1/2}({\mathcal {R}}_2)} \\&\le c(\alpha )\Vert \phi \Vert _{L_2({\mathcal {R}}_2 )}^{(1-2\alpha )/2} \Vert \phi \Vert _{H^1({\mathcal {R}}_2 )}^{(2\alpha +1)/2}\\&\le c(\alpha ,\kappa )\Vert \theta \Vert _{L_2(\Omega _2(v))}^{(1-2\alpha )/2}\Vert \theta \Vert _{H^1(\Omega _2(v))}^{(2\alpha +1)/2}\,. \end{aligned} \end{aligned}$$

The estimate for $\Vert \theta (\cdot ,v+d)\Vert _{H^{\alpha }(D)}$ is proved in a similar way. $\square$

Based on Lemma 4.6 we are in a position to estimate the boundary and transmission terms in the identity provided by Lemma 4.5.

Lemma 4.7

Let $\zeta \in (3/4,1)$ and $\kappa >0$. There is $c(\zeta ,\kappa )>0$ such that, if $v\in {\mathcal {S}} \cap W_\infty ^2(D)$ satisfies $\Vert v\Vert _{H^2(D)}\le \kappa$, then the solution $\chi =\chi _v$ to (4.5) satisfies

(4.22)

and

(4.23)

Proof

To prove (4.22), let us first note that $H^{\zeta -1/2}(D)$ embeds continuously into $L_4(D)$. We use the Cauchy-Schwarz inequality and Lemma 4.6 with $\alpha =\zeta -1/2$ and deduce

$$\begin{aligned} \begin{aligned} \left| \frac{\sigma _2}{2}\int _D \partial _x^2 v(x) \big (\partial _z\chi _2(x,v(x)+d)\big )^2\, \mathrm {d}x\right|&\le \frac{\sigma _2}{2} \Vert \partial _x^2 v\Vert _{L_2(D)}\, \Vert \partial _z\chi _2(\cdot ,v+d)\Vert _{L_4(D)}^2\\&\le c(\kappa )\, \Vert \partial _z\chi _2(\cdot ,v+d)\Vert _{H^{\zeta -1/2}(D)}^2\\&\le c(\zeta ,\kappa )\, \Vert \partial _z\chi _2\Vert _{L_2(\Omega _2(v))}^{2(1-\zeta )}\, \Vert \partial _z\chi _2\Vert _{H^1(\Omega _2(v))}^{2\zeta }\,. \end{aligned} \end{aligned}$$

As for (4.23) we obtain analogously

$$\begin{aligned}&\left| \frac{\sigma _2}{2}\int _D \frac{\partial _x^2 v(x)}{1+(\partial _x v(x))^2}\, \left[ \big (\partial _x\chi _2(x,v(x))\big )^2 + \big (\partial _z\chi _2(x,v(x))\big )^2 \right] \,\mathrm {d}x\right| \nonumber \\&\qquad \qquad \le \frac{\sigma _2}{2} \Vert \partial _x^2 v\Vert _{L_2(D)} \Vert \nabla \chi _2(\cdot ,v)\Vert _{L_4(D)}^2 \nonumber \\&\qquad \qquad \le c(\zeta ,\kappa )\, \Vert \nabla \chi _2\Vert _{L_2(\Omega _2(v))}^{2(1-\zeta )}\, \Vert \nabla \chi _2\Vert _{H^1(\Omega _2(v))}^{2\zeta } \end{aligned}$$

(4.24)

and

$$\begin{aligned} \begin{aligned}&\left| \frac{\sigma _1}{2}\int _D \frac{\partial _x^2 v(x)}{1+(\partial _x v(x))^2}\, \left[ \big (\partial _x\chi _1(x,v(x))\big )^2 + \big (\partial _z\chi _1(x,v(x))\big )^2 \right] \,\mathrm {d}x\right| \\&\qquad \qquad \le \frac{\sigma _1}{2} \Vert \partial _x^2 v\Vert _{L_2(D)} \Vert \nabla \chi _1(\cdot ,v)\Vert _{L_4(D)}^2 \,. \end{aligned} \end{aligned}$$

(4.25)

At this point, we use (4.17) and (4.18) to show that

$$\begin{aligned} \partial _x \chi _1&= \frac{\sigma _1+\sigma _2(\partial _x v)^2}{\sigma _1\big (1+(\partial _x v)^2\big )} \partial _x \chi _2 +\frac{\llbracket \sigma \rrbracket \partial _x v}{\sigma _1\big (1+(\partial _x v)^2\big )}\partial _z\chi _2 \quad \text {on }\ \Sigma (v)\,, \\ \partial _z \chi _1&= \frac{\llbracket \sigma \rrbracket \partial _x v}{\sigma _1\big (1+(\partial _x v)^2\big )}\partial _x\chi _2 + \frac{\sigma _1+\sigma _2(\partial _x v)^2}{\sigma _1\big (1+(\partial _x v)^2\big )} \partial _z \chi _2 \quad \text {on }\ \Sigma (v)\,. \end{aligned}$$

