Analyticity of Steklov eigenvalues of nearly-circular and nearly-spherical domains

TL;DR

This paper proves that Steklov eigenvalues depend analytically on small domain perturbations for nearly-circular and nearly-spherical shapes, using a transformation approach and geometric bounds.

Contribution

It establishes the analyticity of the Dirichlet-to-Neumann operator and Steklov eigenvalues under domain perturbations, extending previous methods to nearly-circular and nearly-spherical domains.

Findings

Steklov eigenvalues are analytic functions of domain perturbation parameters.

The Dirichlet-to-Neumann operator's analyticity is proven for nearly-circular and nearly-spherical domains.

Transformation and geometric bounding techniques are effectively used for the analysis.

Abstract

We consider the Dirichlet-to-Neumann operator (DNO) on nearly-circular and nearly-spherical domains in two and three dimensions, respectively. Treating such domains as perturbations of the ball, we prove the analyticity of the DNO with respect to the domain perturbation parameter. Consequently, the Steklov eigenvalues are also shown to be analytic in the domain perturbation parameter. To obtain these results, we use the strategy of Nicholls and Nigam (2004); we transform the equation on the perturbed domain to a ball and geometrically bound the Neumann expansion of the transformed Dirichlet-to-Neumann operator.

Equations

$${\Omega }_{\epsilon }=\{(r,\hat{\theta }):r\le 1+\epsilon \rho (\hat{\theta }),\hat{\theta }\in S^{d-1}\},$$

$$\Delta u_{\epsilon }$$

$${\partial }_{n_{\epsilon }}{u}_{\epsilon }$$

$$\xi \mapsto G_{\epsilon }\xi ={\partial }_{n_{\epsilon }}{u}_{\epsilon },$$

$$\Delta u_{\epsilon }$$

$$u_{\epsilon }(\hat{\theta })$$

$$\Vert G_n\xi {\Vert }_{H^{s+\frac{1}{2}}(D)}\le K_1\Vert \xi {\Vert }_{H^{s+\frac{3}{2}}(S^{d-1})}{B}^n$$

$$f(\theta )=\frac{1}{\sqrt{2\pi }}\sum _{k\in \mathbb{Z}}\hat{f}(k)e^{ik\theta },\text{where}\hat{f}(k)=\frac{1}{\sqrt{2\pi }}{\int }_0^{2\pi }f(\theta )e^{-ik\theta }d\theta .$$

$$\Vert f{\Vert }_{L^2(S^1)}^2$$

$$\Vert f{\Vert }_{H^1(S^1)}^2$$

$$\left[r^{-1}{\partial }_{r}r{\partial }_r+r^{-2}{\partial }_{\theta }^2\right]v=0$$

$$v(1+\epsilon \rho (\theta ),\theta )=\xi (\theta ).$$

$$(r^{\prime },{\theta }^{\prime })=\left((1+\epsilon \rho (\theta ){)}^{-1}r,\theta \right).$$

$$\frac{\partial }{\partial r}=\frac{1}{1+\epsilon \rho ({\theta }^{\prime })}\frac{\partial }{\partial r^{\prime }}\text{and}\frac{\partial }{\partial \theta }=\frac{\partial }{\partial {\theta }^{\prime }}-\frac{\epsilon r^{\prime }{\rho }^{\prime }({\theta }^{\prime })}{1+\epsilon \rho ({\theta }^{\prime })}\frac{\partial }{\partial r^{\prime }}.$$

$$u_{\epsilon }(r^{\prime },{\theta }^{\prime })=v((1+\epsilon \rho ({\theta }^{\prime }))r^{\prime },{\theta }^{\prime }),$$

$$\frac{1}{r^{\prime }{\left(1+\epsilon \rho ({\theta }^{\prime })\right)}^2}\frac{\partial }{\partial r^{\prime }}{r}^{\prime }\frac{\partial u_{\epsilon }}{\partial r^{\prime }}+\frac{1}{(r^{\prime }{)}^2{\left(1+\epsilon \rho ({\theta }^{\prime })\right)}^2}{\left(\frac{\partial }{\partial {\theta }^{\prime }}-\frac{\epsilon r^{\prime }{\rho }^{\prime }({\theta }^{\prime })}{1+\epsilon \rho ({\theta }^{\prime })}\frac{\partial }{\partial r^{\prime }}\right)}^2{u}_{\epsilon }=0.$$

