Partial data inverse problems for quasilinear conductivity equations.

Introduction and statement of results

We show that the knowledge of the Dirichlet-to-Neumann maps given on an arbitrary open non-empty portion of the boundary of a smooth domain in R n , n ≥ 2 , for classes of semilinear and quasilinear conductivity equations, determines the nonlinear conductivities uniquely. The main ingredient in the proof is a certain L 1 -density result involving sums of products of gradients of harmonic functions which vanish on a closed proper subset of the boundary.

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ , $n\ge 2$ , be a connected bounded open set with $C^{\infty }$ boundary. Let us consider the Dirichlet problem for the following isotropic semilinear conductivity equation,

1.1 $$\begin{array}{c}\hfill \left\{\begin{array}{cc}\text{div}(\gamma (x,u)\mathrm{\nabla }u)=0\hfill & \text{in}\mathrm{\Omega },\hfill \\ u=f\hfill & \text{on}\partial \mathrm{\Omega }.\hfill \end{array}\right)\end{array}$$

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Here we assume that the function $\gamma :\overline{\mathrm{\Omega }}\times \mathbb{C}\to \mathbb{C}$ satisfies the following conditions,

• the map $\mathbb{C}\ni \tau \mapsto \gamma (·,\tau )$ is holomorphic with values in the Hölder space $C^{1,\alpha }(\overline{\mathrm{\Omega }})$ with some $0<\alpha <1$ ,
• $\gamma (x,0)=1$ , for all $x\in \mathrm{\Omega }$ .

The semilinear conductivity equation (1.1) can be viewed as a steady state semilinear heat equation where the conductivity depends on the temperature, and in physics, such models occur, for instance, in nonlinear heat conduction in composite materials, see [[23]].

It is shown in Theorem B.1 that under the assumptions (a) and (b), there exist $\delta >0$ and $C>0$ such that when $f\in B_{\delta }(\partial \mathrm{\Omega }):=\{f\in C^{2,\alpha }{(\partial \mathrm{\Omega }):\Vert f\Vert }_{C^{2,\alpha }(\partial \mathrm{\Omega })}<\delta \}$ , the problem (1.1) has a unique solution $u=u_f\in C^{2,\alpha }(\overline{\mathrm{\Omega }})$ satisfying ${\Vert u\Vert }_{C^{2,\alpha }(\overline{\mathrm{\Omega }})}<C\delta$ . Let $\mathrm{\Gamma }\subset \partial \mathrm{\Omega }$ be an arbitrary non-empty open subset of the boundary $\partial \mathrm{\Omega }$ . Associated to the problem (1.1), we define the partial Dirichlet-to-Neumann map

$$\begin{array}{c}\hfill {\mathrm{\Lambda }}_{\gamma }^{\mathrm{\Gamma }}(f)=(\gamma (x,u){\partial }_{\nu }u){|}_{\mathrm{\Gamma }},\end{array}$$

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where $f\in B_{\delta }(\partial \mathrm{\Omega })$ with $\text{supp}(f)\subset \mathrm{\Gamma }$ . Here $\nu$ is the unit outer normal to the boundary.

We are interested in the following inverse boundary problem for the semilinear conductivity equation (1.1): given the knowledge of the partial Dirichlet-to-Neumann map ${\mathrm{\Lambda }}_{\gamma }^{\mathrm{\Gamma }}$ , determine the semilinear conductivity $\gamma$ in $\overline{\mathrm{\Omega }}\times \mathbb{C}$ . Our first main result gives a complete solution to this problem.

Theorem 1.1

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ , $n\ge 2$ , be a connected bounded open set with $C^{\infty }$ boundary, and let $\mathrm{\Gamma }\subset \partial \mathrm{\Omega }$ be an arbitrary open non-empty subset of the boundary $\partial \mathrm{\Omega }$ . Let ${\gamma }_1,{\gamma }_2:\overline{\mathrm{\Omega }}\times \mathbb{C}\to \mathbb{C}$ satisfy the assumptions (a) and (b). If ${\mathrm{\Lambda }}_{{\gamma }_1}^{\mathrm{\Gamma }}={\mathrm{\Lambda }}_{{\gamma }_2}^{\mathrm{\Gamma }}$ then ${\gamma }_1={\gamma }_2$ in $\overline{\mathrm{\Omega }}\times \mathbb{C}$ .

It is also of great interest and importance to consider inverse boundary problems for nonlinear conductivity equations with conductivities depending not only on the solution u but also on its gradient, $\mathrm{\nabla }u$ . Such equations occur, in particular, in the study of transport properties of non-linear composite materials, see [[36]], as well as in glaciology, when modeling the stationary motion of a glacier, see [[12]]. Furthermore, such equations can be considered as a simple scalar model of the nonlinear elasticity system, see [[44], Section 2]. To this end, we are able to solve partial data inverse boundary problems for a class of quasilinear conductivities of the form $\gamma (x,u,\omega ·\mathrm{\nabla }u)$ , depending on the space variable, the solution, as well as the derivative of the solution in a fixed direction $\omega \in {\mathbb{S}}^{n-1}=\{\omega \in {\mathbb{R}}^n:|\omega |=1\}$ . To state the result, let $\omega \in {\mathbb{S}}^{n-1}$ be fixed and let us consider the Dirichlet problem for the following isotropic quasilinear conductivity equation,

1.2 $$\begin{array}{c}\hfill \left\{\begin{array}{cc}\text{div}(\gamma (x,u,\omega ·\mathrm{\nabla }u)\mathrm{\nabla }u)=0\hfill & \text{in}\mathrm{\Omega },\hfill \\ u=\lambda +f\hfill & \text{on}\partial \mathrm{\Omega }.\hfill \end{array}\right)\end{array}$$

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Here we assume that the function $\gamma :\overline{\mathrm{\Omega }}\times \mathbb{C}\times \mathbb{C}\to \mathbb{C}$ satisfies the following conditions,

• the map $\mathbb{C}\times \mathbb{C}\ni (\tau ,z)\mapsto \gamma (·,\tau ,z)$ is holomorphic with values in $C^{1,\alpha }(\overline{\mathrm{\Omega }})$ with some $0<\alpha <1$ ,
• $\gamma (x,\tau ,0)=1$ , for all $x\in \mathrm{\Omega }$ and all $\tau \in \mathbb{C}$ .

It is established in Theorem B.1 that under the assumptions (i) and (ii) for each $\lambda \in \mathbb{C}$ , there exist ${\delta }_{\lambda }>0$ and $C_{\lambda }>0$ such that when $f\in B_{{\delta }_{\lambda }}(\partial \mathrm{\Omega }):=\{f\in C^{2,\alpha }{(\partial \mathrm{\Omega }):\Vert f\Vert }_{C^{2,\alpha }(\partial \mathrm{\Omega })}<{\delta }_{\lambda }\}$ , the problem (1.2) has a unique solution $u=u_{\lambda ,f}\in C^{2,\alpha }(\overline{\mathrm{\Omega }})$ satisfying ${\Vert u-\lambda \Vert }_{C^{2,\alpha }(\overline{\mathrm{\Omega }})}<C_{\lambda }{\delta }_{\lambda }$ . Associated to the problem (1.2), we define the partial Dirichlet-to-Neumann map

$$\begin{array}{c}\hfill {\mathrm{\Lambda }}_{\gamma }^{\mathrm{\Gamma }}(\lambda +f)=(\gamma (x,u,\omega ·\mathrm{\nabla }u){\partial }_{\nu }u){|}_{\mathrm{\Gamma }},\end{array}$$

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where $f\in B_{{\delta }_{\lambda }}(\partial \mathrm{\Omega })$ with $\text{supp}(f)\subset \mathrm{\Gamma }$ and $\lambda \in \mathbb{C}$ .

Our second main result is as follows.

Theorem 1.2

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ , $n\ge 2$ , be a connected bounded open set with $C^{\infty }$ boundary, and let $\mathrm{\Gamma }\subset \partial \mathrm{\Omega }$ be an arbitrary open non-empty subset of the boundary $\partial \mathrm{\Omega }$ . Let $\omega \in {\mathbb{S}}^{n-1}$ be fixed. Assume that ${\gamma }_1,{\gamma }_2:\overline{\mathrm{\Omega }}\times \mathbb{C}\times \mathbb{C}\to \mathbb{C}$ satisfy the assumptions (i) and (ii). Let $\mathrm{\Sigma }\subset \mathbb{C}$ be a set which has a limit point in $\mathbb{C}$ . Then if for all $\lambda \in \mathrm{\Sigma }$ , we have

$$\begin{array}{c}\hfill {\mathrm{\Lambda }}_{{\gamma }_1}^{\mathrm{\Gamma }}(\lambda +f)={\mathrm{\Lambda }}_{{\gamma }_2}^{\mathrm{\Gamma }}(\lambda +f),\forall f\in B_{{\delta }_{\lambda }}(\partial \mathrm{\Omega }),\text{supp}(f)\subset \mathrm{\Gamma },\end{array}$$

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then ${\gamma }_1={\gamma }_2$ in $\overline{\mathrm{\Omega }}\times \mathbb{C}\times \mathbb{C}$ .

Note that in Theorem 1.2 the Dirichlet-to-Neumann maps ${\mathrm{\Lambda }}_{{\gamma }_j}^{\mathrm{\Gamma }}$ map the Dirichlet data $\lambda +f$ , which is not supported on $\mathrm{\Gamma }$ , unless $\lambda =0$ , to the Neumann data which is measured on $\mathrm{\Gamma }$ .

Remark 1.3

To the best of our knowledge, the partial data results of Theorem 1.1 and Theorem 1.2 are the first partial data results for nonlinear conductivity equations.

Remark 1.4

It might be interesting to note that an analog of the partial data results of Theorem 1.1 and Theorem 1.2 is still not known in the case of the linear conductivity equation in dimensions $n\ge 3$ . We refer to [[17]] for the corresponding partial data result for the linear conductivity equation in dimension $n=2$ .

Remark 1.5

An analog of Theorem 1.1 in the full data case, i.e. when $\mathrm{\Gamma }=\partial \mathrm{\Omega }$ , was proved in [[42]] where instead of working with small Dirichlet data one considers small perturbations of constant Dirichlet data as in (1.2). Furthermore, it was assumed in [[42]] that the semilinear conductivity is strictly positive while no analyticity was required. The proof of [[42]] relies on a first order linearization of the Dirichlet-to-Neumann map at constant Dirichlet boundary values which leads to the inverse boundary problem for the linear conductivity equation and therefore, an application of results of [[47]] and [[35]] for the linear conductivity problem in dimensions $n\ge 3$ and in dimension $n=2$ , respectively, gives the recovery of the semilinear conductivity.

Remark 1.6

To the best of our knowledge Theorem 1.2 is new even in the full data case. Indeed, in the full data case, so far authors have only considered the recovery of conductivities of the form $\gamma (x,u)$ , see e.g. [[42], [46]], or of the form $\gamma (u,\mathrm{\nabla }u)$ , see e.g. [[34], [41]], or conductivities which depend x and $\mathrm{\nabla }u$ in some specific way, see e.g. [[5]]. We obtain in Theorem 1.2, for what seems to be the first time, the recovery of some general class of quasilinear conductivities of the form $\gamma (x,u,\omega ·\mathrm{\nabla }u)$ , depending on the space variable, the solution, as well as the derivative of the solution in a fixed direction.

Remark 1.7

The assumption that the conductivity is holomorphic as a function $\mathbb{C}\ni \tau \mapsto \gamma (·,\tau ,·)$ in Theorem 1.2 is motivated by the proof of the solvability of the forward problem and the differentiability with respect to the boundary data. This assumption could perhaps be weakened and one could show that the full knowledge of the partial Dirichlet-to-Neumann map ${\mathrm{\Lambda }}_{\gamma }^{\mathrm{\Gamma }}$ determines the conductivity $\gamma$ . As the main focus of this paper is on establishing the partial data inverse results, we decided not to elaborate upon this issue further.

We remark that starting with [[27]], it has been known that nonlinearity may be helpful when solving inverse problems for hyperbolic PDE. Analogous phenomena for nonlinear elliptic equations have been revealed and exploited in [[10], [29]], see also [[24]–[26], [28], [30]]. A noteworthy aspect of all of these works is that the presence of a nonlinearity enables one to solve inverse problems for nonlinear PDE in situations where the corresponding inverse problems for linear equations are still open. The present paper is also concerned with illustrating this general phenomenon.

Let us proceed to discuss the main ideas of the proofs of Theorem 1.1 and Theorem 1.2. Using the technique of higher order linearizations of the partial Dirichlet-to-Neumann map, introduced in [[10], [29]], see also [[42], [46]] for the use of the second linearization, we reduce the proof of Theorem 1.2 to the following density result.

Theorem 1.8

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ , $n\ge 2$ , be a connected bounded open set with $C^{\infty }$ boundary, let $\mathrm{\Gamma }\subset \partial \mathrm{\Omega }$ be an open non-empty subset of $\partial \mathrm{\Omega }$ , let $\omega \in {\mathbb{S}}^{n-1}$ be fixed, and let $m=2,3,\cdots ,$ be fixed. Let $f\in L^{\infty }(\mathrm{\Omega })$ be such that

1.3 $$\begin{array}{c}\hfill {\int }_{\mathrm{\Omega }}f(\sum _{k=1}^m\prod _{r=1,r\ne k}^m(\omega ·\mathrm{\nabla }{u}_r)\mathrm{\nabla }{u}_k)·\mathrm{\nabla }{u}_{m+1}dx=0,\end{array}$$

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for all functions $u_l\in C^{\infty }(\overline{\mathrm{\Omega }})$ harmonic in $\mathrm{\Omega }$ with $\text{supp}(u_l{|}_{\partial \mathrm{\Omega }})\subset \mathrm{\Gamma }$ , $l=1,\cdots ,m+1$ . Then $f=0$ in $\mathrm{\Omega }$ .

Similarly, using higher order linearizations of the partial Dirichlet-to-Neumann map, we show that Theorem 1.1 will follow from the following density result.

Theorem 1.9

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ , $n\ge 2$ , be a connected bounded open set with $C^{\infty }$ boundary, let $\mathrm{\Gamma }\subset \partial \mathrm{\Omega }$ be an open non-empty subset of $\partial \mathrm{\Omega }$ , and let $m=2,3,\cdots ,$ be fixed. Let $f\in L^{\infty }(\mathrm{\Omega })$ be such that

1.4 $$\begin{array}{c}\hfill {\int }_{\mathrm{\Omega }}f(\sum _{k=1}^m\prod _{r=1,r\ne k}^m{u}_r\mathrm{\nabla }{u}_k)·\mathrm{\nabla }{u}_{m+1}dx=0,\end{array}$$

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for all functions $u_l\in C^{\infty }(\overline{\mathrm{\Omega }})$ harmonic in $\mathrm{\Omega }$ with $\text{supp}(u_l{|}_{\partial \mathrm{\Omega }})\subset \mathrm{\Gamma }$ , $l=1,\cdots ,m+1$ . Then $f=0$ in $\mathrm{\Omega }$ .

Theorems 1.8 and 1.9 can be viewed as extensions of the results of [[8]] and [[24]]. Indeed, it was proved in [[8]] that the linear span of the set of products of harmonic functions in $\mathrm{\Omega }$ which vanish on a closed proper subset of the boundary is dense in $L^1(\mathrm{\Omega })$ , and this density result was extended in [[24]] by showing that the linear span of the set of scalar products of gradients of harmonic functions in $\mathrm{\Omega }$ which vanish on a closed proper subset of the boundary is also dense in $L^1(\mathrm{\Omega })$ .