Consequently,

$$\begin{aligned} |\partial _x \chi _1|&\le \frac{\max \{\sigma _1,\sigma _2\}}{\sigma _1} \left( |\partial _x \chi _2| + |\partial _z \chi _2| \right) \quad \text {on }\ \Sigma (v)\,, \\ |\partial _z \chi _1|&\le \frac{\max \{\sigma _1,\sigma _2\}}{\sigma _1} \left( |\partial _x \chi _2| + |\partial _z \chi _2| \right) \quad \text {on }\ \Sigma (v)\,, \end{aligned}$$

so that

$$\begin{aligned} \Vert \nabla \chi _1(\cdot ,v)\Vert _{L_4(D)} \le c \Vert \nabla \chi _2(\cdot ,v)\Vert _{L_4(D)}\,. \end{aligned}$$

Owing to (4.25) and the above inequality, we may then argue as in the proof of (4.24) to conclude that

$$\begin{aligned}&\left| \frac{\sigma _1}{2}\int _D \frac{\partial _x^2 v(x)}{1+(\partial _x v(x))^2}\, \left[ \big (\partial _x\chi _1(x,v(x))\big )^2 + \big (\partial _z\chi _1(x,v(x))\big )^2 \right] \,\mathrm {d}x\right| \\&\qquad \qquad \le c(\zeta ,\kappa )\, \Vert \nabla \chi _2\Vert _{L_2(\Omega _2(v))}^{2(1-\zeta )}\, \Vert \nabla \chi _2\Vert _{H^1(\Omega _2(v))}^{2\zeta }\,, \end{aligned}$$

as claimed in (4.23). $\square$

We now gather the previous findings to deduce the following crucial $H^2$-estimate on the solution $\psi _v$ to (4.1) for $v\in {\mathcal {S}}\cap W_\infty ^2(D)$, which only depends on the $H^2(D)$-norm of v (but not on its $W_\infty ^2(D)$-norm).

Proposition 4.8

Let $\kappa >0$ and $v\in {\mathcal {S}}\cap W_\infty ^2(D)$ be such that $\Vert v\Vert _{H^2(D)}\le \kappa$. There is a constant $c_0(\kappa )>0$ such that the solution $\psi _v$ to (4.1) satisfies

$$\begin{aligned} \Vert \chi \Vert _{H^1(\Omega (v))} + \Vert \chi _1\Vert _{H^2(\Omega _1(v))} + \Vert \chi _2\Vert _{H^2(\Omega _2(v))}\le c_0(\kappa ) \end{aligned}$$

(4.26a)

and

$$\begin{aligned} \Vert \psi _v\Vert _{H^1(\Omega (v))}+\Vert \psi _{v,1}\Vert _{H^2(\Omega _1(v))}+\Vert \psi _{v,2}\Vert _{H^2(\Omega _2(v))} \le c_0(\kappa )\,, \end{aligned}$$

(4.26b)

recalling that $\chi =\psi _v-h_v$ and $\chi _i = \chi |_{\Omega _i(v)}$, $i=1,2$.

Proof

Let $v\in {\mathcal {S}} \cap W_\infty ^2(D)$ with $\Vert v\Vert _{H^2(D)}\le \kappa$. Since $\sigma$ is constant on $\Omega _1(v)$ and on $\Omega _2(v)$, it readily follows from (4.5a) that

$$\begin{aligned} \sum _{i=1}^2 \int _{\Omega _i(v)}\sigma |\Delta \chi _i|^2\, \mathrm {d}(x,z)= \sum _{i=1}^2 \int _{\Omega _i(v)}\sigma | \Delta h_{v,i}|^2\, \mathrm {d}(x,z)\,. \end{aligned}$$