$$r^{-1}{\partial }_{r}r{\partial }_r{u}_{\epsilon }+r^{-2}{\left({\partial }_{\theta }-\epsilon r(1+\epsilon \rho (\theta ){)}^{-1}{\rho }^{\prime }(\theta ){\partial }_r\right)}^2{u}_{\epsilon }=0.$$

$$\left(r^{-1}{\partial }_{r}r{\partial }_r+r^{-2}{\partial }_{\theta }^2\right)u_{\epsilon }-\epsilon r^{-1}{\left(1+\epsilon \rho (\theta )\right)}^{-1}\left(2{\rho }^{\prime }(\theta ){\partial }_{\theta }+{\rho }^{\prime \prime }(\theta )\right){\partial }_r{u}_{\epsilon }$$

$$+{\epsilon }^2{r}^{-1}{\left(1+\epsilon \rho (\theta )\right)}^{-2}{\left({\rho }^{\prime }(\theta )\right)}^2{\partial }_r\left(2+r{\partial }_r\right)u_{\epsilon }=0.$$

$$\Delta u_{\epsilon }=\epsilon L_1{u}_{\epsilon }+{\epsilon }^2{L}_2{u}_{\epsilon }$$

$$u_{\epsilon }(1,\theta )=\xi (\theta ).$$

$$\Delta$$

$$L_1$$

$$L_2$$

$$-2({\rho }^{\prime }(\theta ){)}^2{r}^{-1}{\partial }_r-{\rho }^2(\theta )\left[{\partial }_r^2+r^{-1}{\partial }_r+r^{-2}{\partial }_{\theta }^2\right].$$

$$u_{\epsilon }(r,\theta )=\sum _{n=0}^{\infty }{\epsilon }^n{u}_n(r,\theta ).$$

$$\Delta u_0(r,\theta )=0$$

$$u_0(1,\theta )=\xi (\theta ).$$

$$\Delta u_n(r,\theta )=L_1{u}_{n-1}+L_2{u}_{n-2}$$

$$u_n(1,\theta )=0.$$

$$\Delta w(r,\theta )=F(r,\theta )$$

$$w(1,\theta )=\xi (\theta )$$

$$\Vert w{\Vert }_{H^{s+1}(D)}\le K_0\left(\Vert F{\Vert }_{H^{s-1}(D)}+\Vert \xi {\Vert }_{H^{s+\frac{1}{2}}(S^1)}\right).$$

$$\xi (\theta )=\frac{1}{\sqrt{2\pi }}\sum _{k\in \mathbb{Z}}\hat{\xi }(k)e^{ik\theta },\text{where}\hat{\xi }(k)=\frac{1}{\sqrt{2\pi }}{\int }_0^{2\pi }\xi (\theta )e^{-ik\theta }d\theta .$$

$$\Delta \Phi (r,\theta )=0$$

$$\Phi (1,\theta )=\xi (\theta )$$

$$\Delta v(r,\theta )=F(r,\theta )$$

Full text

1. Introduction

Let $\Omega_{\varepsilon}\subset\mathbb{R}^{d}$ for $d=2,3$ be a nearly-circular or nearly-spherical domain of the form

[TABLE]

where the domain perturbation function $\rho\in C^{s+2}(S^{d-1})$ for some $s\in\mathbb{N}$ and the perturbation parameter, $\varepsilon\geq 0$, is assumed to be small in magnitude. We consider the Steklov eigenproblem on the perturbed domain $\Omega_{\varepsilon}$,

[TABLE]

Here $\Delta$ is the Laplacian on $H^{1}(\Omega_{\varepsilon})$ and $\partial_{n_{\varepsilon}}=\hat{n}_{\varepsilon}\cdot\nabla$ denotes the outward normal derivative on the boundary of $\Omega_{\varepsilon}$. It is well-known that the Steklov spectrum is discrete, real, and non-negative; we enumerate the eigenvalues in increasing order, $0=\sigma_{0}(\Omega_{\varepsilon})<\sigma_{1}(\Omega_{\varepsilon})\leq\sigma_{2}(\Omega_{\varepsilon})\cdots\to\infty$. The Steklov spectrum coincides with the spectrum of the Dirichlet-to-Neumann operator (DNO), $G_{\varepsilon}\colon H^{s+\frac{1}{2}}(\partial\Omega_{\varepsilon})\to H^{s-\frac{1}{2}}(\partial\Omega_{\varepsilon})$, which maps