To prove Theorem 1.8, we shall follow the general strategy of the work [[8]], see also [[24]]. We first establish a corresponding local result in a neighborhood of a boundary point in $\mathrm{\Gamma }$ assuming, as we may, that $\mathrm{\Gamma }$ is a small open neighborhood of this point, see Proposition 2.1 below. We then show how to pass from this local result to the global one of Theorem 1.8. The essential difference here compared with the works [[8], [24]] is that working with products of $m+1$ gradients in the orthogonality identity (1.3), we need to prove a certain Runge type approximation theorem in the $W^{1,m+1}$ -topology for any $m=2,3,\cdots$ fixed, as opposed to $L^2$ and $H^1$ approximation results obtained in [[8]] and [[24]], respectively.

We shall only prove Theorem 1.8 as the proof of Theorem 1.9 is obtained by inspection of that proof as the only difference between the orthogonality relations (1.3) and (1.4) is that (1.3) contains $\omega ·\mathrm{\nabla }{u}_r$ with harmonic functions $u_r$ while (1.4) contains $u_r$ instead, and no new difficulties occur.

Remark 1.10

While the present paper was under review, the inverse boundary problem with full data, i.e. when the measurement are performed along the entire boundary $\partial \mathrm{\Omega }$ , was solved in [[6]] for quasilinear isotropic conductivity $\gamma$ of the form $\gamma (x,u,\mathrm{\nabla }u)$ , showing that the quasilinear conductivity $\gamma$ can indeed be uniquely determined from these measurements, provided that the map $\mathbb{C}\times {\mathbb{C}}^n\ni (\rho ,\mu )\mapsto \gamma (·,\rho ,\mu )$ is is holomorphic with values in $C^{1,\alpha }(\overline{\mathrm{\Omega }})$ with some $0<\alpha <1$ , and $0<\gamma (·,0,0)\in C^{\infty }(\overline{\mathrm{\Omega }})$ . It would be interesting to solve the partial data inverse problem for such conductivities to be on par with the full data result of [[6]]. The difficulty here compared with the recovery of the conductivities of the form $\gamma (x,u,\omega ·\mathrm{\nabla }u)$ in Theorem 1.2 is that higher order linearizations of the partial Dirichlet-to-Neumann map lead to a density statement in the spirit of Theorem 1.8 where instead of working with a scalar function f one has to work with a function with values in the space of symmetric tensors of rank $m\in \mathbb{N}$ . Furthermore, a challenge in the proof of partial data result compared with the full data result of [[6]] is that one has to work with harmonic functions which vanish on an arbitrary portion of the boundary in the density statement. It is not quite clear how to extend the analytic microlocal analysis framework of [[8]] to prove the needed density result in this more general situation.

Let us finally remark that inverse boundary problems for nonlinear elliptic PDE have been studied extensively in the literature. We refer to [[4], [7], [10], [15], [18]–[22], [26], [29], [34], [41]–[43], [45]], and the references given there. In particular, inverse boundary problems with partial data were studied for a certain class of semilinear equations of the form $-\mathrm{\Delta }u+V(x,u)=0$ in [[25], [30]] relying on the density result of [[8]], for semilinear equations of the form $-\mathrm{\Delta }u+q(x){(\mathrm{\nabla }u)}^2=0$ in [[24]], and for nonlinear magnetic Schrödinger equations in [[28]].

The paper is organized as follows. In Sect. 2 we establish Theorem 1.8. Theorem 1.2 in proven in Sect. 3. The proof of Theorem 1.1 occupies Sect. 4. In Appendix A we present an alternative simple proof of Theorem 1.2 in the full data case. In Appendix B we show the well-posedness of the Dirichlet problem for our quasilinear conductivity equation, in the case of boundary data close to a constant one.

Proof of Theorem 1.8

We shall proceed by following the general strategy of [[8]]. It suffices to assume that $\mathrm{\Gamma }\subset \partial \mathrm{\Omega }$ is a proper open nonempty subset of $\partial \mathrm{\Omega }$ , and even a small open neighborhood of some boundary point.

Local result

Theorem 1.8 will be obtained as a corollary of the following local result.

Proposition 2.1

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ , $n\ge 2$ , be a bounded open set with $C^{\infty }$ boundary, and let $m=2,3,\cdots ,$ be fixed. Let $x_0\in \partial \mathrm{\Omega }$ , and let $\tilde{\mathrm{\Gamma }}\subset \partial \mathrm{\Omega }$ be the complement of an open boundary neighborhood of $x_0$ . Then there exists $\delta >0$ such that if we have (1.3) for any harmonic functions $u_l\in C^{\infty }(\overline{\mathrm{\Omega }})$ satisfying $u_l{|}_{\tilde{\mathrm{\Gamma }}}=0$ , $l=1,\cdots ,m+1$ , then $f=0$ in $B(x_0,\delta )\cap \mathrm{\Omega }$ .

Proof

It suffices to choose $u_1=\cdots =u_m$ in (1.3). Hence, (1.3) implies that

2.1 $$\begin{array}{c}\hfill {\int }_{\mathrm{\Omega }}f{(\omega ·\mathrm{\nabla }{v}_1)}^{m-1}\mathrm{\nabla }{v}_1·\mathrm{\nabla }{v}_{2}dx=0,\end{array}$$

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for all harmonic functions $v_1,v_2\in C^{\infty }(\overline{\mathrm{\Omega }})$ satisfying $v_l{|}_{\tilde{\mathrm{\Gamma }}}=0$ , $l=1,2$ . Our goal is to show that (2.1) gives that $f=0$ in $B(x_0,\delta )\cap \mathrm{\Omega }$ with $\delta >0$ . Using conformal transformations (in particular Kelvin transforms) of harmonic functions as in [[8], Section 3], and arguing as in that work, we are reduced to the following setting: $x_0=0$ , the tangent plane to $\mathrm{\Omega }$ at $x_0$ is given by $x_1=0$ ,

$$\begin{array}{c}\hfill \mathrm{\Omega }\subset \{x\in {\mathbb{R}}^n:|x+e_1|<1\},\tilde{\mathrm{\Gamma }}=\{x\in \partial \mathrm{\Omega }:x_1\le -2c\},e_1=(1,0,\cdots ,0),\end{array}$$

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for some $c>0$ .

Let $p(\zeta )={\zeta }^2$ , $\zeta \in {\mathbb{C}}^n$ , be the principal symbol of $-\mathrm{\Delta }$ , holomorphically extended to ${\mathbb{C}}^n$ . Let $\zeta \in p^{-1}(0)$ and let $\chi \in C_0^{\infty }({\mathbb{R}}^n)$ be such that $\text{supp}(\chi )\subset \{x\in {\mathbb{R}}^n:x_1\le -c\}$ and $\chi =1$ on $\{x\in \partial \mathrm{\Omega }:x_1\le -2c\}$ . We shall work with harmonic functions of the form

2.2 $$\begin{array}{c}\hfill v(x,\zeta )=e^{-\frac{i}{h}x·\zeta }+r(x,\zeta ),\end{array}$$

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where r is the solution to the Dirichlet problem,

$$\begin{array}{c}\hfill \left\{\begin{array}{c}-\mathrm{\Delta }r=0\text{in}\mathrm{\Omega },\hfill \\ {r|}_{\partial \mathrm{\Omega }}=-(e^{-\frac{i}{h}x·\zeta }\chi ){|}_{\partial \mathrm{\Omega }}.\hfill \end{array}\right)\end{array}$$

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By the boundary elliptic regularity, we have $v\in C^{\infty }(\overline{\mathrm{\Omega }})$ , and furthermore ${v|}_{\tilde{\mathrm{\Gamma }}}=0$ . Since in view of (2.1) we shall work with products of $m+1$ gradients of harmonic functions, we need to have good estimates for the remainder r in $C^1(\overline{\mathrm{\Omega }})$ . To that end, in view of Sobolev's embedding, we would like to bound ${\Vert r\Vert }_{H^k(\mathrm{\Omega })}$ with $k\in \mathbb{N}$ , $k>n/2+1$ . Boundary elliptic regularity gives that for $k\ge 2$ ,

2.3 $$\begin{array}{c}\hfill {\Vert r\Vert }_{H^k(\mathrm{\Omega })}\le C{\Vert e^{-\frac{i}{h}x·\zeta }\chi \Vert }_{H^{k-1/2}(\partial \mathrm{\Omega })},\end{array}$$

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see [[9], Section 24.2]. Now by interpolation, we get

2.4 $$\begin{array}{c}\hfill \Vert e^{-\frac{i}{h}x·\zeta }{\chi \Vert }_{H^{k-1/2}(\partial \mathrm{\Omega })}\le \Vert e^{-\frac{i}{h}x·\zeta }{\chi \Vert }_{H^k(\partial \mathrm{\Omega })}^{1/2}{\Vert e^{-\frac{i}{h}x·\zeta }\chi \Vert }_{H^{k-1}(\partial \mathrm{\Omega })}^{1/2},\end{array}$$

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see [[14], Theorem 7.22, p. 189]. We have

$$\begin{array}{c}\hfill \Vert e^{-\frac{i}{h}x·\zeta }{\chi \Vert }_{L^2(\partial \mathrm{\Omega })}\le C{e}^{\frac{1}{h}{sup}_{x\in K}x·\text{Im}\zeta },\end{array}$$

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where $K=\text{supp}\chi \cap \partial \mathrm{\Omega }$ , and therefore,

2.5 $$\begin{array}{c}\hfill \Vert e^{-\frac{i}{h}x·\zeta }{\chi \Vert }_{H^k(\partial \mathrm{\Omega })}\le C(1+\frac{|\zeta |}{h}+\cdots +\frac{{|\zeta |}^k}{h^k})e^{\frac{1}{h}{sup}_{x\in K}x·\text{Im}\zeta }.\end{array}$$

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It follows from (2.4) and (2.5) that

2.6 $$\begin{array}{c}\hfill \Vert e^{-\frac{i}{h}x·\zeta }{\chi \Vert }_{H^{k-1/2}(\partial \mathrm{\Omega })}\le C(1+\frac{{|\zeta |}^k}{h^k})e^{\frac{1}{h}{sup}_{x\in K}x·\text{Im}\zeta }.\end{array}$$

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Using (2.3) and (2.6), we see that

$$\begin{array}{c}\hfill {\Vert r\Vert }_{H^k(\mathrm{\Omega })}\le C(1+\frac{{|\zeta |}^k}{h^k})e^{\frac{1}{h}{sup}_{x\in K}x·\text{Im}\zeta }.\end{array}$$

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Taking $k>n/2+1$ and using the Sobolev embedding $H^k(\mathrm{\Omega })\subset C^1(\overline{\mathrm{\Omega }})$ , we get

2.7 $$\begin{array}{c}\hfill {\Vert r\Vert }_{C^1(\overline{\mathrm{\Omega }})}\le C(1+\frac{{|\zeta |}^k}{h^k})e^{\frac{1}{h}{sup}_{x\in K}x·\text{Im}\zeta }.\end{array}$$

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Using that $\text{supp}(\chi )\subset \{x\in {\mathbb{R}}^n:x_1\le -c\}$ and $\chi =1$ on $\{x\in \partial \mathrm{\Omega }:x_1\le -2c\}$ , we obtain from (2.7) that

2.8 $$\begin{array}{c}\hfill {\Vert r\Vert }_{C^1(\overline{\mathrm{\Omega }})}\le C(1+\frac{{|\zeta |}^k}{h^k})e^{-\frac{c}{h}\text{Im}{\zeta }_1}{e}^{\frac{1}{h}|\text{Im}{\zeta }^{\prime }|},\end{array}$$

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when $\text{Im}{\zeta }_1\ge 0$ .

Now the identity (2.1) implies that

2.9 $$\begin{array}{c}\hfill {\int }_{\mathrm{\Omega }}f(x){(\omega ·hDv(x,\zeta ))}^{m-1}hDv(x,\zeta )·hDv(x,m\eta )dx=0,\end{array}$$

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for all $\zeta ,\eta \in p^{-1}(0)$ . Here $v(x,\zeta )$ and $v(x,m\eta )$ are harmonic functions of the form (2.2) and $D=i^{-1}\mathrm{\nabla }$ . Using that

$$\begin{array}{cc}& {(\omega ·hDv(x,\zeta ))}^{m-1}={(-\omega ·\zeta e^{-\frac{i}{h}x·\zeta }+\omega ·hDr(x,\zeta ))}^{m-1}\hfill \\ \hfill & ={(-\omega ·\zeta )}^{m-1}{e}^{-\frac{(m-1)i}{h}x·\zeta }+\sum _{l=1}^{m-1}\left(\begin{array}{c}m-1\\ l\end{array}\right){(\omega ·hDr(x,\zeta ))}^l{(-\omega ·\zeta e^{-\frac{i}{h}x·\zeta })}^{m-1-l},\hfill \end{array}$$

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we obtain from (2.9) that

2.10 $$\begin{array}{c}\hfill {\int }_{\mathrm{\Omega }}f(x){(-\omega ·\zeta )}^{m-1}m(\zeta ·\eta )e^{-\frac{\mathit{mi}}{h}x·(\zeta +\eta )}dx=I_1+I_2,\end{array}$$

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where

$$\begin{array}{cc}\hfill I_1=& -{\int }_{\mathrm{\Omega }}f(x){(-\omega ·\zeta )}^{m-1}{e}^{-\frac{(m-1)i}{h}x·\zeta }(-\zeta e^{-\frac{i}{h}x·\zeta }·hDr(x,m\eta )\hfill \\ & -m\eta e^{-\frac{\mathit{mi}}{h}x·\eta }·hDr(x,\zeta )+hDr(x,\zeta )·hDr(x,m\eta ))dx,\hfill \\ \hfill I_2=& -{\int }_{\mathrm{\Omega }}f(x)\sum _{l=1}^{m-1}\left(\begin{array}{c}m-1\\ l\end{array}\right){(\omega ·hDr(x,\zeta ))}^l{(-\omega ·\zeta e^{-\frac{i}{h}x·\zeta })}^{m-1-l}\hfill \\ & (m\zeta ·\eta e^{-\frac{i}{h}x·(\zeta +m\eta )}-\zeta e^{-\frac{i}{h}x·\zeta }·hDr(x,m\eta )-m\eta e^{-\frac{\mathit{mi}}{h}x·\eta }·hDr(x,\zeta )\hfill \\ & +hDr(x,\zeta )·hDr(x,m\eta ))dx.\hfill \end{array}$$

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We shall next proceed to bound the absolute values of $I_1$ and $I_2$ . To that end, first note that when $\text{Im}{\zeta }_1\ge 0$ , using the fact that $\mathrm{\Omega }\subset \{x\in {\mathbb{R}}^n:|x+e_1|<1\}$ , we have

2.11

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Using (2.8) and (2.11), we obtain that for all $\zeta ,\eta \in p^{-1}(0)$ , $\text{Im}{\zeta }_1\ge 0$ , $\text{Im}{\eta }_1\ge 0$ ,