Since

$$\begin{aligned} |\Delta \chi _i|^2 = |\partial _x^2\chi _i|^2 + |\partial _z^2\chi _i|^2 + 2 \partial _x^2\chi _i \partial _z^2 \chi _i\,, \qquad i=1,2\,, \end{aligned}$$

we infer from Lemma 4.5 and the above two formulas that

Using Lemma 4.7 with $\zeta =7/8$, along with the identity

$$\begin{aligned} \begin{aligned}&\sum _{i=1}^2 \int _{\Omega _i(v)}\sigma \big \{|\partial _x^2\chi _i|^2+ 2|\partial _{x}\partial _{z}\chi _i|^2+|\partial _z^2\chi _i|^2\big \}\,\mathrm {d}(x,z) \\& = \sigma _1 \Vert \nabla \chi _1\Vert _{H^1(\Omega _1(v))}^2 + \sigma _2 \Vert \nabla \chi _2\Vert _{H^1(\Omega _2(v))}^2\,, \end{aligned} \end{aligned}$$

we further obtain

$$\begin{aligned}&\sigma _1 \Vert \nabla \chi _1\Vert _{H^1(\Omega _1(v))}^2 + \sigma _2 \Vert \nabla \chi _2\Vert _{H^1(\Omega _2(v))}^2\\&\qquad \qquad \le \sum _{i=1}^2 \int _{\Omega _i(v)}\sigma |\Delta h_{v,i}|^2\,\mathrm {d}(x,z) + c(\kappa )\, \Vert \nabla \chi _2\Vert _{L_2(\Omega _2(v))}^{1/4}\, \Vert \nabla \chi _2\Vert _{H^1(\Omega _2(v))}^{7/4}\,. \end{aligned}$$

Hence, thanks to Young’s inequality,

$$\begin{aligned}&\sigma _1 \Vert \nabla \chi _1\Vert _{H^1(\Omega _1(v))}^2 + \sigma _2 \Vert \nabla \chi _2\Vert _{H^1(\Omega _2(v))}^2\\&\qquad \qquad \le \sum _{i=1}^2 \int _{\Omega _i(v)}\sigma |\Delta h_{v,i}|^2\,\mathrm {d}(x,z) + \frac{\sigma _2}{2}\, \Vert \nabla \chi _2\Vert _{H^1(\Omega _2(v))}^2 + c(\kappa ) \Vert \nabla \chi _2\Vert _{L_2(\Omega _2(v))}^2\,. \end{aligned}$$

Recalling that

$$\begin{aligned} \Vert \nabla \chi _2\Vert _{L_2(\Omega _2(v))}^2&\le \frac{1}{\sigma _2} \int _{\Omega (v)} \sigma |\nabla \chi |^2\, \mathrm {d}(x,z) \le \frac{1}{\sigma _2} \int _{\Omega (v)} \sigma |\nabla h_v|^2\, \mathrm {d}(x,z) \\&\le \frac{\max \{\sigma _1,\sigma _2\}}{\sigma _2} \Vert \nabla h_v\Vert _{L_2(\Omega (v))}^2 \end{aligned}$$

by (4.5) and that $\min \{\sigma _1,\sigma _2\}>0$, we conclude that

$$\begin{aligned} \begin{aligned}&\Vert \nabla \chi _1\Vert _{H^1(\Omega _1(v))}^2 + \Vert \nabla \chi _2\Vert _{H^1(\Omega _2(v))}^2 \\& \le c(\kappa ) \left( \Vert \Delta h_{v,1}\Vert _{L_2(\Omega _1(v))}^2 + \Vert \Delta h_{v,2}\Vert _{L_2(\Omega _2(v))}^2 + \Vert \nabla h_v\Vert _{L_2(\Omega (v))}^2 \right) \,. \end{aligned} \end{aligned}$$

(4.27)

Owing to the continuous embedding of $H^2(D)$ in $C({\bar{D}})$, combining (4.27) and Lemma 3.2 leads us to the estimate

$$\begin{aligned} \begin{aligned}&\Vert \chi \Vert _{H^1(\Omega (v))} + \Vert \chi _{1}\Vert _{H^2(\Omega _1(v))} + \Vert \chi _{2}\Vert _{H^2(\Omega _2(v))} \\& \le c(\kappa ) \big (\Vert \nabla h_v \Vert _{L_2(\Omega (v))} + \Vert \Delta h_{v,1}\Vert _{L_2(\Omega _1(v))}^2 + \Vert \Delta h_{v,2}\Vert _{L_2(\Omega _2(v))}^2 \big )\,. \end{aligned} \end{aligned}$$

The bound (4.26a) then readily follows from the assumptions (2.1a) and (2.1c). Finally, (4.26a), together with (2.1a) and (2.1c), yields (4.26b). $\square$

4.3 $H^2$-regularity and $H^2$-estimates on $\psi _v$ for $v\in \bar{{\mathcal {S}}}$

Finally, we extend Propositions 4.1 and 4.8 by showing the $H^2$-regularity of $\psi _v$ and the corresponding $H^2$-estimates for an arbitrary $v\in \bar{{\mathcal {S}}}$; that is, we drop the additional $W_\infty ^2$-regularity of v assumed in the previous sections and also allow for a non-empty coincidence set.