[TABLE]

where $u_{\varepsilon}$ is the harmonic extension of $\xi$ to $\Omega_{\varepsilon}$, satisfying

[TABLE]

We refer the reader to [3] for a general description of the Steklov spectrum.

The goal of this paper is to prove the analyticity of the Steklov eigenvalues, $\sigma_{\varepsilon}$, in the perturbation parameter $\varepsilon$. Our main result is the following theorem.

Theorem 1.1**.**

Let $d=2$ or $3$ and $s\in\mathbb{N}$. If $\rho\in C^{s+2}(S^{d-1})$, then the Dirichlet-to-Neumann operator (DNO), $G_{\varepsilon}\colon H^{s+\frac{3}{2}}(\partial\Omega_{\varepsilon})\to H^{s+\frac{1}{2}}(\partial\Omega_{\varepsilon})$, is analytic in the domain parameter $\varepsilon$. More precisely, if $\rho\in C^{s+2}(S^{d-1})$, then there exists a Neumann series, $G_{\varepsilon}=\sum_{n=0}^{\infty}\varepsilon^{n}G_{n}$, that converges strongly as an operator from $H^{s+\frac{3}{2}}(S^{d-1})$ to $H^{s+\frac{1}{2}}(S^{d-1})$. That is, there exists constants $K_{1}$ and $C$ such that

[TABLE]

for any $B>C|\rho|_{C^{s+2}}$.

We prove Theorem 1.1 in two and three dimensions separately; these proofs can be found in Sections 2.2 and 3.2, respectively. In both dimensions, our proof of Theorem 1.1 follows the strategy in [5]. We first show the analyticity of the harmonic extension, that is, for fixed $\xi(\hat{\theta})$ the solution $u_{\varepsilon}$ in (3) is analytic in $\varepsilon$. Using this, we then prove that the DNO, $G_{\varepsilon}$, is also analytically dependent on $\varepsilon$, establishing Theorem 1.1.

Using an analyticity result in [4], we obtain the analytic dependence of the Steklov eigenvalues $\{\sigma_{j}(\varepsilon)\}_{j\in\mathbb{N}}$ on $\varepsilon$ within the same disc of convergence as in Theorem 1.1, as stated in the following corollary.

Corollary 1.2**.**

The Steklov eigenvalues, $\sigma_{\varepsilon}$, consist of branches of one or several analytic functions which have at most algebraic singularities near $\varepsilon=0$. The same is true of the corresponding eigenprojections.

The proof of Corollary 1.2 is given in Section 4.

Corollary 1.2 justifies Assumption 1.1 in [7]. Here, the first two terms of the asymptotoic series for $\sigma_{\varepsilon}$ are computed for reflection-symmetric nearly-circular domains. Corollary 1.2 also justifies the computation of the shape derivative that appears in [1]. Here, numerical methods are developed for the eigenvalue optimization problem of maximizing the $k$-th Steklov eigenvalue as a function of the domain with a volume constraint.

2. Two-dimensional nearly-circular domains

Here we consider the Steklov eigenproblem (2) in $\mathbb{R}^{2}$. We will identify $\hat{\theta}$ with its corresponding angle $\theta$ made with the positive $x$-axis, as usual. We write the Fourier series for $f\colon S^{1}\to\mathbb{C}$ as

[TABLE]

Denoting $\langle k\rangle=\sqrt{1+k^{2}}$, we introduce the spaces $L^{2}(S^{1})$ and $H^{1}(S^{1})$ with norms

[TABLE]

Similarly, we define the $H^{s}(S^{1})$ space with norm $\|f\|^{2}_{H^{s}(S^{1})}=\sum_{k\in\mathbb{Z}}\langle k\rangle^{2s}|\hat{f}(k)|^{2}$.

