Exterior Dirichlet and Neumann Problems in Domains with Random Boundaries

Abstract

An approximation of statistical moments of solutions to exterior Dirichlet and Neumann problems with random boundary surfaces is investigated. A rigorous shape calculus approach has been used to approximate these statistical moments by those of the corresponding shape derivatives, which are computed by boundary integral equation methods. Examples illustrate our theoretical results.

1 Introduction

In this paper, we consider the Poisson equation in an unbounded domain exterior to a bounded object in ${{\mathbb {R}}}^3$, subject to a Dirichlet or Neumann condition on the boundary of the object. Given a proper spatial resolution, virtually any natural or manufactured surface (i.e., the boundary of the object) becomes rough. For example, the boundary of a nano-device falls within this category; see, e.g., [5]. Consequently, there is a growing interest in studying probabilistic descriptions of rough surfaces, and of solutions of partial differential equations defined on random domains whose boundaries are these surfaces; see [6, 13,14,15, 20] and the references therein.

A common approach in dealing with random input data is the use of Taylor series expansion to expand all the random fields at hand about the expectation of the random input data. By substituting this expansion into the original problem and disregarding high-order terms, one arrives at deterministic subproblems. Shape calculus [4, 11, 19] is one of the methods that follow this approach and has been employed for elliptic interface problems on bounded and unbounded domains, see [9, 14]. In this paper, we use shape calculus to solve exterior Dirichlet and Neumann problems on random unbounded domains.

We now give a more detailed description of the problems to be considered. Let $\Gamma ^\epsilon (\omega )$ be a random closed surface in ${{\mathbb {R}}}^3$ where $\omega$ and $\epsilon$ indicate, respectively, the randomness and perturbation amplitude. A more precise definition of $\Gamma ^\epsilon (\omega )$ is presented in Sect. 2.2. We consider the Poisson equation in the unbounded domain $D^\epsilon (\omega )$ whose boundary is $\Gamma ^\epsilon (\omega )$, namely

$$\begin{aligned} -\Delta u^\epsilon ({\varvec{x}},\omega ) = f({\varvec{x}}), \quad {\varvec{x}}\in D^\epsilon (\omega ). \end{aligned}$$

(1.1)

Equation (1.1) is supplemented with either a Dirichlet boundary condition

$$\begin{aligned} u^\epsilon ({\varvec{x}},\omega ) = g_D({\varvec{x}}), \quad {\varvec{x}}\in \Gamma ^\epsilon (\omega ), \end{aligned}$$

(1.2)

or a Neumann boundary condition

$$\begin{aligned} \frac{\partial u^\epsilon }{\partial {\varvec{n}}^\epsilon }({\varvec{x}},\omega ) = g_N({\varvec{x}}), \quad {\varvec{x}}\in \Gamma ^\epsilon (\omega ), \end{aligned}$$

(1.3)

where ${\varvec{n}}^\epsilon (\cdot , \omega )$ denotes the outward unit normal vector of the surface $\Gamma ^\epsilon (\omega )$ and $\partial /\partial {\varvec{n}}^\epsilon$ denotes the normal derivative. For both problems we further impose a vanishing condition at infinity, namely

$$\begin{aligned} u^\epsilon ({\varvec{x}},\omega ) = O(\left\| {{\varvec{x}}}\right\| _{}^{-1}) \quad \text {as} \quad \left\| {{\varvec{x}}}\right\| _{} \rightarrow \infty . \end{aligned}$$

(1.4)

Here, the given function f, defined in an unbounded fixed domain containing all unbounded domains $D^\epsilon (\omega )$, is assumed to be independent of $\omega$. Similarly, $g_D$ and $g_N$ assumed to be independent of $\omega$ are defined in a bounded domain containing all surfaces $\Gamma ^\epsilon (\omega )$.

The uncertainty in the random surface $\Gamma ^\epsilon (\omega )$ (and the domain $D^\epsilon (\omega )$) results in the uncertainty in the solution $u^\epsilon (\cdot ,\omega )$. The dependence of the solution on the random event is nonlinear, and shape calculus is used to linearize the problem of computing the mean, covariance and other statistical moments of the solutions.