Fig. 3

Geometry of $\Omega (w)$ for a state $w\in \bar{{\mathcal {S}}}$ with non-empty and disconnected coincidence set

Proposition 4.9

Let $\kappa >0$ and $v\in \bar{{\mathcal {S}}}$ be such that $\Vert v\Vert _{H^2(D)} \le \kappa$.

• (a) The unique minimizer $\psi _v\in {\mathcal {A}}(v)$ of ${\mathcal {J}}(v)$ on ${\mathcal {A}}(v)$ provided by Lemma 3.1 satisfies

  $$\begin{aligned} \psi _{v,i} = \psi _v|_{\Omega _i(v)} \in H^2(\Omega _i(v))\,, \qquad i=1,2\,, \end{aligned}$$

  and is a strong solution to the transmission problem (4.1). Moreover, there is $c_1(\kappa )>0$ such that

  $$\begin{aligned} \Vert \psi _v\Vert _{H^1(\Omega (v))} + \Vert \psi _{v,1}\Vert _{H^2(\Omega _1(v))} + \Vert \psi _{v,2}\Vert _{H^2(\Omega _2(v))}\le c_1(\kappa )\,. \end{aligned}$$

  (4.28)

• (b) Consider a sequence $(v_n)_{n\ge 1}$ in $\bar{{\mathcal {S}}}$ satisfying

  $$\begin{aligned} \Vert v_n\Vert _{H^2(D)}\le \kappa \,, \quad n\ge 1\,, \;\;\text { and }\;\; \lim _{n\rightarrow \infty } \Vert v_n-v\Vert _{H^1(D)} = 0. \end{aligned}$$

  (4.29)

  If $i\in \{1,2\}$ and $U_i$ is an open subset of $\Omega _i(v)$ such that ${{\bar{U}}}_i$ is a compact subset of $\Omega _i(v)$, then

  $$\begin{aligned} \psi _{v_n,i}\rightharpoonup \psi _{v,i} \quad \text {in}\quad H^2(U_i)\,, \end{aligned}$$

  recalling that $\psi _{v_n,i} = \psi _{v_n}|_{\Omega _i(v_n)}$.

The proof involves three steps: we first establish Proposition 4.9 (b) under the additional assumption

$$\begin{aligned} \sup _{n\ge 1}\left\{ \Vert \psi _{v_n,1}\Vert _{H^2(\Omega _1(v_n))} + \Vert \psi _{v_n,2}\Vert _{H^2(\Omega _2(v_n))} \right\} < \infty \,. \end{aligned}$$

Building upon this result, we take advantage of the density of ${\mathcal {S}}\cap W_\infty ^2(D)$ in $\bar{{\mathcal {S}}}$ and of the estimates derived in Proposition 4.8 to verify Proposition 4.9 (a) by a compactness argument. Combining the previous steps leads us finally to a complete proof of Proposition 4.9 (b). We thus start with the proof of Proposition 4.9 (b) when the solutions $(\psi _{v_n})_{n\ge 1}$ to (4.1) associated with the sequence $(v_n)_{n\ge 1}$ satisfies the above additional bound. We state this result as a separate lemma for definiteness.

Lemma 4.10

Let $\kappa >0$ and $v\in \bar{{\mathcal {S}}}$ be such that $\Vert v\Vert _{H^2(D)} \le \kappa$ and consider a sequence $(v_n)_{n\ge 1}$ in $\bar{{\mathcal {S}}}$ satisfying (4.29). Assume further that, for each $n\ge 1$, $(\psi _{v_n,1},\psi _{v_n,2})$ belongs to $H^2(\Omega _1(v_n))\times H^2(\Omega _2(v_n))$ and that there is $\mu >0$ such that

$$\begin{aligned} \Vert \psi _{v_n,1}\Vert _{H^2(\Omega _1(v_n))} + \Vert \psi _{v_n,2}\Vert _{H^2(\Omega _2(v_n))} \le \mu \,, \qquad n\ge 1\,. \end{aligned}$$

(4.30)

Then $\psi _{v,i} \in H^2(\Omega _i(v))$, $i=1,2$. In addition, if $i\in \{1,2\}$ and $U_i$ is an open subset of $\Omega _i(v)$ such that ${{\bar{U}}}_i$ is a compact subset of $\Omega _i(v)$, then

$$\begin{aligned} \psi _{v_n,i}\rightharpoonup \psi _{v,i} \quad \text {in}\quad H^2(U_i) \end{aligned}$$

and

$$\begin{aligned} \Vert \psi _{v,1}\Vert _{H^2(\Omega _1(v))} + \Vert \psi _{v,2}\Vert _{H^2(\Omega _2(v))} \le \mu \,. \end{aligned}$$

(4.31)

The proof is very close to that of [7, Proposition 3.13 & Corollary 3.14], so that we omit the details here and refer to the extended version of this paper [8] instead.