2.12 $$\begin{array}{c}\hfill \begin{array}{cc}& |I_1{|\le C\Vert f\Vert }_{L^{\infty }}{e}^{\frac{m(|\text{Im}{\zeta }^{\prime }|+|\text{Im}{\eta }^{\prime }|)}{h}}{e}^{-\frac{c}{h}min(\text{Im}{\zeta }_1,\text{Im}{\eta }_1)}{|\zeta |}^{m-1}\hfill \\ \hfill & (|\zeta |h(1+\frac{{|m\eta |}^k}{h^k})+m|\eta |h(1+\frac{{|\zeta |}^k}{h^k})+h^2(1+\frac{{|m\eta |}^k}{h^k})(1+\frac{{|\zeta |}^k}{h^k})),\hfill \end{array}\end{array}$$

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and

2.13 $$\begin{array}{cc}\hfill |I_2|\le & {C\Vert f\Vert }_{L^{\infty }}{e}^{\frac{m(|\text{Im}{\zeta }^{\prime }|+|\text{Im}{\eta }^{\prime }|)}{h}}{e}^{-\frac{c}{h}min(\text{Im}{\zeta }_1,\text{Im}{\eta }_1)}h(1+\frac{{|\zeta |}^k}{h^k}{)}^{m-1}{(1+|\zeta |}^{m-2})\hfill \\ \hfill & (m|\zeta ||\eta |+|\zeta |h(1+\frac{{|m\eta |}^k}{h^k})+m|\eta |h(1+\frac{{|\zeta |}^k}{h^k})\hfill \\ \hfill & +h^2(1+\frac{{|m\eta |}^k}{h^k})(1+\frac{{|\zeta |}^k}{h^k})).\hfill \end{array}$$

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As noticed in [[8]], the differential of the map

$$\begin{array}{c}\hfill s:p^{-1}(0)\times p^{-1}(0)\to {\mathbb{C}}^n,(\zeta ,\eta )\mapsto \zeta +\eta .\end{array}$$

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at a point $({\zeta }_0,{\eta }_0)$ is surjective, provided that ${\zeta }_0$ and ${\eta }_0$ are linearly independent. The latter holds if ${\zeta }_0=\gamma$ and ${\eta }_0=-\overline{\gamma }$ with $\gamma \in {\mathbb{C}}^n$ given as follows. Recall that $\omega =({\omega }_1,\cdots ,{\omega }_n)\in {\mathbb{S}}^{n-1}$ is fixed. Then there exists ${\omega }_k\ne 0$ , and if $2\le k\le n$ we set $\gamma =(i,0,\cdots ,0,1,0,\cdots ,0)$ where 1 is on the kth position. If ${\omega }_1\ne 0$ then we set $\gamma =(i,1,0,\cdots ,0)\in {\mathbb{C}}^n$ .

Note that $\gamma ·\omega \ne 0$ and ${\zeta }_0+{\eta }_0=2i{e}_1$ . An application of the inverse function theorem gives that there exists $\epsilon >0$ small such that any $z\in {\mathbb{C}}^n$ , $|z-2i{e}_1|<2\epsilon$ , may be decomposed as $z=\zeta +\eta$ where $\zeta ,\eta \in p^{-1}(0)$ , $|\zeta -\gamma |<C_1\epsilon$ and $|\eta +\overline{\gamma }|<C_1\epsilon$ with some $C_1>0$ . We obtain that any $z\in {\mathbb{C}}^n$ such that $|z-2ia{e}_1|<2\epsilon a$ for some $a>0$ , may be decomposed as

2.14 $$\begin{array}{c}\hfill z=\zeta +\eta ,\zeta ,\eta \in p^{-1}(0),|\zeta -a\gamma |<C_{1}a\epsilon ,|\eta +a\overline{\gamma }|<C_{1}a\epsilon .\end{array}$$

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It follows from (2.14) that

2.15 $$\begin{array}{c}\hfill |\text{Im}{\zeta }^{\prime }|<C_{1}a\epsilon ,|\text{Im}{\eta }^{\prime }|<C_{1}a\epsilon ,|\zeta |\le Ca,|\eta |\le Ca.\end{array}$$

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We also conclude from (2.14) that for $\epsilon >0$ small enough,

2.16 $$\begin{array}{c}\hfill \text{Im}{\zeta }_1>a/2,\text{Im}{\eta }_1>a/2,|\zeta ·\eta |\ge a^2,|\omega ·\zeta |>\frac{a}{2}\sqrt{{\omega }_1^2+{\omega }_k^2}.\end{array}$$

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Hence, assuming that $a>1$ , we obtain from (2.10) with the help of (2.12), (2.13), (2.14), (2.15), (2.16) that

2.17 $$\begin{array}{c}\hfill \begin{array}{cc}\hfill |{\int }_{\mathrm{\Omega }}f(x)e^{-\frac{\mathit{mi}}{h}x·z}dx|& \le {C\Vert f\Vert }_{L^{\infty }}{e}^{-\frac{\mathit{ca}}{2h}}{e}^{\frac{2m{C}_{1}a\epsilon }{h}}(\frac{a}{h}{)}^N\hfill \\ \hfill & \le {C\Vert f\Vert }_{L^{\infty }}{e}^{-\frac{\mathit{ca}}{4h}}{e}^{\frac{2m{C}_{1}a\epsilon }{h}},\hfill \end{array}\end{array}$$

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for all $z\in {\mathbb{C}}^n$ such that $|z-2ia{e}_1|<2\epsilon a$ and $\epsilon >0$ sufficiently small. Here N is a fixed integer which depends on k and m. The estimate (2.17) is completely analogous to the bound (3.8) in [[8]], and hence, the proof of Proposition 2.1 is completed by repeating the arguments of [[8]] exactly as they stand. The idea is to extrapolate the exponential decay to more values of the frequency variable z which is achieved in [[8]] by using a variant of the proof of the Watermelon theorem. $\square$

Next in order to pass from this local result to the global one of Theorem 1.8, we need a Runge type approximation theorem in the $W^{1,m+1}$ -topology, $m=2,3,\cdots$ , which will extend [[8], Lemma 2.2] and [[24], Lemma 2.2], where approximations in the $L^2$ and $H^1$ topologies were established, respectively. To prove such an approximation theorem, we need to recall some facts about $L^p$ based Sobolev spaces which we shall now proceed to do.

Some facts about Lp based Sobolev spaces

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ , $n\ge 2$ , be a bounded open set with $C^{\infty }$ boundary, and let $1<p<\infty$ . Then we have for the dual space of the Sobolev space $W^{1,p}(\mathrm{\Omega })$ ,

$$\begin{array}{c}\hfill {(W^{1,p}(\mathrm{\Omega }))}^{\ast }={\tilde{W}}^{-1,p^{\prime }}(\mathrm{\Omega }),\end{array}$$

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where

$$\begin{array}{c}\hfill {\tilde{W}}^{-1,p^{\prime }}(\mathrm{\Omega })=\{u\in W^{-1,p^{\prime }}({\mathbb{R}}^n):\text{supp}(u)\subset \overline{\mathrm{\Omega }}\},\end{array}$$

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and $\frac{1}{p}+\frac{1}{p^{\prime }}=1$ , see [[3], page 163], [[38], Section 4.3.2]. The duality pairing is defined as follows: if $v\in {\tilde{W}}^{-1,p^{\prime }}(\mathrm{\Omega })$ and $u\in W^{1,p}(\mathrm{\Omega })$ , we set

2.18 $$\begin{array}{c}\hfill {(v,u)}_{{\tilde{W}}^{-1,p^{\prime }}(\mathrm{\Omega }),W^{1,p}(\mathrm{\Omega })}:={(v,\text{Ext}(u))}_{W^{-1,p^{\prime }}({\mathbb{R}}^n),W^{1,p}({\mathbb{R}}^n)},\end{array}$$

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where $\text{Ext}(u)\in W^{1,p}({\mathbb{R}}^n)$ is an arbitrary extension of u, see [[2], Theorem 9.7] for the existence of such an extension, and ${(·,·)}_{W^{-1,p^{\prime }}({\mathbb{R}}^n),W^{1,p}({\mathbb{R}}^n)}$ is the extension of $L^2$ scalar product ${(\phi ,\psi )}_{L^2({\mathbb{R}}^n)}={\int }_{{\mathbb{R}}^n}\phi (x)\overline{\psi (x)}dx$ . One can show that the definition (2.18) is independent of the choice of an extension.

We shall also need the following fact, see [[38], Section 4.3.2, p. 318].

Proposition 2.2

$C_0^{\infty }(\mathrm{\Omega })$ is dense in ${\tilde{W}}^{-1,p^{\prime }}(\mathrm{\Omega })$ with respect to $W^{-1,p^{\prime }}({\mathbb{R}}^n)$ topology.

We have the following result concerning the solvability of the Dirichlet problem for the Laplacian, see [[32], Theorem 7.10.2, p. 494].

Theorem 2.3

Let $v\in W^{-1,p}(\mathrm{\Omega })$ and $g\in W^{1-1/p,p}(\partial \mathrm{\Omega })$ with $1<p<\infty$ . Then the Dirichlet problem

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\mathrm{\Delta }u=v\hfill & \text{in}\mathrm{\Omega },\hfill \\ {u|}_{\partial \mathrm{\Omega }}=g,\hfill \end{array}\right)\end{array}$$

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has a unique solution $u\in W^{1,p}(\mathrm{\Omega })$ . Moreover,

$$\begin{array}{c}\hfill {\Vert u\Vert }_{W^{1,p}(\mathrm{\Omega })}\le {C(\Vert v\Vert }_{W^{-1,p}(\mathrm{\Omega })}+{\Vert g\Vert }_{W^{1-1/p,p}(\partial \mathrm{\Omega })}).\end{array}$$

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We shall also need the following result about the structure of distributions in $W^{-1,p}({\mathbb{R}}^n)$ supported by a smooth hypersurface in ${\mathbb{R}}^n$ . We refer to [[1], Theorem 5.1.13], [[31], Lemma 3.39] for this result in the case of distributions in $H^{-1}({\mathbb{R}}^n)$ . Since we did not find a reference for the case of distributions in $W^{-1,p}({\mathbb{R}}^n)$ with $1<p<\infty$ , we shall present the proof of this result here.

Proposition 2.4

Let F be a smooth compact hypersurface in ${\mathbb{R}}^n$ . Let $u\in W^{-1,p}({\mathbb{R}}^n)$ , with some $1<p<\infty$ , be such that $\text{supp}(u)\subset F$ . Then

$$\begin{array}{c}\hfill u=v\otimes {\delta }_F,v\in {(W^{1-1/p^{\prime },p^{\prime }}(F))}^{\ast }=B_{p,p}^{-(1-1/p^{\prime })}(F).\end{array}$$

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Here $\frac{1}{p}+\frac{1}{p^{\prime }}=1$ and $B_{p,p}^{-(1-1/p^{\prime })}(F)$ is the Besov space on the manifold F, see [[38], Section 2.3.1, p. 169], [[39]] for the definition, and for any $\phi \in C_0^{\infty }({\mathbb{R}}^n)$ , $u(\phi )=(v\otimes {\delta }_F)(\phi )=v(\phi {|}_F)$ .

Proof

Introducing a partition of unity and making a smooth change of variables, we see that it suffices to establish the following local result: let $u\in W^{-1,p}({\mathbb{R}}^n)$ , $1<p<\infty$ , such that $\text{supp}(u)\subset \{x_n=0\}$ , then $u=v\otimes {\delta }_{x_n=0}$ with $v\in {(W^{1-1/p^{\prime },p^{\prime }}({\mathbb{R}}^{n-1}))}^{\ast }=B_{p,p}^{-(1-1/p^{\prime })}({\mathbb{R}}^{n-1})$ . In order to prove this result we follow [[31], Lemma 3.39].

First we claim that if $\phi \in C_0^{\infty }({\mathbb{R}}^n)$ is such that ${\phi |}_{x_n=0}=0$ then $u(\phi )=0$ . To that end, we let

$$\begin{array}{c}\hfill {\phi }_{\pm }(x)=\left\{\begin{array}{cc}\phi (x),\hfill & \text{if}x\in {\mathbb{R}}_{\pm }^n=\{x\in {\mathbb{R}}^n:\pm x_n>0\},\hfill \\ 0,\hfill & \text{otherwise}.\hfill \end{array}\right)\end{array}$$

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Then ${\phi }_{\pm }\in W^{1,p^{\prime }}({\mathbb{R}}^n)$ and therefore, by [[2], Proposition 9.18], ${\phi }_{\pm }\in W_0^{1,p^{\prime }}({\mathbb{R}}_{\pm }^n)$ . Thus, there exist sequences ${\phi }_{j,\pm }\in C_0^{\infty }({\mathbb{R}}_{\pm }^n)$ such that ${\phi }_{j,\pm }\to {\phi }_{\pm }$ in $W^{1,p^{\prime }}({\mathbb{R}}_{\pm }^n)$ as $j\to \infty$ . Letting

$$\begin{array}{c}\hfill {\chi }_j(x)=\left\{\begin{array}{cc}{\phi }_{j,+}(x),\hfill & \text{if}x\in {\mathbb{R}}_{+}^n,\hfill \\ {\phi }_{j,-}(x),\hfill & \text{if}x\in {\mathbb{R}}_{-}^n,\hfill \end{array}\right)\end{array}$$

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we see that ${\chi }_j\in C_0^{\infty }({\mathbb{R}}^n)$ , ${\chi }_j=0$ near $\{x_n=0\}$ , and ${\chi }_j\to \phi$ in $W^{1,p^{\prime }}({\mathbb{R}}^n)$ . Hence, we have $0=u({\chi }_j)\to u(\phi )$ , and therefore, $u(\phi )=0$ , establishing the claim.

To proceed we need the following result, see [[33], [13], Theorem 1.5.1.1, p. 37]. The trace operator ${u\mapsto u|}_{x_n=0}$ , which is defined on $C_0^{\infty }({\mathbb{R}}^n)$ , has a unique continuous extension as an operator,

$$\begin{array}{c}\hfill \gamma :W^{1,p^{\prime }}({\mathbb{R}}^n)\to W^{1-1/p^{\prime },p^{\prime }}({\mathbb{R}}^{n-1}),1<p^{\prime }<\infty .\end{array}$$

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This operator has a right continuous inverse, the extension operator,

$$\begin{array}{c}\hfill E:W^{1-1/p^{\prime },p^{\prime }}({\mathbb{R}}^{n-1})\to W^{1,p^{\prime }}({\mathbb{R}}^n)\end{array}$$

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so that $\gamma (E\psi )=\psi$ for all $\psi \in W^{1-1/p^{\prime },p^{\prime }}({\mathbb{R}}^{n-1})$ .

Now we define

2.19 $$\begin{array}{c}\hfill v(\phi )=u(E\phi ),\phi \in C_0^{\infty }({\mathbb{R}}^{n-1}).\end{array}$$

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We have

$$\begin{array}{c}\hfill |v(\phi )|\le {\Vert u\Vert }_{W^{-1,p}({\mathbb{R}}^n)}{\Vert E\phi \Vert }_{W^{1,p^{\prime }}({\mathbb{R}}^n)}\le {C\Vert u\Vert }_{W^{-1,p}({\mathbb{R}}^n)}{\Vert \phi \Vert }_{W^{1-1/p^{\prime },p^{\prime }}({\mathbb{R}}^{n-1})},\end{array}$$

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and therefore, $v\in {(W^{1-1/p^{\prime },p^{\prime }}({\mathbb{R}}^{n-1}))}^{\ast }$ . Note that when $1<p^{\prime }<\infty$ ,

$$\begin{array}{c}\hfill W^{1-1/p^{\prime },p^{\prime }}({\mathbb{R}}^{n-1})=B_{p^{\prime },p^{\prime }}^{1-1/p^{\prime }}({\mathbb{R}}^{n-1}),{(B_{p^{\prime },p^{\prime }}^{1-1/p^{\prime }}({\mathbb{R}}^{n-1}))}^{\ast }=B_{p,p}^{-(1-1/p^{\prime })}({\mathbb{R}}^{n-1}),\end{array}$$

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see [[38], Section 2.5, p. 190, and Section 2.6.1, p. 198].