2.1. Analyticity of the harmonic extension for nearly-circular domains

We first consider the problem of harmonically extending a function $\xi(\theta)$ from $\partial\Omega_{\varepsilon}$ to $\Omega_{\varepsilon}$,

[TABLE]

Mapping $\Omega_{\varepsilon}$ to the unit disk, $D=\Omega_{0}$, we make the change of variables

[TABLE]

The partial derivatives in the new coordinates are given by

[TABLE]

Applying this change of coordinates to the Laplace equation (4) and setting

[TABLE]

we obtain the problem

[TABLE]

Multiplying both sides by $\left(1+\varepsilon\rho(\theta^{\prime})\right)^{2}$ and dropping the primes on the transformed variables yields

[TABLE]

Expanding the operator in the second term on the left hand side, we obtain

[TABLE]

Again multiplying both sides by $\left(1+\varepsilon\rho(\theta)\right)^{2}$, we obtain the transformed Laplace equation,

[TABLE]

where

[TABLE]

We formally expand the solution, $u_{\varepsilon}$, in powers of $\varepsilon$,

[TABLE]

Next, we collect terms in powers of $\varepsilon$. At $O(\varepsilon^{0})$, we obtain

[TABLE]

At $O(\varepsilon^{n})$ for $n>0$, we obtain

[TABLE]

We next show that there exists a unique solution of (6) of the form in (7). The following Lemma is analogous to [5, Lemma 4].

Lemma 2.1**.**

For $s\in\mathbb{N}$, there is a constant $K_{0}>0$ such that for any $F\in H^{s-1}(D)$ and $\xi\in H^{s+\frac{1}{2}}(S^{1})$, the solution of

[TABLE]

satisfies

[TABLE]

Proof.

We will prove the result for $s=0$. Since $\xi\in H^{\frac{1}{2}}(S^{1})$, we have the Fourier series

[TABLE]

Setting $v=w-\Phi$, where $\Phi(r,\theta)=\frac{1}{\sqrt{2\pi}}\sum_{k\in\mathbb{Z}}\hat{\xi}(k)r^{|k|}e^{ik\theta}$, we have that

[TABLE]

and

[TABLE]

Using $\hat{\Phi}(r,k)=\hat{\xi}(k)r^{|k|}$, a straightforward integration yields

[TABLE]

for some constant $C_{\frac{1}{2}}>0$.

Multiplying (8a) by $\overline{v}$, integrating by parts, and using (8b) yields

[TABLE]

By the duality of $H_{0}^{1}(D)$ and $H^{-1}(D)$, we have $\int_{D}|F\overline{v}|\leq\|F\|_{H^{-1}(D)}\|v\|_{H_{0}^{1}(D)}$. Since $v\in H_{0}^{1}(D)$, by the Poincaré inequality, there exists a constant $C_{D}$ such that $\|v\|_{H^{1}(D)}\leq C_{D}\|v\|_{H^{1}_{0}(D)}$ and we conclude that

[TABLE]

Using the decomposition $w=v+\Phi$ and using (9) and (10), we obtain

[TABLE]

Taking $K_{0}=\max\{C_{\frac{1}{2}},C_{D}\}$ yields the desired result for $s=0$. The proof for $s\geq 1$ is similar. ∎

The next Lemma will be used to prove the inductive step in the proof of Theorem 2.3 and is analogous to [5, Lemma 5]. In the proof, we use the following result [6, 5]. For $\varepsilon\geq 0$, $s\in\mathbb{N}$, $f\in C^{s}(S^{d-1})$, $u\in H^{s}(B^{d})$, $g\in C^{s+\frac{1}{2}+\delta}(S^{d-1})$, and $\mu\in H^{s+\frac{1}{2}}(S^{d-1})$, there exists a constant $M=M(s,d)$ so that

[TABLE]

Lemma 2.2**.**

Let $s\in\mathbb{N}$ and let $\rho\in C^{s+2}(S^{1})$. Assume that $K_{1}$ and $B$ are constants so that

[TABLE]

If $B>|\rho|_{C^{s+2}}$, then there exists a $C_{0}$ such that

[TABLE]

Proof.