Proof of Proposition 4.9 (a)

Let $v\in {\bar{{\mathcal {S}}}}$ be such that $\Vert v\Vert _{H^2(D)}\le \kappa$. We may choose a sequence $(v_n)_{n\ge 1}$ in ${\mathcal {S}}\cap W_\infty ^2(D)$ satisfying

$$\begin{aligned} v_n\rightarrow v \ \text { in }\ H^2(D)\,,\qquad \sup _{n\ge 1}\,\Vert v_n\Vert _{H^2(D)}\le 2\kappa \,. \end{aligned}$$

(4.32)

Owing to (4.32) and the regularity property $v_n\in {\mathcal {S}}\cap W_\infty ^2(D)$, $n\ge 1$, Proposition 3.3 guarantees that $(\psi _{v_n,1},\psi _{v_n,2})$ belongs to $H^2(\Omega _1(v_n))\times H^2(\Omega _2(v_n))$ and $(\psi _{v_n})_{n\ge 1}$ satisfies (4.30) with $\mu =c_0(2\kappa )$. We then infer from Lemma 4.10 that $(\psi _{v,1},\psi _{v,2})$ belongs to $H^2(\Omega _1(v))\times H^2(\Omega _2(v))$ and satisfies

$$\begin{aligned} \Vert \psi _{v,1}\Vert _{H^2(\Omega _1(v))} + \Vert \psi _{v,2}\Vert _{H^2(\Omega _2(v))} \le c_0(2\kappa )\,. \end{aligned}$$

Combining the above bound with (2.1d) and Lemma 4.10 gives (4.28). Checking that $\psi _v$ is a strong solution to (4.1) is then done as in [7, Corollary 3.14], see also the extended version of this paper [8] for a complete proof. $\square$

Proof of Proposition 4.9 (b)

Proposition 4.9 (b) is now a straightforward consequence of Proposition 4.9 (a) and Lemma 4.10. $\square$

Proof of Theorem 1.1

The proof of Theorem 1.1 readily follows from Proposition 4.9 (a). $\square$

We supplement the $H^2$-weak continuity of $\psi _v$ with respect to v reported in Proposition 4.9 with the continuity of the traces of $\nabla \psi _{v,2}$ on the upper and lower boundaries of $\Omega _2(v)$.

Proposition 4.11

Let $\kappa >0$ and $v\in \bar{{\mathcal {S}}}$ be such that $\Vert v\Vert _{H^2}(D)\le \kappa$ and consider a sequence $(v_n)_{n\ge 1}$ in $\bar{{\mathcal {S}}}$ satisfying (4.29). Then, for $p\in [1,\infty )$,

$$\begin{aligned} \nabla \psi _{v_n,2}(\cdot ,v_n)&\rightarrow \nabla \psi _{v,2}(\cdot ,v) \quad \text { in }\quad L_p(D,\mathbb {R}^2)\,, \end{aligned}$$

(4.33)

$$\begin{aligned} \nabla \psi _{v_n,2}(\cdot ,v_n+d)&\rightarrow \nabla \psi _{v,2}(\cdot ,v+d) \quad \text { in }\quad L_p(D,\mathbb {R}^2)\,, \end{aligned}$$

(4.34)

and

$$\begin{aligned} \Vert \nabla \psi _{v,2}(\cdot ,v) \Vert _{L_p(D,\mathbb {R}^2)} + \Vert \nabla \psi _{v,2}(\cdot ,v+d) \Vert _{L_p(D,\mathbb {R}^2)} \le c(p,\kappa )\,. \end{aligned}$$

(4.35)

Proof

Recall first from (4.28) that

$$\begin{aligned} \Vert \psi _{v_n,2}\Vert _{H^2(\Omega _2(v_n))} \le c_1(\kappa )\,,\qquad n\ge 1\,. \end{aligned}$$

(4.36)

As in the proof of Lemma 4.6 we map $\Omega _2(v)$ onto the rectangle ${\mathcal {R}}_2 =D\times (1,1+d)$ and define, for $(x,\eta )\in {\mathcal {R}}_2$ and $n\ge 1$,

$$\begin{aligned} \phi _n(x,\eta ) := \psi _{v_n,2}(x,\eta +v_n(x)-1)\,,\qquad \phi (x,\eta ) := \psi _{v,2}(x,\eta +v(x)-1)\,. \end{aligned}$$