Finally, we claim that $u-v\otimes {\delta }_{x_n=0}=0$ . Indeed, letting $\phi \in C_0^{\infty }({\mathbb{R}}^n)$ and using (2.19) and our first claim, we get

$$\begin{array}{c}\hfill (u-v\otimes {\delta }_{x_n=0})(\phi )=u(\phi )-v(\phi {|}_{x_n=0})=u(\phi -E(\phi {|}_{x_n=0}))=0.\end{array}$$

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This completes the proof of Proposition 2.4. $\square$

Runge type approximation

Let ${\mathrm{\Omega }}_1\subset {\mathrm{\Omega }}_2\subset {\mathbb{R}}^n$ , $n\ge 2$ , be two bounded open sets with $C^{\infty }$ boundaries such that ${\mathrm{\Omega }}_2\backslash \overline{{\mathrm{\Omega }}_1}\ne \mathrm{\varnothing }$ . Suppose that $\partial {\mathrm{\Omega }}_1\cap \partial {\mathrm{\Omega }}_2=\overline{U}$ where $U\subset \partial {\mathrm{\Omega }}_1$ is open with $C^{\infty }$ boundary. Let $\mathcal{G}:C^{\infty }(\overline{{\mathrm{\Omega }}_2})\to C^{\infty }(\overline{{\mathrm{\Omega }}_2})$ , $a\mapsto w$ , be the solution operator to the Dirichlet problem,

$$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\mathrm{\Delta }w=a\hfill & \text{in}{\mathrm{\Omega }}_2,\hfill \\ {w|}_{\partial {\mathrm{\Omega }}_2}=0.\hfill \end{array}\right)\end{array}$$

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The following result is an extension of [[8], Lemma 2.2] and [[24], Lemma 2.2], where the similar density results were obtained in the $L^2$ and $H^1$ topologies, respectively.

Lemma 2.5

The space

$$\begin{array}{c}\hfill W:=\{\mathcal{G}a{|}_{{\mathrm{\Omega }}_1}:a\in C^{\infty }(\overline{{\mathrm{\Omega }}_2}),\text{supp}(a)\subset {\mathrm{\Omega }}_2\backslash \overline{{\mathrm{\Omega }}_1}\}\end{array}$$

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is dense in the space

$$\begin{array}{c}\hfill S:=\{u\in C^{\infty }(\overline{{\mathrm{\Omega }}_1}):-\mathrm{\Delta }u=0\text{in}{\mathrm{\Omega }}_1,u{|}_{\partial {\mathrm{\Omega }}_1\cap \partial {\mathrm{\Omega }}_2}=0\},\end{array}$$

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with respect to the $W^{1,p}({\mathrm{\Omega }}_1)$ -topology, for any $1<p<\infty$ .

Proof

We shall follow the proof of [[24], Lemma 2.2] closely, adapting it to the $L^p$ based Sobolev spaces. Let $v\in {\tilde{W}}^{-1,p^{\prime }}({\mathrm{\Omega }}_1)$ , $\frac{1}{p}+\frac{1}{p^{\prime }}=1$ , be such that

2.20 $$\begin{array}{c}\hfill {(v,\mathcal{G}a{|}_{{\mathrm{\Omega }}_1})}_{{\tilde{W}}^{-1,p^{\prime }}({\mathrm{\Omega }}_1),W^{1,p}({\mathrm{\Omega }}_1)}=0\end{array}$$

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for any $a\in C^{\infty }(\overline{{\mathrm{\Omega }}_2})$ , $\text{supp}(a)\subset {\mathrm{\Omega }}_2\backslash \overline{{\mathrm{\Omega }}_1}$ . In view of the Hahn–Banach theorem, we have to prove that

$$\begin{array}{c}\hfill {(v,u)}_{{\tilde{W}}^{-1,p^{\prime }}({\mathrm{\Omega }}_1),W^{1,p}({\mathrm{\Omega }}_1)}=0,\end{array}$$

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for any $u\in S$ .

To that end, we first note that as $\mathcal{G}a\in C^{\infty }(\overline{{\mathrm{\Omega }}_2})$ and ${\mathcal{G}a|}_{\partial {\mathrm{\Omega }}_2}=0$ , we have $\mathcal{G}a\in W_0^{1,p}({\mathrm{\Omega }}_2)$ . By [[2], Proposition 9.18], we can view $\mathcal{G}a$ as an element of $W^{1,p}({\mathbb{R}}^n)$ via an extension by 0 to ${\mathbb{R}}^n\backslash {\mathrm{\Omega }}_2$ . By the definition of $W_0^{1,p}({\mathrm{\Omega }}_2)$ , there exists a sequence ${\phi }_j\in C_0^{\infty }({\mathrm{\Omega }}_2)$ such that ${\phi }_j\to \mathcal{G}a$ in $W^{1,p}({\mathbb{R}}^n)$ . We have in view of (2.20) that

2.21 $$\begin{array}{c}\hfill \begin{array}{c}\hfill 0={(v,\mathcal{G}a)}_{W^{-1,p^{\prime }}({\mathbb{R}}^n),W^{1,p}({\mathbb{R}}^n)}=\underset{j\to \infty }{lim}{(v,{\phi }_j)}_{W^{-1,p^{\prime }}({\mathbb{R}}^n),W^{1,p}({\mathbb{R}}^n)}\\ \hfill =\underset{j\to \infty }{lim}{(v,{\phi }_j)}_{W^{-1,p^{\prime }}({\mathrm{\Omega }}_2),W_0^{1,p}({\mathrm{\Omega }}_2)}={(v,\mathcal{G}a)}_{W^{-1,p^{\prime }}({\mathrm{\Omega }}_2),W_0^{1,p}({\mathrm{\Omega }}_2)}.\end{array}\\ \hfill \end{array}$$

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Next, Proposition 2.2 implies that there is a sequence $v_j\in C_0^{\infty }({\mathrm{\Omega }}_1)$ such that $v_j\to v$ in $W^{-1,p^{\prime }}({\mathbb{R}}^n)$ . Consider the following Dirichlet problems,

2.22 $$\begin{array}{c}\hfill \left\{\begin{array}{cc}{-\mathrm{\Delta }f=v|}_{{\mathrm{\Omega }}_2}\in W^{-1,p^{\prime }}({\mathrm{\Omega }}_2)\hfill & \text{in}{\mathrm{\Omega }}_2,\hfill \\ f=0\hfill & \text{on}\partial {\mathrm{\Omega }}_2,\hfill \end{array}\right)\left\{\begin{array}{cc}-\mathrm{\Delta }{f}_j=v_j\hfill & \text{in}{\mathrm{\Omega }}_2,\hfill \\ f_j=0\hfill & \text{on}\partial {\mathrm{\Omega }}_2.\hfill \end{array}\right)\end{array}$$

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By Theorem 2.3, the problems (2.22) have unique solutions $f\in W_0^{1,p^{\prime }}({\mathrm{\Omega }}_2)$ and $f_j\in C^{\infty }(\overline{{\mathrm{\Omega }}_2})\cap W_0^{1,p^{\prime }}({\mathrm{\Omega }}_2)$ , respectively.

Using (2.21), (2.22), we get

2.23 $$\begin{array}{cc}\hfill 0=& {(v,\mathcal{G}a)}_{W^{-1,p^{\prime }}({\mathrm{\Omega }}_2),W_0^{1,p}({\mathrm{\Omega }}_2)}=\underset{j\to \infty }{lim}{(v_j,\mathcal{G}a)}_{W^{-1,p^{\prime }}({\mathrm{\Omega }}_2),W_0^{1,p}({\mathrm{\Omega }}_2)}\hfill \\ \hfill =& \underset{j\to \infty }{lim}{(-\mathrm{\Delta }{f}_j,\mathcal{G}a)}_{W^{-1,p^{\prime }}({\mathrm{\Omega }}_2),W_0^{1,p}({\mathrm{\Omega }}_2)}=\underset{j\to \infty }{lim}{\int }_{{\mathrm{\Omega }}_2}(-\mathrm{\Delta }{f}_j)\overline{\mathcal{G}a}dx\hfill \\ \hfill =& \underset{j\to \infty }{lim}{\int }_{{\mathrm{\Omega }}_2}{f}_j\overline{a}dx={\int }_{{\mathrm{\Omega }}_2}f\overline{a}dx.\hfill \end{array}$$

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Here we have used Green's formula, the fact that $f_j{{|}_{\partial {\mathrm{\Omega }}_2}=\mathcal{G}a|}_{\partial {\mathrm{\Omega }}_2}=0$ , and that

$$\begin{array}{c}\hfill \Vert f-f_j{\Vert }_{W^{1,p^{\prime }}({\mathrm{\Omega }}_2)}\le C{\Vert v-v_j\Vert }_{W^{-1,p^{\prime }}({\mathbb{R}}^n)},\end{array}$$

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which is a consequence of Theorem 2.3.

It follows from (2.23) that $f=0$ in ${\mathrm{\Omega }}_2\backslash \overline{{\mathrm{\Omega }}_1}$ . This together with the fact that $f\in W_0^{1,p^{\prime }}({\mathrm{\Omega }}_2)$ , in view of [[2], Proposition 9.18], allows us to conclude that $f\in W_0^{1,p^{\prime }}({\mathrm{\Omega }}_1)$ . Thus, there exists a sequence ${\widehat{f}}_j\in C_0^{\infty }({\mathrm{\Omega }}_1)$ be such that ${\widehat{f}}_j\to f$ in $W^{1,p^{\prime }}({\mathbb{R}}^n)$ , and therefore, $-\mathrm{\Delta }{\widehat{f}}_j\to -\mathrm{\Delta }f$ in $W^{-1,p^{\prime }}({\mathbb{R}}^n)$ .

Let $u\in S$ and let $\text{Ext}(u)\in W^{1,p}({\mathbb{R}}^n)$ be an extension of u. Using Green's formula, we get

2.24 $$\begin{array}{cc}\hfill {(-\mathrm{\Delta }f,\text{Ext}(u))}_{W^{-1,p^{\prime }}({\mathbb{R}}^n),W^{1,p}({\mathbb{R}}^n)}=& \underset{j\to \infty }{lim}{((-\mathrm{\Delta }{\widehat{f}}_j),\text{Ext}(u))}_{W^{-1,p^{\prime }}({\mathbb{R}}^n),W^{1,p}({\mathbb{R}}^n)}\hfill \\ \hfill =& \underset{j\to \infty }{lim}{\int }_{{\mathrm{\Omega }}_1}(-\mathrm{\Delta }{\widehat{f}}_j)\overline{u}dx=0.\hfill \end{array}$$

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Let $g=-\mathrm{\Delta }f-v\in W^{-1,p^{\prime }}({\mathbb{R}}^n)$ . We have that $\text{supp}(g)\subset \partial {\mathrm{\Omega }}_1$ , in view of the fact that $\text{supp}(v),\text{supp}(f)\subset \overline{{\mathrm{\Omega }}_1}$ , and (2.22). An application of Proposition 2.4 gives therefore

$$\begin{array}{c}\hfill g=h\otimes {\delta }_{\partial {\mathrm{\Omega }}_1},h\in B_{p^{\prime },p^{\prime }}^{-(1-1/p)}(\partial {\mathrm{\Omega }}_1).\end{array}$$

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It also follows from (2.22) that $\text{supp}(g)\subset \partial {\mathrm{\Omega }}_1\cap \partial {\mathrm{\Omega }}_2=\overline{U}$ , and hence, $\text{supp}(h)\subset \overline{U}$ . Here $U\subset \partial {\mathrm{\Omega }}_1$ is a bounded open set with $C^{\infty }$ boundary, and therefore, there exists a sequence $h_j\in C_0^{\infty }(U)$ such that $h_j\to h$ in $B_{p^{\prime },p^{\prime }}^{-(1-1/p)}(\partial {\mathrm{\Omega }}_1)$ , see [[38], Section 4.3.2, p. 318]. Thus, we get

2.25 $$\begin{array}{c}\hfill \begin{array}{cc}& {(g,\text{Ext}(u))}_{W^{-1,p^{\prime }}({\mathbb{R}}^n),W^{1,p}({\mathbb{R}}^n)}={(h,u{|}_{\partial {\mathrm{\Omega }}_1})}_{B_{p^{\prime },p^{\prime }}^{-(1-1/p)}(\partial {\mathrm{\Omega }}_1),W^{1-1/p,p}(\partial {\mathrm{\Omega }}_1)}\hfill \\ \hfill & =\underset{j\to \infty }{lim}{(h_j,u{|}_{\partial {\mathrm{\Omega }}_1})}_{B_{p^{\prime },p^{\prime }}^{-(1-1/p)}(\partial {\mathrm{\Omega }}_1),B_{p,p}^{1-1/p}(\partial {\mathrm{\Omega }}_1)}=\underset{j\to \infty }{lim}{\int }_{\partial {\mathrm{\Omega }}_1}{h}_j\overline{u}dS=0,\hfill \end{array}\\ \hfill \end{array}$$

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where the last equality follows from the fact that ${u|}_{\partial {\mathrm{\Omega }}_1\cap \partial {\mathrm{\Omega }}_2}=0$ . Combining (2.24) and (2.25), we see that

$$\begin{array}{cc}\hfill (v,& {u)}_{{\tilde{W}}^{-1,p^{\prime }}({\mathrm{\Omega }}_1),W^{1,p}({\mathrm{\Omega }}_1)}\hfill \\ \hfill & ={(-\mathrm{\Delta }f,\text{Ext}(u))}_{W^{-1,p^{\prime }}({\mathbb{R}}^n),W^{1,p}({\mathbb{R}}^n)}-{(g,\text{Ext}(u))}_{W^{-1,p^{\prime }}({\mathbb{R}}^n),W^{1,p}({\mathbb{R}}^n)}=0.\hfill \end{array}$$

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$\square$

From local to global results. Completion of proof of Theorem 1.8

We follow [[8]]. Let $\tilde{\mathrm{\Gamma }}=\partial \mathrm{\Omega }\backslash \mathrm{\Gamma }$ . Assuming that f satisfies (1.3) and using Proposition 2.1, we would like to show that f vanishes inside $\mathrm{\Omega }$ . To that end, let $x_0\in \mathrm{\Gamma }$ and let us fix a point $x_1\in \mathrm{\Omega }$ . Let $\theta :[0,1]\to \overline{\mathrm{\Omega }}$ be a $C^1$ curve joining $x_0$ to $x_1$ such that $\theta (0)=x_0$ , ${\theta }^{\prime }(0)$ is the interior normal to $\partial \mathrm{\Omega }$ at $x_0$ and $\theta (t)\in \mathrm{\Omega }$ , for all $t\in (0,1]$ . We set

$$\begin{array}{c}\hfill {\mathrm{\Theta }}_{\epsilon }(t)=\{x\in \overline{\mathrm{\Omega }}:d(x,\theta ([0,t]))\le \epsilon \}\end{array}$$

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and

$$\begin{array}{c}\hfill I=\{t\in [0,1]:f\text{vanishes a.e. on}{\mathrm{\Theta }}_{\epsilon }(t)\cap \mathrm{\Omega }\}.\end{array}$$

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By Proposition 2.1, we have $0\in I$ if $\epsilon >0$ is small enough. First as in [[8]], I is a closed subset of [0, 1]. If one proves that I is open then $I=[0,1]$ due to the fact that [0, 1] is connected. This implies that $x_1\notin \text{supp}(f)$ , and as $x_1$ is an arbitrary point of $\mathrm{\Omega }$ , we conclude that $f=0$ in $\mathrm{\Omega }$ , and this will complete the proof of Theorem 1.8. Hence, we only need to prove that the set I is open in [0, 1].