As the main results of this article, we prove that

• the shape derivatives of the solutions to the exterior Dirichlet and Neumann problems exist (Theorems 3.7 and 3.10), and these derivatives are solutions to the corresponding exterior problems on domains with fixed (deterministic) boundaries with random boundary data;

• the k-moment of the solutions to the exterior Dirichlet and Neumann problems (the precise definition of which is given in Sect. 2) can be approximated by that of their shape derivatives;

• the k-moment of the shape derivatives can be calculated by solving boundary integral equations on the deterministic boundaries.

In summary, we design a complete efficient process to compute the k-moments of the solutions $u^\epsilon (\cdot ,\omega )$. Examples support our theory.

The paper is organized as follows. In Sect. 2 we introduce the function spaces to be used and definitions of k-moments. Section 3 discusses the existence and characterization of shape derivatives of solutions to the Dirichlet and Neumann problems in consideration. Section 4 discusses the use of shape derivatives to approximate statistical moments of the solutions. Examples to illustrate the theory are given in Sect. 5.

2 Preliminaries

2.1 Statistical Moments

Throughout this paper, we denote by $(\Omega , \Sigma ,{\mathbb {P}})$ a generic complete probability space and let X be a separable Hilbert space. For any $1\le k \le \infty$, the Bochner space $L^k(\Omega , X)$ is defined as usual by

$$\begin{aligned} L^k(\Omega ,X): = \big \{v: \Omega \rightarrow X \text { measurable}: \Vert v\Vert _{L^k(\Omega ,X)} <\infty \big \} \end{aligned}$$

(2.1)

which is equipped with a norm

$$\begin{aligned} \left\| {v}\right\| _{L^k(\Omega , X)} := \left\{ \begin{array}{ll} \displaystyle \left( \int _{\Omega }\left\| {v(\omega )}\right\| _{X}^k d{\mathbb {P}}(\omega )\right) ^{1/k}, &{}\quad 1\le k < \infty ,\\ \mathop {\mathrm{ess sup}}\limits _{\omega \in \Omega }\left\| {v(\omega )}\right\| _{X}, &{}\quad k=\infty . \end{array} \right. \end{aligned}$$

(2.2)

The elements of $L^k(\Omega ,X)$ are called random fields.

We denote by $X^{(k)}$ the k-times tensor product of X, i.e.,

$$\begin{aligned} X^{(k)} := \underbrace{X \otimes \cdots \otimes X}_{k\ \text {times}}, \end{aligned}$$

(2.3)

being the closure of the algebraic tensor product $X \otimes \cdots \otimes X$ under the norm induced by the natural inner product

$$\begin{aligned} \left\langle \left\langle {v_1 \otimes \cdots \otimes v_k},{w_1 \otimes \cdots \otimes w_k}\right\rangle \right\rangle = \langle v_1, w_1\rangle \cdots \langle v_k, w_k \rangle . \end{aligned}$$

If ${\mathcal {T}}:X\rightarrow Y$ is a linear operator from a Hilbert space X to a Hilbert space Y, we denote by ${\mathcal {T}}^{(k)}: X^{(k)} \rightarrow Y^{(k)}$ the k-times tensor operator of ${\mathcal {T}}$. Note here that for any $v_1\otimes v_2\otimes \cdots \otimes v_k\in X^{(k)}$, we have ${\mathcal {T}}^{(k)} (v_1\otimes v_2\otimes \cdots \otimes v_k) = {\mathcal {T}}(v_1)\otimes {\mathcal {T}}(v_2)\otimes \cdots \otimes {\mathcal {T}}(v_k)\in Y^{(k)}$.