Let $q\in (1,2)$. Since

$$\begin{aligned} \nabla \phi _n(x,\eta )&= \Big ( \partial _x \psi _{v_n} + \partial _x v_n \partial _z \psi _{v_n} \,,\, \partial _z \psi _{v_n} \Big )(x,\eta +v_n(x)-1)\,, \\ \partial _x^2 \phi _n(x,\eta )&= \Big ( \partial _x^2 \psi _{v_n} + 2\partial _x v_n \partial _x \partial _z \psi _{v_n} + (\partial _x v_n)^2 \partial _z^2 \psi _{v_n} + \partial _x^2 v_n \partial _z \psi _{v_n} \Big )(x,\eta +v_n(x)-1)\,, \\ \partial _x\partial _\eta \phi _n(x,\eta )&= \Big ( \partial _x\partial _z \psi _{v_n} + \partial _x v_n \partial _z^2 \psi _{v_n}\Big )(x,\eta +v_n(x)-1)\,, \\ \partial _\eta ^2 \phi _n(x,\eta )&= \partial _z^2\psi _{v_n}(x,\eta +v_n(x)-1)\,, \end{aligned}$$

it follows from (4.29), (4.36), the continuous embedding of $H^2(D)$ in $C^1({\bar{D}})$, and that of $H^1({\mathcal {R}}_2)$ in $L^{2q/(2-q)}({\mathcal {R}}_2)$ that

$$\begin{aligned} \phi _n\in W_q^2({\mathcal {R}}_2)\;\;\text { with }\;\; \Vert \phi _n\Vert _{W_q^2({\mathcal {R}}_2)}\le c(q,\kappa )\,,\qquad n\ge 1\,. \end{aligned}$$

(4.37)

Now, given $p\in [1,\infty )$, we choose $q\in (1,\min \{2,p\})$ satisfying $1<2/q<1 + 1/p$ and $s\in (2/q-1/p,1)$. Since

$$\begin{aligned} \phi _n \rightharpoonup \phi \quad \text { in }\quad W_q^2({\mathcal {R}}_2) \end{aligned}$$

by (2.1d), (4.37), and Proposition 4.9, the continuity of the trace as a mapping from $W_q^1({\mathcal {R}}_2)$ to $W_q^{1-1/q}(D\times \{1\})$ and the compactness of the embedding of $W_q^{1-1/q}(D)$ in $L_p(D)$ imply that

$$\begin{aligned} \nabla \phi _n (\cdot ,1) \rightarrow \nabla \phi (\cdot ,1) \quad \text { in }\quad W_q^{s-1/q}(D) \end{aligned}$$

(4.38)

and

$$\begin{aligned} \Vert \nabla \phi (\cdot ,1)\Vert _{L_p(D)} \le c(p,\kappa )\,. \end{aligned}$$

(4.39)

That is,

$$\begin{aligned} \partial _z\psi _{v_n,2}(\cdot ,v_n)= \partial _\eta \phi _n (\cdot ,1) \rightarrow \partial _\eta \phi (\cdot ,1)=\partial _z\psi _{v,2}(\cdot ,v) \quad \text { in }\quad L_p(D) \end{aligned}$$

and, recalling (4.29) and the continuous embedding of $H^2(D)$ in $C^1({\bar{D}})$,

$$\begin{aligned} \begin{aligned} \partial _x\psi _{v_n,2}(\cdot ,v_n)= \partial _x \phi _n (\cdot ,1)&-\partial _x v_n\partial _\eta \phi _n (\cdot ,1)\\&\rightarrow \partial _x \phi (\cdot ,1)-\partial _x v\partial _\eta \phi (\cdot ,1) =\partial _x\psi _{v,2}(\cdot ,v) \quad \text { in }\quad L_p(D)\,. \end{aligned} \end{aligned}$$

Furthermore, (4.38) and (4.39), along with the bound $\Vert v\Vert _{H^2(D)}\le \kappa$ and the continuous embedding of $H^2(D)$ in $C^1({\bar{D}})$, entail that

$$\begin{aligned} \Vert \nabla \psi _{v,2}(\cdot ,v)\Vert _{L_p(D)} \le c(p,\kappa )\,, \end{aligned}$$

which proves (4.33) and the first bound in (4.35). Clearly, (4.34) and the second bound in (4.35) are shown in the same way. $\square$

Proof of Theorem 1.3

The proof of Theorem 1.3 is now a consequence of Proposition 3.3 for (1.4a), Proposition 4.9  (b) for (1.4b), and Proposition 4.11 for (1.4c). $\square$

Appendix A. The Identity (4.12)

This appendix is devoted to the proof of the identity (4.12), which can be seen as a variant of [4, Lemma 4.3.1.2] with piecewise constant linear constraints on the boundaries instead of constant ones.