To this end, let $t\in I$ and $\epsilon >0$ be small enough so that $\partial {\mathrm{\Theta }}_{\epsilon }(t)\cap \partial \mathrm{\Omega }\subset \mathrm{\Gamma }$ . Arguing as in [[8], [24]], we smooth out $\mathrm{\Omega }\backslash {\mathrm{\Theta }}_{\epsilon }(t)$ into an open subset ${\mathrm{\Omega }}_1$ of $\mathrm{\Omega }$ with smooth boundary such that

$$\begin{array}{c}\hfill {\mathrm{\Omega }}_1\supset \mathrm{\Omega }\backslash {\mathrm{\Theta }}_{\epsilon }(t),\partial \mathrm{\Omega }\cap \partial {\mathrm{\Omega }}_1\supset \tilde{\mathrm{\Gamma }},\end{array}$$

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and $\partial {\mathrm{\Omega }}_1\cap \partial \mathrm{\Omega }=\overline{U}$ where $U\subset \partial {\mathrm{\Omega }}_1$ is an open set with $C^{\infty }$ boundary. By smoothing out the set $\mathrm{\Omega }\cup B(x_0,{\epsilon }^{\prime })$ , with $0<{\epsilon }^{\prime }\ll \epsilon$ sufficiently small, we enlarge the set $\mathrm{\Omega }$ into an open set ${\mathrm{\Omega }}_2$ with smooth boundary so that

$$\begin{array}{c}\hfill \partial {\mathrm{\Omega }}_2\cap \partial \mathrm{\Omega }\supset \partial {\mathrm{\Omega }}_1\cap \partial \mathrm{\Omega }=\partial {\mathrm{\Omega }}_1\cap \partial {\mathrm{\Omega }}_2\supset \tilde{\mathrm{\Gamma }}.\end{array}$$

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Let $G_{{\mathrm{\Omega }}_2}$ be the Green kernel associated to the open set ${\mathrm{\Omega }}_2$ ,

$$\begin{array}{c}\hfill -{\mathrm{\Delta }}_y{G}_{{\mathrm{\Omega }}_2}(x,y)=\delta (x-y),G_{{\mathrm{\Omega }}_2}{(x,·)|}_{\partial {\mathrm{\Omega }}_2}=0.\end{array}$$

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We have $G_{{\mathrm{\Omega }}_2}(x,y)\in C(\mathrm{\Omega }\times \overline{\mathrm{\Omega }}\backslash \{x=y\})$ , see [[40], Section 8.1]. Let us consider

$$\begin{array}{cc}& v(x^{(1)},\cdots ,x^{(m+1)})\hfill \\ \hfill & ={\int }_{{\mathrm{\Omega }}_1}f(y)(\sum _{k=1}^m\prod _{r=1,r\ne k}^m(\omega ·{\mathrm{\nabla }}_y{G}_{{\mathrm{\Omega }}_2}(x^{(r)},y)){\mathrm{\nabla }}_y{G}_{{\mathrm{\Omega }}_2}(x^{(k)},y))\hfill \\ \hfill & ·{\mathrm{\nabla }}_y{G}_{{\mathrm{\Omega }}_2}(x^{(m+1)},y)dy,\hfill \end{array}$$

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where $x^{(1)},\cdots ,x^{(m+1)}\in {\mathrm{\Omega }}_2\backslash \overline{{\mathrm{\Omega }}_1}$ . The function v is harmonic in all variables $x^{(1)},\cdots ,x^{(m+1)}\in {\mathrm{\Omega }}_2\backslash \overline{{\mathrm{\Omega }}_1}$ . Since $f=0$ on ${\mathrm{\Theta }}_{\epsilon }(t)\cap \mathrm{\Omega }$ , we have

$$\begin{array}{cc}& v(x^{(1)},\cdots ,x^{(m+1)})\hfill \\ \hfill & ={\int }_{\mathrm{\Omega }}f(y)(\sum _{k=1}^m\prod _{r=1,r\ne k}^m(\omega ·{\mathrm{\nabla }}_y{G}_{{\mathrm{\Omega }}_2}(x^{(r)},y)){\mathrm{\nabla }}_y{G}_{{\mathrm{\Omega }}_2}(x^{(k)},y))\hfill \\ \hfill & ·{\mathrm{\nabla }}_y{G}_{{\mathrm{\Omega }}_2}(x^{(m+1)},y)dy,\hfill \end{array}$$

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where $x^{(1)},\cdots ,x^{(m+1)}\in {\mathrm{\Omega }}_2\backslash \overline{{\mathrm{\Omega }}_1}$ . Now when $x^{(l)}\in {\mathrm{\Omega }}_2\backslash \overline{\mathrm{\Omega }}$ , the Green function $G_{{\mathrm{\Omega }}_2}(x^{(l)},·)\in C^{\infty }(\overline{\mathrm{\Omega }})$ is harmonic on $\mathrm{\Omega }$ , and $G_{{\mathrm{\Omega }}_2}(x^{(l)},·){|}_{\tilde{\mathrm{\Gamma }}}=0$ . By the orthogonality condition (1.3), we have $v(x^{(1)},\cdots ,x^{(m+1)})=0$ when $x^{(l)}\in {\mathrm{\Omega }}_2\backslash \overline{\mathrm{\Omega }}$ , $l=1,\cdots ,m+1$ .

As $v(x^{(1)},\cdots ,x^{(m+1)})$ is harmonic in all variables $x^{(1)},\cdots ,x^{(m+1)}\in {\mathrm{\Omega }}_2\backslash \overline{{\mathrm{\Omega }}_1}$ , and ${\mathrm{\Omega }}_2\backslash \overline{{\mathrm{\Omega }}_1}$ is connected, by unique continuation, we get that $v(x^{(1)},\cdots ,x^{(m+1)})=0$ when $x^{(1)},\cdots ,x^{(m+1)}\in {\mathrm{\Omega }}_2\backslash \overline{{\mathrm{\Omega }}_1}$ , i.e.

2.26 $$\begin{array}{cc}& {\int }_{{\mathrm{\Omega }}_1}f(y)(\sum _{k=1}^m\prod _{r=1,r\ne k}^m(\omega ·{\mathrm{\nabla }}_y{G}_{{\mathrm{\Omega }}_2}(x^{(r)},y)){\mathrm{\nabla }}_y{G}_{{\mathrm{\Omega }}_2}(x^{(k)},y))\hfill \\ \hfill & ·{\mathrm{\nabla }}_y{G}_{{\mathrm{\Omega }}_2}(x^{(m+1)},y)dy\hfill \\ \hfill & =0,x^{(1)},\cdots ,x^{(m+1)}\in {\mathrm{\Omega }}_2\backslash \overline{{\mathrm{\Omega }}_1}.\hfill \end{array}$$

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Let $a_l\in C^{\infty }(\overline{{\mathrm{\Omega }}_2})$ , $\text{supp}(a_l)\subset {\mathrm{\Omega }}_2\backslash \overline{{\mathrm{\Omega }}_1}$ , $l=1,\cdots ,m+1$ . Multiplying (2.26) by $a_1(x^{(1)})\cdots a_{m+1}(x^{(m+1)})$ , and integrating, we get

2.27 $$\begin{array}{cc}& {\int }_{{\mathrm{\Omega }}_1}f(y)(\sum _{k=1}^m\prod _{r=1,r\ne k}^m{\int }_{{\mathrm{\Omega }}_2}(\omega ·{\mathrm{\nabla }}_y{G}_{{\mathrm{\Omega }}_2}(x^{(r)},y))a_r(x^{(r)})d{x}^{(r)}\hfill \\ \hfill & {\int }_{{\mathrm{\Omega }}_2}{\mathrm{\nabla }}_y{G}_{{\mathrm{\Omega }}_2}(x^{(k)},y)a_k(x^{(k)})d{x}^{(k)})\hfill \\ \hfill & ·{\int }_{{\mathrm{\Omega }}_2}{\mathrm{\nabla }}_y{G}_{{\mathrm{\Omega }}_2}(x^{(m+1)},y)a_{m+1}(x^{(m+1)})d{x}^{(m+1)}dy=0.\hfill \end{array}$$

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Now it follows from the definition of W in Lemma 2.5 that any $v\in W$ is given by

$$\begin{array}{c}\hfill v(y)={\int }_{{\mathrm{\Omega }}_2}{G}_{{\mathrm{\Omega }}_2}(x,y)a(x)dx,y\in {\mathrm{\Omega }}_1,\end{array}$$

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where $a\in C^{\infty }(\overline{{\mathrm{\Omega }}_2})$ , $\text{supp}(a)\subset {\mathrm{\Omega }}_2\backslash \overline{{\mathrm{\Omega }}_1}$ . This together with (2.27) gives that

2.28 $$\begin{array}{c}\hfill \begin{array}{c}\hfill {\int }_{{\mathrm{\Omega }}_1}f(y)(\sum _{k=1}^m\prod _{r=1,r\ne k}^m(\omega ·\mathrm{\nabla }{v}^{(r)})\mathrm{\nabla }{v}^{(k)})·\mathrm{\nabla }{v}^{(m+1)}dy=0,\end{array}\end{array}$$

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for all $v^{(1)},\cdots ,v^{(m+1)}\in W$ .

The $(m+1)$ -linear form,

$$\begin{array}{cc}& W^{1,m+1}({\mathrm{\Omega }}_1)\times \cdots \times W^{1,m+1}({\mathrm{\Omega }}_1)\to \mathbb{C},\hfill \\ \hfill & (v^{(1)},\cdots ,v^{(m)})\mapsto {\int }_{{\mathrm{\Omega }}_1}f(y)(\sum _{k=1}^m\prod _{r=1,r\ne k}^m(\omega ·\mathrm{\nabla }{v}^{(r)})\mathrm{\nabla }{v}^{(k)})·\mathrm{\nabla }{v}^{(m+1)}dy\hfill \end{array}$$

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is continuous in view of Hölder's inequality. An application of Lemma 2.5 with $p=m+1$ shows that (2.28) holds for all $v^{(1)},\cdots ,v^{(m)}\in C^{\infty }(\overline{{\mathrm{\Omega }}_1})$ harmonic in ${\mathrm{\Omega }}_1$ which vanish on $\partial {\mathrm{\Omega }}_1\cap \partial {\mathrm{\Omega }}_2$ . Proposition 2.1 implies that f vanishes on a neighborhood of $\partial {\mathrm{\Omega }}_1\backslash (\partial {\mathrm{\Omega }}_1\cap \partial {\mathrm{\Omega }}_2)$ , and therefore, I is an open set. The proof of Theorem 1.8 is complete.

Proof of Theorem 1.2

First it follows from (i) and (ii) that for each $\tau \in \mathbb{C}$ fixed, $\gamma$ can be expanded into a power series

3.1 $$\begin{array}{c}\hfill \gamma (x,\tau ,z)=1+\sum _{k=1}^{\infty }{\partial }_z^k\gamma (x,\tau ,0)\frac{z^k}{k!},{\partial }_z^k\gamma (x,\tau ,0)\in C^{1,\alpha }(\overline{\mathrm{\Omega }}),\tau ,z\in \mathbb{C},\\ \hfill \end{array}$$

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converging in the $C^{\alpha }(\overline{\mathrm{\Omega }})$ topology. Furthermore, the map $\mathbb{C}\ni \tau \mapsto {\partial }_z^k\gamma (x,\tau ,0)$ is holomorphic with values in $C^{\alpha }(\overline{\mathrm{\Omega }})$ .

Let $\lambda \in \mathrm{\Sigma }$ be arbitrary but fixed. Let $\epsilon =({\epsilon }_1,\cdots ,{\epsilon }_m)\in {\mathbb{C}}^m$ , $m\ge 2$ , and consider the Dirichlet problem (1.2) with

3.2 $$\begin{array}{c}\hfill f=\sum _{k=1}^m{\epsilon }_k{f}_k,f_k\in C^{\infty }(\partial \mathrm{\Omega }),\text{supp}(f_k)\subset \mathrm{\Gamma },k=1,\cdots ,m.\end{array}$$

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Then for all $|\epsilon |$ sufficiently small, the problem (1.2) has a unique solution $u(·;\epsilon )\in C^{2,\alpha }(\overline{\mathrm{\Omega }})$ close to $\lambda$ in $C^{2,\alpha }(\overline{\mathrm{\Omega }})$ -topology, which depends holomorphically on $\epsilon \in \text{neigh}(0,{\mathbb{C}}^m)$ , with values in $C^{2,\alpha }(\overline{\mathrm{\Omega }})$ .

We shall use an induction argument on $m\ge 2$ to prove that the equality

$$\begin{array}{c}\hfill {\mathrm{\Lambda }}_{{\gamma }_1}^{\mathrm{\Gamma }}(\lambda +\sum _{k=1}^m{\epsilon }_k{f}_k)={\mathrm{\Lambda }}_{{\gamma }_2}^{\mathrm{\Gamma }}(\lambda +\sum _{k=1}^m{\epsilon }_k{f}_k),\end{array}$$

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for all $|\epsilon |$ sufficiently small and all $f_k\in C^{\infty }(\partial \mathrm{\Omega })$ , $\text{supp}(f_k)\subset \mathrm{\Gamma }$ , $k=1,\cdots ,m$ , gives that ${\partial }_z^{m-1}{\gamma }_1(x,\lambda ,0)={\partial }_z^{m-1}{\gamma }_1(x,\lambda ,0)$ .

First let $m=2$ and we proceed to carry out a second order linearization of the partial Dirichlet-to-Neumann map. Let $u_j=u_j(x;\epsilon )\in C^{2,\alpha }(\overline{\mathrm{\Omega }})$ be the unique solution close to $\lambda$ in $C^{2,\alpha }(\overline{\mathrm{\Omega }})$ -topology of the Dirichlet problem,

3.3 $$\begin{array}{c}\hfill \left\{\begin{array}{cc}\mathrm{\Delta }{u}_j+\text{div}({\sum }_{k=1}^{\infty }{\partial }_z^k{\gamma }_j(x,u_j,0)\frac{{(\omega ·\mathrm{\nabla }{u}_j)}^k}{k!}\mathrm{\nabla }{u}_j)=0\hfill & \text{in}\mathrm{\Omega },\hfill \\ u_j=\lambda +{\epsilon }_1{f}_1+{\epsilon }_2{f}_2\hfill & \text{on}\partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$

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for $j=1,2$ . The solution $u_j$ is $C^{\infty }$ with respect to $\epsilon$ for $|\epsilon |$ sufficiently small in view of Theorem B.1. Applying ${\partial }_{{\epsilon }_l}{|}_{\epsilon =0}$ , $l=1,2$ , to (3.3), and using that $u_j(x,0)=\lambda$ , we get

3.4 $$\begin{array}{c}\hfill \left\{\begin{array}{cc}\mathrm{\Delta }{v}_j^{(l)}=0\hfill & \text{in}\mathrm{\Omega },\hfill \\ v_j^{(l)}=f_l\hfill & \text{on}\partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$

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where $v_j^{(l)}={\partial }_{{\epsilon }_l}{u}_j{|}_{\epsilon =0}$ . It follows that $v^{(l)}:=v_1^{(l)}=v_2^{(l)}\in C^{\infty }(\overline{\mathrm{\Omega }})$ .