We begin by rewriting

[TABLE]

as well as

[TABLE]

First, we measure $L_{1}u_{N-1}$ in $H^{s}(D)$ and use the triangle inequality and (11) to obtain:

[TABLE]

Here, in the third inequality, we have used that all operators acting on $u_{N-1}$ are second order. Similarly, we estimate $L_{2}u_{N-2}$ in $H^{s}(D)$:

[TABLE]

∎

The following Theorem justifies the convergnece of (7) for sufficiently small $\varepsilon>0$ and is analogous to [5, Theorem 3].

Theorem 2.3**.**

Given $s\in\mathbb{N}$, if $\rho\in C^{s+2}(S^{1})$ and $\xi\in H^{s+\frac{3}{2}}(S^{1})$, there exists constants $C_{0}$ and $K_{0}$ and a unique solution of (6) such that

[TABLE]

for any $B>2K_{0}C_{0}|\rho|_{C^{s+2}}$.

Proof.

We proceed by induction. For $n=0$, we use Lemma 2.1 to we see

[TABLE]

as desired to show (12). We now define $K_{1}=K_{0}\|\xi\|_{H^{s+\frac{3}{2}}(S^{1})}$ for the remainder of the proof to be used in Lemma 2.2.

Suppose inequality (12) holds for $n<N$. Then by Lemma 2.1,

[TABLE]

By Lemma 2.2, we may bound $\|L_{1}u_{N-1}\|_{H^{s}}$ and $\|L_{2}u_{N-2}\|_{H^{s}}$ so that

[TABLE]

provided $B>2K_{0}C_{0}|\rho|_{C^{s+2}}$. ∎

2.2. Proof of Theorem 1.1 in two dimensions: Analyticity of the Dirichlet to Neumann operator

The Dirichlet to Neumann operator (DNO), $G\colon H^{s+1}(\partial\Omega_{\varepsilon})\to H^{s}(\partial\Omega_{\varepsilon})$, is given by

[TABLE]

where $v$ is the harmonic extension of $\xi$ from $\partial\Omega_{\varepsilon}$ to $\Omega_{\varepsilon}$, satisfying (4). Making the change of coordinates given in (5), we obtain the transformed DNO, $G(1+\varepsilon\rho)\colon H^{s+1}(S^{1})\to H^{s}(S^{1})$, given by

[TABLE]

where $u_{\varepsilon}$ satisfies (6) and

[TABLE]

Since $M_{\rho}(\varepsilon)$ is clearly analytic in $\varepsilon$, we need only show the analyticity of $\hat{G}(1+\varepsilon\rho(\theta^{\prime}))$. Dropping the prime notation on the new variables, we obtain

[TABLE]

We expand the non-normalized DNO, $\hat{G}_{\rho,\varepsilon}$, as a power series in $\varepsilon$

[TABLE]

which yields the following recursive formula:

[TABLE]

We now prove the following theorem, which proves Theorem 1.1 and guarantees the uniform convergence of the series (13) for suitably small $\varepsilon$.

Theorem 2.4**.**

Let $\xi\in H^{s+3/2}(S^{1})$. Then

[TABLE]

for $B>C|\rho|_{C^{s+2}}$.

Proof.

We will proceed via induction. First, we show (14) fo $n=0$:

[TABLE]

In the second inequality of the first line, we have used the trace theorem, while Theorem 2.3 is used in the second line. Now suppose that (14) holds for $n<N$. Then we have the following estimate:

[TABLE]

for $B>C|\rho|_{C^{s+2}}$, where $C$ is independent of $u$, $N$, $\xi$, and $\rho$. Here we have used the second inequality in (11), as well as the trace theorem, Theorem 2.3, and the inductive hypothesis on $\hat{G}_{\rho,N-2}$ and $\hat{G}_{\rho,N-2}$. ∎

3. Three dimensional nearly-spherical domains

Here we consider (2) in dimension $d=3$. We identify $\hat{\theta}\in\mathbb{S}^{2}$ with the inclination, $\theta\in[0,\pi]$, and azimuth, $\phi\in[0,2\pi]$. Let $\Omega_{\varepsilon}$ be a nearly-spherical domain where the perturbation function is expanded in the basis of real spherical harmonics,