Lemma A.1

Let ${\mathcal {R}} = D \times (0,1+d)$ and consider $(V,W)\in H^1({\mathcal {R}},{\mathbb {R}}^2)$ satisfying

$$\begin{aligned} V(x,0) = V(x,1+d)&= 0\,, \qquad x\in D=(-L,L)\,, \end{aligned}$$

(A.1a)

$$\begin{aligned} W(\pm L, \eta ) + \tau ^\pm (\eta ) V(\pm L,\eta )&= 0\,, \qquad \eta \in (0,1+d)\,, \end{aligned}$$

(A.1b)

where $\tau ^{\pm }$ are piecewise constant functions of the form

$$\begin{aligned} \tau ^\pm = \tau _1^\pm {\mathbf {1}}_{(0,1)} + \tau _2^\pm {\mathbf {1}}_{(1,1+d)} \end{aligned}$$

(A.2)

with $(\tau _1^+,\tau _1^-,\tau _2^+,\tau _2^-)\in {\mathbb {R}}^4$, featuring possibly a jump discontinuity at $\eta =1$. Then

$$\begin{aligned} \int _{\mathcal {R}}\partial _x V \partial _\eta W \,\mathrm {d}(x,\eta ) = \int _{\mathcal {R}} \partial _{\eta }V \partial _x W\,\mathrm {d}(x,\eta )\,. \end{aligned}$$

When $\tau _1^\pm = \tau _2^\pm$, Lemma A.1 is a straightforward consequence of [4, Lemma 4.3.1.2]. The novelty here is the possibility of handling the jump discontinuity in (A.1b) when $\tau _1^\pm \ne \tau _2^\pm$ in (A.2).

The proof follows the lines of that of [4, Lemma 4.3.1.2]. For $s\ge 1$, we introduce the space

$$\begin{aligned} {\mathcal {G}}^s({\mathcal {R}}) := \{ (V,W)\in H^s({\mathcal {R}},{\mathbb {R}}^2)\,:\, (V,W) \;\text { satisfies } (A.1)\}\, \end{aligned}$$

and first report the density of ${\mathcal {G}}^2({\mathcal {R}})$ in ${\mathcal {G}}^1({\mathcal {R}})$.

Lemma A.2

${\mathcal {G}}^2({\mathcal {R}})$ is dense in ${\mathcal {G}}^1({\mathcal {R}})$.

As in the proof of [4, Lemma 4.3.1.3], the core of the proof of Lemma A.2 is to establish the density of the space ${\mathcal {Z}}^2(\partial {\mathcal {R}})$ of traces of functions in ${\mathcal {G}}^2({\mathcal {R}})$ in the space ${\mathcal {Z}}^1(\partial {\mathcal {R}})$ of traces of functions in ${\mathcal {G}}^1({\mathcal {R}})$, after identifying these two trace spaces. Since the proof is almost identical to that of [4, Lemma 4.3.1.3], we omit it here, but refer to the extended version of this paper [8].

Proof of Lemma A.1

Due to Lemma A.2, it suffices to prove the identity in Lemma A.1 when (V, W) belongs to ${\mathcal {G}}^2({\mathcal {R}})$. This additional regularity allows us to use integration by parts to interchange the derivatives and guarantees the continuity of both V and W on $\bar{{\mathcal {R}}}$. Indeed, $H^2({\mathcal {R}})$ embeds continuously in $C^\alpha (\bar{{\mathcal {R}}})$ for all $\alpha \in (0,1)$ by [12, Chapter 2, Theorem 3.8] and we deduce that

$$\begin{aligned} (V,W)\in C({\bar{\mathcal {R}}},{\mathbb {R}}^2)\,. \end{aligned}$$

(A.3)

Next, after integrating by parts,

$$\begin{aligned} J(V,W) :=&\, \int _{\mathcal {R}} \big ( \partial _x V \partial _\eta W - \partial _\eta V \partial _x W \big )\ \mathrm {d}(x,\eta ) \\ =&\, \int _0^{1+d} \Big [ (V \partial _\eta W)(x,\eta ) \Big ]_{x=-L}^{x=L}\ \mathrm {d}\eta - \int _{\mathcal {R}} V \partial _x\partial _\eta W\ \mathrm {d}(x,\eta ) \\&\qquad - \int _D \Big [ (V\partial _x W)(x,\eta ) \Big ]_{\eta =0}^{\eta =1+d} + \int _{\mathcal {R}} V \partial _x\partial _\eta W\ \mathrm {d}(x,\eta )\,. \end{aligned}$$