Applying ${\partial }_{{\epsilon }_1}{\partial }_{{\epsilon }_2}{|}_{\epsilon =0}$ to (3.3) and letting $w_j={\partial }_{{\epsilon }_1}{\partial }_{{\epsilon }_2}{u}_j{|}_{\epsilon =0}$ , we obtain that

3.5 $$\begin{array}{c}\hfill \left\{\begin{array}{cc}\mathrm{\Delta }{w}_j+\text{div}({\partial }_z{\gamma }_j(x,\lambda ,0)((\omega ·\mathrm{\nabla }{v}^{(1)})\mathrm{\nabla }{v}^{(2)}+(\omega ·\mathrm{\nabla }{v}^{(2)})\mathrm{\nabla }{v}^{(1)}))=0\hfill & \text{in}\mathrm{\Omega },\hfill \\ w_j=0\hfill & \text{on}\partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$

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$j=1,2$ .

The fact that ${\mathrm{\Lambda }}_{{\gamma }_1}^{\mathrm{\Gamma }}(\lambda +{\epsilon }_1{f}_1+{\epsilon }_2{f}_2)={\mathrm{\Lambda }}_{{\gamma }_1}^{\mathrm{\Gamma }}(\lambda +{\epsilon }_1{f}_1+{\epsilon }_2{f}_2)$ for all small $\epsilon$ , and all $f_1,f_2\in C^{\infty }(\partial \mathrm{\Omega })$ with $\text{supp}(f_1),\text{supp}(f_2)\subset \mathrm{\Gamma }$ , gives that

3.6

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An application of ${\partial }_{{\epsilon }_1}{\partial }_{{\epsilon }_2}{|}_{\epsilon =0}$ to (3.6) yields that

3.7 $$\begin{array}{c}\hfill \begin{array}{cc}& ({\partial }_{\nu }{w}_1-{\partial }_{\nu }{w}_2){|}_{\mathrm{\Gamma }}+({\partial }_z{\gamma }_1(x,\lambda ,0)-{\partial }_z{\gamma }_2(x,\lambda ,0))\hfill \\ \hfill & \times ((\omega ·\mathrm{\nabla }{v}^{(1)}){\partial }_{\nu }{v}^{(2)}+(\omega ·\mathrm{\nabla }{v}^{(2)}){\partial }_{\nu }{v}^{(1)}){|}_{\mathrm{\Gamma }}=0.\hfill \end{array}\end{array}$$

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Multiplying the difference of two equations in (3.5) by $v^{(3)}\in C^{\infty }(\overline{\mathrm{\Omega }})$ harmonic in $\mathrm{\Omega }$ , integrating over $\mathrm{\Omega }$ , using Green's formula and (3.7), we obtain that

3.8 $$\begin{array}{cc}& {\int }_{\mathrm{\Omega }}({\partial }_z{\gamma }_1(x,\lambda ,0)-{\partial }_z{\gamma }_2(x,\lambda ,0))((\omega ·\mathrm{\nabla }{v}^{(1)})\mathrm{\nabla }{v}^{(2)}+(\omega ·\mathrm{\nabla }{v}^{(2)})\mathrm{\nabla }{v}^{(1)})·\mathrm{\nabla }{v}^{(3)}dx\hfill \\ \hfill & ={\int }_{\partial \mathrm{\Omega }\backslash \mathrm{\Gamma }}({\partial }_z{\gamma }_1(x,\lambda ,0)-{\partial }_z{\gamma }_2(x,\lambda ,0))((\omega ·\mathrm{\nabla }{v}^{(1)}){\partial }_{\nu }{v}^{(2)}+(\omega ·\mathrm{\nabla }{v}^{(2)}){\partial }_{\nu }{v}^{(1)})v^{(3)}dS\hfill \\ \hfill & +{\int }_{\partial \mathrm{\Omega }\backslash \mathrm{\Gamma }}({\partial }_{\nu }{w}_1-{\partial }_{\nu }{w}_2)v^{(3)}dS=0,\hfill \end{array}$$

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provided that $\text{supp}(v^{(3)}{|}_{\partial \mathrm{\Omega }})\subset \mathrm{\Gamma }$ . It follows from (3.8) that

3.9 $$\begin{array}{c}\hfill {\int }_{\mathrm{\Omega }}({\partial }_z{\gamma }_1(x,\lambda ,0)-{\partial }_z{\gamma }_2(x,\lambda ,0))((\omega ·\mathrm{\nabla }{v}^{(1)})\mathrm{\nabla }{v}^{(2)}+(\omega ·\mathrm{\nabla }{v}^{(2)})\mathrm{\nabla }{v}^{(1)})·\mathrm{\nabla }{v}^{(3)}dx=0,\end{array}$$

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for all $v^{(l)}\in C^{\infty }(\overline{\mathrm{\Omega }})$ harmonic in $\mathrm{\Omega }$ such that $\text{supp}(v^{(l)}{|}_{\partial \mathrm{\Omega }})\subset \mathrm{\Gamma }$ , $l=1,2,3$ . An application of Theorem 1.8 with $m=2$ allows us to conclude that ${\partial }_z{\gamma }_1(·,\lambda ,0)={\partial }_z{\gamma }_2(·,\lambda ,0)$ in $\mathrm{\Omega }$ . Now as $\lambda \in \mathrm{\Sigma }$ is arbitrary and the functions $\mathbb{C}\ni \tau \to {\partial }_z{\gamma }_j(x,\tau ,0)$ , $j=1,2$ , are holomorphic, by the uniqueness properties of holomorphic functions, we have ${\partial }_z{\gamma }_1(·,·,0)={\partial }_z{\gamma }_2(·,·,0)$ in $\overline{\mathrm{\Omega }}\times \mathbb{C}$ .

Let $m\ge 3$ and assume that

3.10 $$\begin{array}{c}\hfill {\partial }_z^k{\gamma }_1(·,·,0)={\partial }_z^k{\gamma }_2(·,·,0)\text{in}\overline{\mathrm{\Omega }}\times \mathbb{C},\end{array}$$

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for all $k=1,\cdots ,m-2$ . Let $\lambda \in \mathrm{\Sigma }$ be arbitrary but fixed. To prove that ${\partial }_z^{m-1}{\gamma }_1(·,\lambda ,0)={\partial }_z^{m-1}{\gamma }_2(·,\lambda ,0)$ in $\overline{\mathrm{\Omega }}$ , we carry out the mth order linearization of the partial Dirichlet-to-Neumann map. In doing so, we let $u_j=u_j(x;\epsilon )\in C^{2,\alpha }(\overline{\mathrm{\Omega }})$ be the unique solution close to $\lambda$ in $C^{2,\alpha }(\overline{\mathrm{\Omega }})$ -topology of the Dirichlet problem,

3.11 $$\begin{array}{c}\hfill \left\{\begin{array}{cc}\mathrm{\Delta }{u}_j+\text{div}({\sum }_{k=1}^{\infty }{\partial }_z^k{\gamma }_j(x,u_j,0)\frac{{(\omega ·\mathrm{\nabla }{u}_j)}^k}{k!}\mathrm{\nabla }{u}_j)=0\hfill & \text{in}\mathrm{\Omega },\hfill \\ u_j=\lambda +{\epsilon }_1{f}_1+\cdots +{\epsilon }_m{f}_m\hfill & \text{on}\partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$

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for $j=1,2$ . We shall next apply ${\partial }_{{\epsilon }_1}\cdots {\partial }_{{\epsilon }_m}{|}_{\epsilon =0}$ to (3.11). To this end, we first note that ${\partial }_{{\epsilon }_1}\cdots {\partial }_{{\epsilon }_m}({\sum }_{k=m}^{\infty }{\partial }_z^k{\gamma }_j(x,u_j,0)\frac{{(\omega ·\mathrm{\nabla }{u}_j)}^k}{k!}\mathrm{\nabla }{u}_j)$ is a sum of terms each of them containing positive powers of $\mathrm{\nabla }{u}_j$ , which vanishes when $\epsilon =0$ . The only term in ${\partial }_{{\epsilon }_1}\cdots {\partial }_{{\epsilon }_m}({\partial }_z^{m-1}{\gamma }_j(x,u_j,0)\frac{{(\omega ·\mathrm{\nabla }{u}_j)}^{m-1}}{(m-1)!}\mathrm{\nabla }{u}_j)$ which does not contain a positive power of $\mathrm{\nabla }{u}_j$ is

3.12 $$\begin{array}{c}\hfill {\partial }_z^{m-1}{\gamma }_j(x,u_j,0)(\sum _{k=1}^m\prod _{r=1,r\ne k}^m(\omega ·\mathrm{\nabla }{\partial }_{{\epsilon }_r}{u}_j)\mathrm{\nabla }{\partial }_{{\epsilon }_k}{u}_j).\end{array}$$

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Finally, the expression ${\partial }_{{\epsilon }_1}\cdots {\partial }_{{\epsilon }_m}({\sum }_{k=1}^{m-2}{\partial }_z^k{\gamma }_j(x,u_j,0)\frac{{(\omega ·\mathrm{\nabla }{u}_j)}^k}{k!}\mathrm{\nabla }{u}_j){|}_{\epsilon =0}$ is independent of $j=1,2$ . Indeed, this follows from (3.10), the fact that this expression contains only the derivatives of $u_j$ of the form ${\partial }_{{\epsilon }_{l_1},\cdots ,{\epsilon }_{l_s}}^s{u}_j{|}_{\epsilon =0}$ with $s=1,\cdots ,m-1$ , ${\epsilon }_{l_1},\cdots ,{\epsilon }_{l_s}\in \{{\epsilon }_1,\cdots ,{\epsilon }_m\}$ , and the fact that

3.13 $$\begin{array}{c}\hfill {\partial }_{{\epsilon }_{l_1},\cdots ,{\epsilon }_{l_s}}^s{u}_1{{|}_{\epsilon =0}={\partial }_{{\epsilon }_{l_1},\cdots ,{\epsilon }_{l_s}}^s{u}_2|}_{\epsilon =0},\end{array}$$

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for $s=1,\cdots ,m-1$ , ${\epsilon }_{l_1},\cdots ,{\epsilon }_{l_s}\in \{{\epsilon }_1,\cdots ,{\epsilon }_m\}$ . The latter can be seen by induction on s, applying the operator ${\partial }_{{\epsilon }_{l_1},\cdots ,{\epsilon }_{l_s}}^s{|}_{\epsilon =0}$ to (3.11) and using (3.10) as well as the unique solvability of the Dirichlet problem for the Laplacian. Thus, an application ${\partial }_{{\epsilon }_1}\cdots {\partial }_{{\epsilon }_m}{|}_{\epsilon =0}$ to (3.11) gives

3.14 $$\begin{array}{c}\hfill \left\{\begin{array}{cc}\mathrm{\Delta }{w}_j+\text{div}({\partial }_z^{m-1}{\gamma }_j(x,\lambda ,0)({\sum }_{k=1}^m\prod _{r=1,r\ne k}^m(\omega ·\mathrm{\nabla }{v}^{(r)})\mathrm{\nabla }{v}^{(k)}))=H_m\hfill & \text{in}\mathrm{\Omega },\hfill \\ w_j=0\hfill & \text{on}\partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$

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cf. (3.12). Here $w_j={\partial }_{{\epsilon }_1}\cdots {\partial }_{{\epsilon }_m}{u}_j{|}_{\epsilon =0}$ and

$$\begin{array}{c}\hfill H_m(x,\lambda ):=-\text{div}({\partial }_{{\epsilon }_1}\cdots {\partial }_{{\epsilon }_m}(\sum _{k=1}^{m-2}{\partial }_z^k{\gamma }_j(x,u_j,0)\frac{{(\omega ·\mathrm{\nabla }{u}_j)}^k}{k!}\mathrm{\nabla }{u}_j){|}_{\epsilon =0}).\end{array}$$

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The fact that ${\mathrm{\Lambda }}_{{\gamma }_1}^{\mathrm{\Gamma }}(\lambda +{\epsilon }_1{f}_1+\cdots +{\epsilon }_m{f}_m)={\mathrm{\Lambda }}_{{\gamma }_1}^{\mathrm{\Gamma }}(\lambda +{\epsilon }_1{f}_1+\cdots +{\epsilon }_m{f}_m)$ for all small $\epsilon$ and all $f_k\in C^{\infty }(\partial \mathrm{\Omega })$ with $\text{supp}(f_k),\subset \mathrm{\Gamma }$ , $k=1,\cdots ,m$ , yields (3.6). Applying of ${\partial }_{{\epsilon }_1}\cdots {\partial }_{{\epsilon }_m}{|}_{\epsilon =0}$ to (3.6), using (3.10) and (3.13), we obtain that

3.15 $$\begin{array}{c}\hfill \begin{array}{cc}& ({\partial }_{\nu }{w}_1-{\partial }_{\nu }{w}_2){|}_{\mathrm{\Gamma }}+({\partial }_z^{m-1}{\gamma }_1(x,\lambda ,0)\hfill \\ \hfill & -{\partial }_z^{m-1}{\gamma }_2(x,\lambda ,0))(\sum _{k=1}^m\prod _{r=1,r\ne k}^m(\omega ·\mathrm{\nabla }{v}^{(r)}){\partial }_{\nu }{v}^{(k)}){|}_{\mathrm{\Gamma }}=0.\hfill \end{array}\end{array}$$

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Using (3.14), (3.15), and proceeding as in the case $m=2$ , we get

3.16 $$\begin{array}{cc}& {\int }_{\mathrm{\Omega }}({\partial }_z^{m-1}{\gamma }_1(x,\lambda ,0)\hfill \\ \hfill & -{\partial }_z^{m-1}{\gamma }_1(x,\lambda ,0))(\sum _{k=1}^m\prod _{r=1,r\ne k}^m(\omega ·\mathrm{\nabla }{v}^{(r)})\mathrm{\nabla }{v}^{(k)})·\mathrm{\nabla }{v}^{(m+1)}dx=0,\hfill \\ \hfill \end{array}$$

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for all $v^{(l)}\in C^{\infty }(\overline{\mathrm{\Omega }})$ harmonic in $\mathrm{\Omega }$ such that $\text{supp}(v^{(l)}{|}_{\partial \mathrm{\Omega }})\subset \mathrm{\Gamma }$ , $l=1,\cdots ,m+1$ . Applying Theorem 1.8, we conclude that ${\partial }_z^{m-1}{\gamma }_1(·,\lambda ,0)={\partial }_z^{m-1}{\gamma }_2(·,\lambda ,0)$ in $\overline{\mathrm{\Omega }}$ . Now as $\lambda \in \mathrm{\Sigma }$ is arbitrary and the functions $\mathbb{C}\ni \tau \to {\partial }_z^{m-1}{\gamma }_j(x,\tau ,0)$ , $j=1,2$ , are holomorphic, we have ${\partial }_z^{m-1}{\gamma }_1(·,·,0)={\partial }_z^{m-1}{\gamma }_2(·,·,0)$ in $\overline{\mathrm{\Omega }}\times \mathbb{C}$ . This completes the proof of Theorem 1.2.