[TABLE]

Here, $Y_{\ell,m}$ denote the real spherical harmonics, which are obtained from the complex spherical harmonics as follows. Define the complex spherical harmonic by

[TABLE]

where $P_{\ell}^{m}$ is the associated Legendre polynomial, which can be defined through the Rodrigues formula, $P_{\ell}^{m}(x)=\frac{(-1)^{m}}{2^{\ell}\ell!}(1-x^{2})^{\frac{m}{2}}\frac{d^{m+\ell}}{dx^{m+\ell}}(x^{2}-1)^{\ell}$. For $\ell\geq 0$ and $|m|\leq\ell$, the real spherical harmonics are then defined by

[TABLE]

3.1. Analyticity of the harmonic extension for nearly-spherical domains

As in Section 2.1, we first consider the problem of harmonically extending a function $\xi(\theta,\phi)$ from $\partial\Omega_{\varepsilon}$ to $\Omega_{\varepsilon}$,

[TABLE]

Mapping $\Omega_{\varepsilon}$ to the unit ball, $B=\Omega_{0}$, we make the change of variables

[TABLE]

The partial derivatives in the new coordinates are given by

[TABLE]

Applying this change of coordinates to the Laplace equation (18a), setting

[TABLE]

multiplying by $(1+\varepsilon\rho)^{4}$, and dropping the primes on the transformed variables yields

[TABLE]

Defining the operators:

[TABLE]

The function $u_{\varepsilon}$ satisfies

[TABLE]

Lemma 3.1**.**

For $s\in\mathbb{N}$, there is a constant $K_{0}>0$ such that for any $F\in H^{s-1}(B)$ and $\xi\in H^{s+\frac{1}{2}}(S^{2})$, the solution of

[TABLE]

satisfies

[TABLE]

Proof.

We will prove the result for $s=0$. Since $\xi\in H^{\frac{1}{2}}(S^{2})$, we have the spherical harmonic transform,

[TABLE]

and $Y_{\ell}^{m}$ are the (complex) spherical harmonics. Set $v=w-\Phi$, where $\Phi$ solves

[TABLE]

Then v satisfies

[TABLE]

We have

[TABLE]

and defining

[TABLE]

we see that

[TABLE]

We thus calculate:

[TABLE]

for some constant $C_{\frac{1}{2}}>0$.

Multiplying (21) by $\overline{v}$ and integrating by parts yields

[TABLE]

By the duality of $H_{0}^{1}(B)$ and $H^{-1}(B)$, we have $\int_{B}|F\overline{v}|\leq\|F\|_{H^{-1}(B)}\|v\|_{H_{0}^{1}(B)}$. Since $v\in H_{0}^{1}(B)$, by the Poincaré inequality, there exists a constant $C_{B}$ such that $\|v\|_{H^{1}(B)}\leq C_{B}\|v\|_{H^{1}_{0}(B)}$ and we conclude that

[TABLE]

Using the decomposition $w=v+\Phi$ and using (22) and (23), we obtain

[TABLE]

Taking $K_{0}=\max\{C_{\frac{1}{2}},C_{B}\}$ yields the desired result for $s=0$. The proof for $s\geq 1$ is similar. ∎

Let us make the ansatz

[TABLE]

Then, by (20), we have the recursive formula

[TABLE]

Lemma 3.2**.**

Let $s\in\mathbb{N}$ and let $\rho\in C^{s+2}(S^{2})$. Assume that $K_{1}$ and $A$ are constants so that

[TABLE]

If $A>|\rho|_{C^{s+2}}$, then there exists a $C_{0}$ such that

[TABLE]

Proof.

Using the triangle inequality and (11), we calculate:

[TABLE]

In the second inequality, we have also used that all operators acting on $u_{N-1}$ are second order. We similarly estimate $\|L_{2}u_{N-2}\|_{H^{s}(B)}$:

[TABLE]

Taking $C_{0}=12M(s)$ completes the proof. ∎

The following theorem justifies the convergence of (24) for suitably small $\varepsilon>0$.