Since $V(x,0) = V(x,1+d) = 0$ for $x\in D$ by (A.1a) and the second and fourth terms cancel each other out, we obtain

$$\begin{aligned} J(V,W) = \int _0^{1+d} V(L,\eta ) \partial _\eta W(L,\eta )\ \mathrm {d}\eta - \int _0^{1+d} V(-L,\eta ) \partial _\eta W(-L,\eta )\ \mathrm {d}\eta \,. \end{aligned}$$

Now, according to (A.1b) and the regularity of V and W,

$$\begin{aligned} \partial _\eta W(\pm L,\eta )&= - \tau _1^\pm \partial _\eta V(\pm L,\eta )\,, \qquad \eta \in (0,1)\,, \\ \partial _\eta W(\pm L,\eta )&= - \tau _2^\pm \partial _\eta V(\pm L,\eta )\,, \qquad \eta \in (1,1+d)\,, \end{aligned}$$

so that, since $[\eta \mapsto V(\pm L,\eta )] \in C([0,1+d])$ by (A.3),

$$\begin{aligned} J(V,W)&= - \tau _1^+ \int _0^1 (V \partial _\eta V)(L,\eta )\ \mathrm {d}\eta - \tau _2^+ \int _1^{1+d} (V \partial _\eta V)(L,\eta )\ \mathrm {d}\eta \nonumber \\&\qquad + \tau _1^- \int _0^1 (V \partial _\eta V)(-L,\eta )\ \mathrm {d}\eta + \tau _2^- \int _1^{1+d} (V \partial _\eta V)(-L,\eta )\ \mathrm {d}\eta \nonumber \\&= - \tau _1^+ \frac{V(L,1)^2 - V(L,0)^2}{2} - \tau _2^+ \frac{V(L,1+d)^2 - V(L,1)^2}{2} \nonumber \\&\qquad + \tau _1^- \frac{V(-L,1)^2 - V(-L,0)^2}{2} + \tau _2^- \frac{V(-L,1+d)^2 - V(-L,1)^2}{2} \nonumber \\&= \frac{\tau _1^+}{2} V(L,0)^2 - \frac{\tau _1^-}{2} V(-L,0)^2 - \frac{\tau _2^+}{2} V(L,1+d)^2 + \frac{\tau _2^-}{2} V(-L,1+d)^2 \\&\qquad - \frac{\tau _1^+ - \tau _2^+}{2} V(L,1)^2 + \frac{\tau _1^- - \tau _2^-}{2} V(-L,1)^2\,. \nonumber \end{aligned}$$

(A.4)

On the one hand, it follows from (A.1) and the continuity (A.3) of V that

$$\begin{aligned} \begin{aligned} V(\pm L,0)&= \lim _{x\rightarrow \pm L} V(x,0) = 0\,, \\ V(\pm L,1+d)&= \lim _{x\rightarrow \pm L} V(x,1+d) = 0\,. \end{aligned} \end{aligned}$$

(A.5)

On the other hand, using (A.1b) along with the continuity (A.3) gives

$$\begin{aligned} \tau _1^\pm V(\pm L, 1)&= \lim _{\eta \nearrow 1} \tau ^\pm (\eta ) V(\pm L,\eta ) = - \lim _{\eta \nearrow 1} W(\pm L, \eta ) \\&= - W(\pm L,1) = - \lim _{\eta \searrow 1} W(\pm L, \eta ) = \lim _{\eta \searrow 1} \tau ^\pm (\eta ) V(\pm L,\eta ) \\&= \tau _2^\pm V(\pm L, 1)\,. \end{aligned}$$

Consequently,

$$\begin{aligned} \left( \tau _1^\pm - \tau _2^\pm \right) V(\pm L, 1)= 0\,. \end{aligned}$$

(A.6)

Combining (A.4), (A.5), and (A.6) leads us to $J(V,W)=0$ and we have proved that

$$\begin{aligned} J(V,W) = 0\,, \qquad (V,W)\in {\mathcal {G}}^2({\mathcal {R}})\,. \end{aligned}$$

(A.7)

In other words, the identity stated in Lemma A.1 is valid for $(V,W)\in {\mathcal {G}}^2({\mathcal {R}})$. $\square$