Proof of Theorem 1.1

First it follows from (a) and (b) that $\gamma$ can be expanded into the following power series,

4.1 $$\begin{array}{c}\hfill \gamma (x,\lambda )=1+\sum _{k=1}^{\infty }{\partial }_{\lambda }^k\gamma (x,0)\frac{{\lambda }^k}{k!},{\partial }_{\lambda }^k\gamma (x,0)\in C^{1,\alpha }(\overline{\mathrm{\Omega }}),\lambda \in \mathbb{C},\end{array}$$

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converging in the $C^{1,\alpha }(\overline{\mathrm{\Omega }})$ topology.

Let $\epsilon =({\epsilon }_1,\cdots ,{\epsilon }_m)\in {\mathbb{C}}^m$ , $m\ge 2$ , and consider the Dirichlet problem (1.1) with f given by (3.2). For all $|\epsilon |$ sufficiently small, the problem (1.1) has a unique small solution $u(·;\epsilon )\in C^{2,\alpha }(\overline{\mathrm{\Omega }})$ , which depends holomorphically on $\epsilon \in \text{neigh}(0,{\mathbb{C}}^m)$ .

As in the proof of Theorem 1.2, we use an induction argument on $m\ge 2$ to show that ${\mathrm{\Lambda }}_{{\gamma }_1}^{\mathrm{\Gamma }}={\mathrm{\Lambda }}_{{\gamma }_2}^{\mathrm{\Gamma }}$ implies that ${\partial }_{\lambda }^{m-1}{\gamma }_1(x,0)={\partial }_{\lambda }^{m-1}{\gamma }_1(x,0)$ .

First let $m=2$ and we perform a second order linearization of the partial Dirichlet-to-Neumann map. Let $u_j=u_j(x;\epsilon )\in C^{2,\alpha }(\overline{\mathrm{\Omega }})$ be the unique solution small solution of the Dirichlet problem,

4.2 $$\begin{array}{c}\hfill \left\{\begin{array}{cc}\mathrm{\Delta }{u}_j+\text{div}({\sum }_{k=1}^{\infty }{\partial }_{\lambda }^k{\gamma }_j(x,0)\frac{u_j^k}{k!}\mathrm{\nabla }{u}_j)=0\hfill & \text{in}\mathrm{\Omega },\hfill \\ u_j={\epsilon }_1{f}_1+{\epsilon }_2{f}_2\hfill & \text{on}\partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$

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for $j=1,2$ . Applying ${\partial }_{{\epsilon }_l}{|}_{\epsilon =0}$ , $l=1,2$ , to (4.2), and using that $u_j(x,0)=0$ , we see that

4.3 $$\begin{array}{c}\hfill \left\{\begin{array}{cc}\mathrm{\Delta }{v}_j^{(l)}=0\hfill & \text{in}\mathrm{\Omega },\hfill \\ v_j^{(l)}=f_l\hfill & \text{on}\partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$

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where $v_j^{(l)}={\partial }_{{\epsilon }_l}{u}_j{|}_{\epsilon =0}$ . We have therefore $v^{(l)}:=v_1^{(l)}=v_2^{(l)}\in C^{\infty }(\overline{\mathrm{\Omega }})$ .

Applying ${\partial }_{{\epsilon }_1}{\partial }_{{\epsilon }_2}{|}_{\epsilon =0}$ to (4.2) and setting $w_j={\partial }_{{\epsilon }_1}{\partial }_{{\epsilon }_2}{u}_j{|}_{\epsilon =0}$ , we get

4.4 $$\begin{array}{c}\hfill \left\{\begin{array}{cc}\mathrm{\Delta }{w}_j+\text{div}({\partial }_{\lambda }{\gamma }_j(x,0)(v^{(1)}\mathrm{\nabla }{v}^{(2)}+v^{(2)}\mathrm{\nabla }{v}^{(1)}))=0\hfill & \text{in}\mathrm{\Omega },\hfill \\ w_j=0\hfill & \text{on}\partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$

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$j=1,2$ . The fact that ${\mathrm{\Lambda }}_{{\gamma }_1}^{\mathrm{\Gamma }}({\epsilon }_1{f}_1+{\epsilon }_2{f}_2)={\mathrm{\Lambda }}_{{\gamma }_1}^{\mathrm{\Gamma }}({\epsilon }_1{f}_1+{\epsilon }_2{f}_2)$ for all small $\epsilon$ , and all $f_1,f_2\in C^{\infty }(\partial \mathrm{\Omega })$ with $\text{supp}(f_1),\text{supp}(f_2)\subset \mathrm{\Gamma }$ , implies that

4.5 $$\begin{array}{c}\hfill (1+\sum _{k=1}^{\infty }{\partial }_{\lambda }^k{\gamma }_1(x,0)\frac{u_1^k}{k!}){\partial }_{\nu }{u}_1{|}_{\mathrm{\Gamma }}=(1+\sum _{k=1}^{\infty }{\partial }_z^k{\gamma }_2(x,0)\frac{u_2^k}{k!}){\partial }_{\nu }{u}_2{|}_{\mathrm{\Gamma }}.\end{array}$$

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Applying ${\partial }_{{\epsilon }_1}{\partial }_{{\epsilon }_2}{|}_{\epsilon =0}$ to (4.5), we get

4.6 $$\begin{array}{c}\hfill ({\partial }_{\nu }{w}_1-{\partial }_{\nu }{w}_2){|}_{\mathrm{\Gamma }}+({\partial }_{\lambda }{\gamma }_1(x,0)-{\partial }_{\lambda }{\gamma }_2(x,0))(v^{(1)}{\partial }_{\nu }{v}^{(2)}+v^{(2)}{\partial }_{\nu }{v}^{(1)}){|}_{\mathrm{\Gamma }}=0.\\ \hfill \end{array}$$

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Multiplying the difference of two equations in (4.4) by $v^{(3)}\in C^{\infty }(\overline{\mathrm{\Omega }})$ harmonic in $\mathrm{\Omega }$ , integrating over $\mathrm{\Omega }$ , using Green's formula and (4.6), we obtain that

4.7 $$\begin{array}{cc}& {\int }_{\mathrm{\Omega }}({\partial }_{\lambda }{\gamma }_1(x,0)-{\partial }_{\lambda }{\gamma }_2(x,0))(v^{(1)}\mathrm{\nabla }{v}^{(2)}+v^{(2)}\mathrm{\nabla }{v}^{(1)})·\mathrm{\nabla }{v}^{(3)}dx\hfill \\ \hfill & ={\int }_{\partial \mathrm{\Omega }\backslash \mathrm{\Gamma }}({\partial }_{\lambda }{\gamma }_1(x,0)-{\partial }_{\lambda }{\gamma }_2(x,0))(v^{(1)}{\partial }_{\nu }{v}^{(2)}+v^{(2)}{\partial }_{\nu }{v}^{(1)})v^{(3)}dS\hfill \\ \hfill & +{\int }_{\partial \mathrm{\Omega }\backslash \mathrm{\Gamma }}({\partial }_{\nu }{w}_1-{\partial }_{\nu }{w}_2)v^{(3)}dS=0,\hfill \end{array}$$

Graph

provided that $\text{supp}(v^{(3)}{|}_{\partial \mathrm{\Omega }})\subset \mathrm{\Gamma }$ . Thus, (4.7) gives that

$$\begin{array}{c}\hfill {\int }_{\mathrm{\Omega }}({\partial }_{\lambda }{\gamma }_1(x,0)-{\partial }_{\lambda }{\gamma }_2(x,0))(v^{(1)}\mathrm{\nabla }{v}^{(2)}+v^{(2)}\mathrm{\nabla }{v}^{(1)})·\mathrm{\nabla }{v}^{(3)}dx=0,\end{array}$$

Graph

for all $v^{(l)}\in C^{\infty }(\overline{\mathrm{\Omega }})$ harmonic in $\mathrm{\Omega }$ such that $\text{supp}(v^{(l)}{|}_{\partial \mathrm{\Omega }})\subset \mathrm{\Gamma }$ , $l=1,2,3$ . By Theorem 1.9 with $m=2$ , we get ${\partial }_{\lambda }{\gamma }_1(·,0)={\partial }_{\lambda }{\gamma }_2(·,0)$ in $\overline{\mathrm{\Omega }}$ .

Let $m\ge 3$ and assume that ${\partial }_{\lambda }^k{\gamma }_1(·,0)={\partial }_{\lambda }^k{\gamma }_2(·,0)\text{in}\overline{\mathrm{\Omega }}$ , for all $k=1,\cdots ,m-2$ . To prove that ${\partial }_{\lambda }^{m-1}{\gamma }_1(·,0)={\partial }_{\lambda }^{m-1}{\gamma }_2(·,·,0)$ in $\overline{\mathrm{\Omega }}$ , we perform the mth order linearization of the partial Dirichlet-to-Neumann map. In doing so, we let $u_j=u_j(x;\epsilon )\in C^{2,\alpha }(\overline{\mathrm{\Omega }})$ be the unique small solution of the Dirichlet problem,

4.8 $$\begin{array}{c}\hfill \left\{\begin{array}{cc}\mathrm{\Delta }{u}_j+\text{div}({\sum }_{k=1}^{\infty }{\partial }_{\lambda }^k{\gamma }_j(x,0)\frac{u_j^k}{k!}\mathrm{\nabla }{u}_j)=0\hfill & \text{in}\mathrm{\Omega },\hfill \\ u_j={\epsilon }_1{f}_1+\cdots +{\epsilon }_m{f}_m\hfill & \text{on}\partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$

Graph

for $j=1,2$ . Applying ${\partial }_{{\epsilon }_1}\cdots {\partial }_{{\epsilon }_m}{|}_{\epsilon =0}$ to (4.8), and arguing as in Theorem 1.2, we obtain that

4.9 $$\begin{array}{c}\hfill \left\{\begin{array}{cc}\mathrm{\Delta }{w}_j+\text{div}({\partial }_{\lambda }^{m-1}{\gamma }_j(x,0)({\sum }_{k=1}^m\prod _{r=1,r\ne k}^m{v}^{(r)}\mathrm{\nabla }{v}^{(k)}))=H_m\hfill & \text{in}\mathrm{\Omega },\hfill \\ w_j=0\hfill & \text{on}\partial \mathrm{\Omega }.\hfill \end{array}\right)\end{array}$$

Graph

Here $w_j={\partial }_{{\epsilon }_1}\cdots {\partial }_{{\epsilon }_m}{u}_j{|}_{\epsilon =0}$ and

$$\begin{array}{c}\hfill H_m(x):=-\text{div}({\partial }_{{\epsilon }_1}\cdots {\partial }_{{\epsilon }_m}(\sum _{k=1}^{m-2}{\partial }_{\lambda }^k{\gamma }_j(x,0)\frac{u_j^k}{k!}\mathrm{\nabla }{u}_j){|}_{\epsilon =0}),\end{array}$$

Graph

which is independent of j.

Now the equality ${\mathrm{\Lambda }}_{{\gamma }_1}^{\mathrm{\Gamma }}({\epsilon }_1{f}_1+\cdots +{\epsilon }_m{f}_m)={\mathrm{\Lambda }}_{{\gamma }_1}^{\mathrm{\Gamma }}({\epsilon }_1{f}_1+\cdots +{\epsilon }_m{f}_m)$ for all small $\epsilon$ and all $f_k\in C^{\infty }(\partial \mathrm{\Omega })$ with $\text{supp}(f_k),\subset \mathrm{\Gamma }$ , $k=1,\cdots ,m$ , implies (4.5). Applying of ${\partial }_{{\epsilon }_1}\cdots {\partial }_{{\epsilon }_m}{|}_{\epsilon =0}$ to (4.5), we obtain that

4.10 $$\begin{array}{cc}\hfill ({\partial }_{\nu }{w}_1& -{\partial }_{\nu }{w}_2{)|}_{\mathrm{\Gamma }}+({\partial }_{\lambda }^{m-1}{\gamma }_1(x,0)-{\partial }_{\lambda }^{m-1}{\gamma }_2(x,0))(\sum _{k=1}^m\prod _{r=1,r\ne k}^m{v}^{(r)}{\partial }_{\nu }{v}^{(k)}){|}_{\mathrm{\Gamma }}=0.\hfill \\ \hfill \end{array}$$

Graph

Proceeding as in the case $m=2$ , and using (4.9), (4.10), we get

$$\begin{array}{c}\hfill {\int }_{\mathrm{\Omega }}({\partial }_{\lambda }^{m-1}{\gamma }_1(x,0)-{\partial }_{\lambda }^{m-1}{\gamma }_1(x,,0))(\sum _{k=1}^m\prod _{r=1,r\ne k}^m{v}^{(r)}\mathrm{\nabla }{v}^{(k)})·\mathrm{\nabla }{v}^{(m+1)}dx=0,\end{array}$$

Graph

for all $v^{(l)}\in C^{\infty }(\overline{\mathrm{\Omega }})$ harmonic in $\mathrm{\Omega }$ such that $\text{supp}(v^{(l)}{|}_{\partial \mathrm{\Omega }})\subset \mathrm{\Gamma }$ , $l=1,\cdots ,m+1$ . An application of Theorem 1.9 allows us to conclude that ${\partial }_{\lambda }^{m-1}{\gamma }_1(·,0)={\partial }_{\lambda }^{m-1}{\gamma }_2(·,0)$ in $\overline{\mathrm{\Omega }}$ . This completes the proof of Theorem 1.1.

Acknowledgements

We are very grateful to the referees for helpful comments and suggestions. The work of Y.K is partially supported by the French National Research Agency ANR (project MultiOnde) grant ANR-17-CE40-0029. The research of K.K. is partially supported by the National Science Foundation (DMS 1815922, DMS 2109199). The research of G.U. is partially supported by NSF, a Walker Professorship at UW and a Si-Yuan Professorship at IAS, HKUST. Part of the work was supported by the NSF grant DMS-1440140 while K.K. and G.U. were in residence at MSRI in Berkeley, California, during Fall 2019 semester.

Data availability statement

Data sharing not applicable to this article as no datasets were generated or analysed during the current study.

Declarations

Conflict of interest

The authors have no conflicts of interest to declare that are relevant to this article.

Appendix A. Proof of Theorem 1.2 in the case of full data

Note that the result of Theorem 1.2 is new even in the case of full data, i.e. $\mathrm{\Gamma }=\partial \mathrm{\Omega }$ , and the purpose of this appendix is to present an alternative simple proof in this case.

Using the linearization of the Dirichlet-to-Neumann map ${\mathrm{\Lambda }}_{\gamma }^{\partial \mathrm{\Omega }}$ , we shall see below that the proof of Theorem 1.2 in the full data case will be a consequence of the following density result.