Theorem 3.3**.**

Given $s\in\mathbb{N}$, if $\rho\in C^{s+2}(S^{2})$ and $\xi\in H^{s+\frac{3}{2}}(S^{2})$, there exists constants $C_{0}$ and $K_{0}$ and a unique solution $u_{\varepsilon}$ of (20) satisfying (24) such that

[TABLE]

for any $A>2K_{0}C_{0}|\rho|_{C^{s+2}}$.

Proof.

We proceed by induction. For $n=0$, we use Lemma 3.1 to we see

[TABLE]

as desired to show (26). We now define $K_{1}=K_{0}\|\xi\|_{H^{s+\frac{3}{2}}(S^{2})}$ for the remainder of the proof to be used in Lemma 3.2.

Suppose inequality (26) holds for $n<N$. Then by Lemma 3.1,

[TABLE]

By Lemma 3.2, we may bound $\|L_{1}u_{N-1}\|_{H^{s}}$ and $\|L_{2}u_{N-2}\|_{H^{s}}$ so that

[TABLE]

provided $A>2K_{0}C_{0}|\rho|_{C^{s+2}}$. ∎

3.2. Proof of Theorem 1.1 in three dimensions: Analyticity of the Dirichlet to Neumann operator

Denote the Dirichlet-to-Neumann operator $G_{\rho,\varepsilon}\colon H^{s+1/2}(\partial\Omega_{\varepsilon})\rightarrow H^{s-1/2}(\partial\Omega_{\varepsilon})$ which is defined

[TABLE]

where $v$ satisfies (18) and

[TABLE]

is the unit-length normal vector on $\partial\Omega_{\varepsilon}$. Here the spherical coordinate vectors $\hat{r},\hat{\theta},\hat{\phi}$ are given by

[TABLE]

Making the change of variables in (19), we obtain

[TABLE]

Since $M_{\rho}(\varepsilon)$ is clearly analytic near $\varepsilon=0$, we need only show the analyticity of $\hat{G}_{\rho,\varepsilon}$ near $\varepsilon=0$. to verify that $G_{\rho,\varepsilon}$ is analytic as well. Note that $\hat{G}_{\rho,\varepsilon}$ satisfies

[TABLE]

We now make a power series ansatz for the non-normalized DNO $\hat{G}_{\rho,\varepsilon}$:

[TABLE]

for $\xi\in H^{s+\frac{1}{2}}(S^{2})$ and $s\in\mathbb{N}$. By (24) and (28), we obtain the recursive relationship,

[TABLE]

The following theorem proves Theorem 1.1 in three dimensions and justifies the convergence of (29) for suitably small $\varepsilon>0$.

Theorem 3.4**.**

Let $\xi\in H^{s+3/2}(S^{2})$. Then

[TABLE]

for $A>C|\rho|_{C^{s+2}}$

Proof.

We will proceed via induction. First, we show (31) fo $n=0$:

[TABLE]

In the second inequality of the first line, we have used the standard Trace theorem, while Theorem 3.3 is used in the second line. Now suppose that (31) holds for $n<N$. Then we have the following estimate:

[TABLE]

for $K_{1}=\max\{2C_{1}K_{0},2C_{1}K_{0}M(s)\}$ and $A>C|\rho|_{C^{s+2}}$. ∎

4. Proof of Corollary 1.2: Analyticity of the Steklov eigenvalues

We now have all of the ingredients to prove Corollary 1.2.

Proof of Corollary 1.2..

Theorems 2.4 and 3.4 show that the expansion of the non-normalized DNO $\hat{G}_{\rho,\varepsilon}$ given in (13) and (29) is uniformly convergent for small $\varepsilon$. It follows that $G_{\rho,\varepsilon}\colon H^{s+\frac{3}{2}}(S^{1})\to H^{s+\frac{1}{2}}(S^{1})$ is analytic for small $\varepsilon$. The DNO operator $G_{\rho,\varepsilon}\colon L^{2}(S^{d-1})\to L^{2}(S^{d-1})$ is self-adjoint [2], hence closed. The result now follows from [4, Ch. 7, Thm 1.8, p. 370]. ∎

Acknowledgments

We would like to thank Nilima Nigam and Fadil Santosa for helpful discussions. B. Osting is partially supported by NSF DMS 16-19755 and 17-52202.