Proposition A.1

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ , $n\ge 2$ , be a bounded open set with $C^{\infty }$ boundary, let $\omega \in {\mathbb{S}}^{n-1}$ be fixed and let $m=2,3,\cdots ,$ be fixed. Let $f\in L^{\infty }(\mathrm{\Omega })$ be such that

A.1 $$\begin{array}{c}\hfill {\int }_{\mathrm{\Omega }}f{(\omega ·\mathrm{\nabla }{v}_1)}^{m-1}\mathrm{\nabla }{v}_1·\mathrm{\nabla }{v}_{2}dx=0,\end{array}$$

Graph

for all functions $v_1,v_2\in C^{\infty }(\overline{\mathrm{\Omega }})$ harmonic in $\mathrm{\Omega }$ . Then $f=0$ in $\mathrm{\Omega }$ .

Proof

Let $\xi \in {\mathbb{S}}^{n-1}$ and consider $k\in {\mathbb{S}}^{n-1}$ such that $\xi ·k=0$ . Let $h>0$ . Setting

$$\begin{array}{c}\hfill v_1(x)=e^{\frac{1}{h}x·(k+i\xi )},v_2(x)=e^{\frac{m}{h}x·(-k+i\xi )},\end{array}$$

Graph

so that $v_1,v_2\in C^{\infty }({\mathbb{R}}^n)$ and harmonic. Substituting $v_1$ and $v_2$ into (A.1) and using that $(k+i\xi )·(-k+i\xi )=-2$ , we get

$$\begin{array}{c}\hfill {(\omega ·(k+i\xi ))}^{m-1}{\int }_{\mathrm{\Omega }}f(x)e^{\frac{2m}{h}ix·\xi }dx=0,\end{array}$$

Graph

and therefore, we have

$$\begin{array}{c}\hfill {\int }_{\mathrm{\Omega }}f(x)e^{\frac{2m}{h}ix·\xi }dx=0,\end{array}$$

Graph

for all $\xi \in {\mathbb{S}}^{n-1}$ , $\xi ·\omega \ne 0$ , and all $h>0$ . Hence, $f=0$ . $\square$

Let $\lambda \in \mathrm{\Sigma }$ be arbitrary but fixed. We shall use an induction argument on $m\ge 2$ to prove that the equality

$$\begin{array}{c}\hfill {\mathrm{\Lambda }}_{{\gamma }_1}^{\mathrm{\Gamma }}(\lambda +\sum _{k=1}^m{\epsilon }_k{f}_k)={\mathrm{\Lambda }}_{{\gamma }_2}^{\mathrm{\Gamma }}(\lambda +\sum _{k=1}^m{\epsilon }_k{f}_k),\end{array}$$

Graph

for all $|\epsilon |$ sufficiently small and all $f_k\in C^{\infty }(\partial \mathrm{\Omega })$ , $\text{supp}(f_k)\subset \mathrm{\Gamma }$ , $k=1,\cdots ,m$ , gives that ${\partial }_z^{m-1}{\gamma }_1(x,\lambda ,0)={\partial }_z^{m-1}{\gamma }_1(x,\lambda ,0)$ .

First when $m=2$ , taking $v^{(1)}=v^{(2)}$ in (3.9) and using Proposition A.1 with $m=2$ , we get ${\partial }_z{\gamma }_1(·,\lambda ,0)={\partial }_z{\gamma }_2(·,\lambda ,0)$ in $\mathrm{\Omega }$ . Now as $\lambda \in \mathrm{\Sigma }$ is arbitrary, we have ${\partial }_z{\gamma }_1(·,·,0)={\partial }_z{\gamma }_2(·,·,0)$ in $\overline{\mathrm{\Omega }}\times \mathbb{C}$ .

Let $m=3,4,\cdots$ . Let $\lambda \in \mathrm{\Sigma }$ be arbitrary but fixed. Letting $v^{(1)}=\cdots =v^{(m)}$ in (3.16) and using Proposition A.1, we see that ${\partial }_z^{m-1}{\gamma }_1(·,\lambda ,0)={\partial }_z^{m-1}{\gamma }_2(·,\lambda ,0)$ in $\mathrm{\Omega }$ . Again, as $\lambda \in \mathrm{\Sigma }$ is arbitrary, we get ${\partial }_z^{m-1}{\gamma }_1(·,·,0)={\partial }_z^{m-1}{\gamma }_2(·,·,0)$ in $\overline{\mathrm{\Omega }}\times \mathbb{C}$ . This completes the proof of Theorem 1.2 in the full data case.

Appendix B. Well-posedness of the Dirichlet problem for a quasilinear conductivity equation

In this appendix we shall recall a standard argument for showing the well-posedness of the Dirichlet problem for a quasilinear conductivity equation.

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ , $n\ge 2$ , be a bounded open set with $C^{\infty }$ boundary. Let $k\in \mathbb{N}\cup \{0\}$ and $0<\alpha <1$ and let $C^{k,\alpha }(\overline{\mathrm{\Omega }})$ be the standard Hölder space on $\mathrm{\Omega }$ , see [[16], [24]]. We observe that $C^{k,\alpha }(\overline{\mathrm{\Omega }})$ is an algebra under pointwise multiplication, with

B.1 $$\begin{array}{c}\hfill {\Vert uv\Vert }_{C^{k,\alpha }(\overline{\mathrm{\Omega }})}\le {C(\Vert u\Vert }_{C^{k,\alpha }(\overline{\mathrm{\Omega }})}{\Vert v\Vert }_{L^{\infty }(\mathrm{\Omega })}+{\Vert u\Vert }_{L^{\infty }(\mathrm{\Omega })}{\Vert v\Vert }_{C^{k,\alpha }(\overline{\mathrm{\Omega }})}),u,v\in C^{k,\alpha }(\overline{\mathrm{\Omega }}),\\ \hfill \end{array}$$

Graph

see [[16], Theorem A.7]. We write $C^{\alpha }(\overline{\mathrm{\Omega }})=C^{0,\alpha }(\overline{\mathrm{\Omega }})$ .

Let $\omega \in {\mathbb{S}}^{n-1}=\{\omega \in {\mathbb{R}}^n,|\omega |=1\}$ , be fixed. Consider the Dirichlet problem for the following isotropic quasilinear conductivity equation,

B.2 $$\begin{array}{c}\hfill \left\{\begin{array}{cc}\text{div}(\gamma (x,u,\omega ·\mathrm{\nabla }u)\mathrm{\nabla }u)=0\hfill & \text{in}\mathrm{\Omega },\hfill \\ u=\lambda +f\hfill & \text{on}\partial \mathrm{\Omega },\hfill \end{array}\right)\end{array}$$

Graph

with $\lambda \in \mathbb{C}$ . We assume that the function $\gamma :\overline{\mathrm{\Omega }}\times \mathbb{C}\times \mathbb{C}\to \mathbb{C}$ satisfies the following conditions,

• the map $\mathbb{C}\times \mathbb{C}\ni (\tau ,z)\mapsto \gamma (·,\tau ,z)$ is holomorphic with values in $C^{1,\alpha }(\overline{\mathrm{\Omega }})$ with some $0<\alpha <1$ ,
• $\gamma (x,0,0)=1$ .

It follows from (i) and (ii) that $\gamma$ can be expand into a power series

B.3 $$\begin{array}{c}\hfill \gamma (x,\tau ,z)=1+\sum _{j+k\ge 1,j\ge 0,k\ge 0}{\partial }_{\tau }^j{\partial }_z^k\gamma (x,0,0)\frac{{\tau }^j{z}^k}{j!k!},{\partial }_{\tau }^j{\partial }_z^k\gamma (x,0,0)\in C^{1,\alpha }(\overline{\mathrm{\Omega }}),\\ \hfill \end{array}$$

Graph

converging in the $C^{1,\alpha }(\overline{\mathrm{\Omega }})$ topology.

We have the following result.

Theorem B.1

Let $\lambda \in \mathbb{C}$ be fixed. Then under the above assumptions, there exist $\delta >0$ , $C>0$ such that for any $f\in B_{\delta }(\partial \mathrm{\Omega }):=\{f\in C^{2,\alpha }{(\partial \mathrm{\Omega }):\Vert f\Vert }_{C^{2,\alpha }(\partial \mathrm{\Omega })}<\delta \}$ , the problem (B.2) has a solution $u=u_{\lambda ,f}\in C^{2,\alpha }(\overline{\mathrm{\Omega }})$ which satisfies

$$\begin{array}{c}\hfill {\Vert u-\lambda \Vert }_{C^{2,\alpha }(\overline{\mathrm{\Omega }})}\le C{\Vert f\Vert }_{C^{2,\alpha }(\partial \mathrm{\Omega })}.\end{array}$$

Graph

The solution u is unique within the class $\{u\in C^{2,\alpha }(\overline{\mathrm{\Omega }}):\Vert u-\lambda {\Vert }_{C^{2,\alpha }(\overline{\mathrm{\Omega }})}<C\delta \}$ and it is depends holomorphically on $f\in B_{\delta }(\partial \mathrm{\Omega })$ . Furthermore, the map

$$\begin{array}{c}\hfill B_{\delta }(\partial \mathrm{\Omega })\to C^{1,\alpha }(\overline{\mathrm{\Omega }}),f\mapsto {\partial }_{\nu }{u|}_{\partial \mathrm{\Omega }}\end{array}$$

Graph

is holomorphic.

Proof

Let $\lambda \in \mathbb{C}$ be fixed, and let

$$\begin{array}{c}\hfill B_1=C^{2,\alpha }(\partial \mathrm{\Omega }),B_2=C^{2,\alpha }(\overline{\mathrm{\Omega }}),B_3=C^{\alpha }(\overline{\mathrm{\Omega }})\times C^{2,\alpha }(\partial \mathrm{\Omega }).\end{array}$$

Graph

Consider the map,

B.4 $$\begin{array}{c}\hfill F:B_1\times B_2\to B_3,F(f,u)=(\text{div}(\gamma (x,u,\omega ·\mathrm{\nabla }u)\mathrm{\nabla }u),u{|}_{\partial \mathrm{\Omega }}-\lambda -f).\\ \hfill \end{array}$$

Graph

Following [[29]], we shall make use of the implicit function theorem for holomorphic maps between complex Banach spaces, see [[37], p. 144]. First we check that F enjoys the mapping property (B.4). To that end in view of the fact that $C^{1,\alpha }(\overline{\mathrm{\Omega }})$ is an algebra under pointwise multiplication, we only need to show that $\gamma (x,u,\omega ·\mathrm{\nabla }u)\in C^{1,\alpha }(\overline{\mathrm{\Omega }})$ . In doing so, by Cauchy's estimates, we get

B.5 $$\begin{array}{c}\hfill \Vert {\partial }_{\tau }^j{\partial }_z^k{\gamma (x,0,0)\Vert }_{C^{1,\alpha }(\overline{\mathrm{\Omega }})}\le \frac{j!k!}{R_1^j{R}_2^k}\underset{|\tau |=R_1,|z|=R_2}{sup}{\Vert \gamma (·,\tau ,z)\Vert }_{C^{1,\alpha }(\overline{\mathrm{\Omega }})},R_1,R_2>0,\\ \hfill \end{array}$$

Graph

for all $j\ge 0$ , $k\ge 0$ , and $j+k\ge 1$ . With the help of (B.1) and (B.5), we obtain that

B.6 $$\begin{array}{c}\hfill \begin{array}{cc}& \Vert {\partial }_{\tau }^j{\partial }_z^k\gamma (x,0,0)\frac{u^j{(\omega ·\mathrm{\nabla }u)}^k}{j!k!}{\Vert }_{C^{1,\alpha }(\overline{\mathrm{\Omega }})}\hfill \\ \hfill & \le \frac{C^{j+k}}{R_1^j{R}_2^k}{\Vert u\Vert }_{C^{1,\alpha }(\overline{\mathrm{\Omega }})}^j{\Vert \omega ·\mathrm{\nabla }u\Vert }_{C^{1,\alpha }(\overline{\mathrm{\Omega }})}^k\underset{|\tau |=R_1,|z|=R_2}{sup}{\Vert \gamma (·,\tau ,z)\Vert }_{C^{1,\alpha }(\overline{\mathrm{\Omega }})}.\hfill \end{array}\end{array}$$

Graph

Taking $R_1=2C{\Vert u\Vert }_{C^{1,\alpha }(\overline{\mathrm{\Omega }})}$ and $R_2=2C{\Vert \omega ·\mathrm{\nabla }u\Vert }_{C^{1,\alpha }(\overline{\mathrm{\Omega }})}$ , we see that the series

$$\begin{array}{c}\hfill \sum _{j+k\ge 1,j\ge 0,k\ge 0}{\partial }_{\tau }^j{\partial }_z^k\gamma (x,0,0)\frac{u^j{(\omega ·\mathrm{\nabla }u)}^k}{j!k!}\end{array}$$

Graph

converges in $C^{1,\alpha }(\overline{\mathrm{\Omega }})$ . Hence, in view of (B.3), $\gamma (x,u,\omega ·\mathrm{\nabla }u)\in C^{1,\alpha }(\overline{\mathrm{\Omega }})$ .

Let us show that F in (B.4) is holomorphic. First F is locally bounded as it is continuous in (f, u). Hence, we only need to check that F is weak holomorphic, see [[37], p. 133]. To that end, letting $(f_0,u_0),(f_1,u_1)\in B_1\times B_2$ , we show that the map

Graph

is holomorphic in $\mathbb{C}$ with values in $B_3$ . Clearly, we only have to check that the map $\mu \mapsto \gamma (x,u_0(x)+\mu u_1(x),\omega ·(\mathrm{\nabla }{u}_0(x)+\mu \mathrm{\nabla }{u}_1(x)))$ is holomorphic in $\mathbb{C}$ with values in $C^{1,\alpha }(\overline{\mathrm{\Omega }})$ . This is a consequence of the fact that the series

$$\begin{array}{c}\hfill \sum _{j+k\ge 1,\mathrm{j}\ge 0,k\ge 0}{\partial }_{\tau }^j{\partial }_z^k\gamma (x,0,0)\frac{{(u_0+\mu u_1)}^j{(\omega ·\mathrm{\nabla }(u_0+\mu u_1))}^k}{j!k!}\end{array}$$

Graph

converges in $C^{1,\alpha }(\overline{\mathrm{\Omega }})$ , locally uniformly in $\lambda \in \mathbb{C}$ , in view of (B.6).

We have $F(0,\lambda )=0$ and the partial differential ${\partial }_{u}F(0,\lambda ):B_2\to B_3$ is given by

$$\begin{array}{c}\hfill {\partial }_{u}F(0,\lambda )v=(\mathrm{\Delta }v,v{|}_{\partial \mathrm{\Omega }}).\end{array}$$

Graph

It follows from [[11], Theorem 6.15] that the map ${\partial }_{u}F(0,\lambda ):B_2\to B_3$ is a linear isomorphism.

An application of the implicit function theorem, see [[37], p. 144], shows that there exists $\delta >0$ and a unique holomorphic map $S:B_{\delta }(\partial \mathrm{\Omega })\to C^{2,\alpha }(\overline{\mathrm{\Omega }})$ such that $S(0)=\lambda$ and $F(f,S(f))=0$ for all $f\in B_{\delta }(\partial \mathrm{\Omega })$ . Letting $u=S(f)$ and using that S is Lipschitz continuous and $S(0)=\lambda$ , we have

$$\begin{array}{c}\hfill {\Vert u-\lambda \Vert }_{C^{2,\alpha }(\overline{\mathrm{\Omega }})}\le C{\Vert f\Vert }_{C^{2,\alpha }(\partial \mathrm{\Omega })}.\end{array}$$

Graph

$\square$

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