The k-moment ${\mathcal {M}}^{k} [v]$ of a random field $v\in L^k(\Omega , X)$ is an element of the tensor product defined by

$$\begin{aligned} {\mathcal {M}}^{k} [v] := \int _{\Omega } \big ( \underbrace{v(\omega )\otimes \cdots \otimes v(\omega )}_{k\text {-times}} \big ) \,d{\mathbb {P}}(\omega ). \end{aligned}$$

(2.4)

In the case $k=1$, the statistical moment ${\mathcal {M}}^1 [v]$ coincides with the mean value of v and is denoted by ${\mathbb {E}}[v]$. If $k\ge 2$, the statistical moment ${\mathcal {M}}^k [v]$ is the k-point autocorrelation function of v. The quantity ${\mathcal {M}}^k [v - {\mathbb {E}}[v]]$ is termed the k-th central moment of v. We distinguish in particular second-order moments: the correlation and covariance defined by

$$\begin{aligned} \mathrm{{Cor}}[v] := {\mathcal {M}}^2[v] \quad \text {and} \quad \mathrm{{Covar}}[v] := {\mathcal {M}}^2[v - {\mathbb {E}}[v]]. \end{aligned}$$

(2.5)

In this paper, we work with X being Sobolev spaces of real-valued functions defined on a domain $U\subset {{\mathbb {R}}}^3$ yielding, in particular, the representation

$$\begin{aligned} {\mathrm{{Cor}}[v]({\varvec{x}},{\varvec{y}})} := \int _{\Omega } v({\varvec{x}},\omega ) v({\varvec{y}},\omega ) \,d{\mathbb {P}}(\omega ), \quad {\varvec{x}},{\varvec{y}}\in U. \end{aligned}$$

(2.6)

We observe that $\mathrm{{Cor}}[v]$ is defined on the Cartesian product $U \times U$. Similarly, ${\mathcal {M}}^k[v]$ is defined on the k-fold Cartesian product $U \times \cdots \times U$. Here, the dimension of the underlying domain grows rapidly with increasing moment order k.

2.2 Random Surfaces

We now give a precise description of the random surfaces $\Gamma ^\epsilon (\omega )$. Let $\Gamma ^0$ be a closed surface in ${{\mathbb {R}}}^3$ which separates ${{\mathbb {R}}}^3$ into an interior bounded domain and an exterior unbounded domain which is denoted by $D^0$. For the subsequent analysis, we assume that $\Gamma ^0$ is a smooth closed curve. The surface $\Gamma ^0$ will be fixed throughout the paper and will be called the nominal surface.

In the present paper, we utilize the domain perturbation model based on the speed method (see, e.g., the monograph [19] and references therein) and random domain perturbation model from [8, 10, 13,14,15]. Suppose $\kappa \in L^k(\Omega , {C^{0,1}(\Gamma ^0)})$ is a random field, i.e., for almost all realizations $\omega \in \Omega$, we have $\kappa (\cdot ,\omega ) \in {C^{0,1}(\Gamma ^0)}$. We consider for sufficiently small and nonnegative $\epsilon$ a family of random surfaces of the form

$$\begin{aligned} \Gamma ^\epsilon (\omega ) = \{ {\varvec{x}}+ \epsilon \kappa ({\varvec{x}},\omega ){\varvec{n}}^0({\varvec{x}}): {\varvec{x}}\in \Gamma ^0 \}, \quad \omega \in \Omega . \end{aligned}$$

(2.7)

Here, the uncertainty of the surfaces $\Gamma ^\epsilon (\omega )$ is represented by the uncertainty in $\kappa (\cdot ,\omega )$. Notice that the surface $\Gamma ^\epsilon (\omega )|_{\epsilon = 0}$ is identical with $\Gamma ^0$ and therefore is a deterministic closed surface. Moreover, the limit $\Gamma ^\epsilon (\omega ) \rightarrow \Gamma ^0$ as $\epsilon \rightarrow 0$ is well defined in $L^k(\Omega ,C^{0,1})$. More precisely, if we identify $\Gamma ^\epsilon$ and $\Gamma ^0$ with the functions defining their graphs, then

$$\begin{aligned} \begin{aligned} \Vert \Gamma ^\epsilon - \Gamma ^0\Vert _{L^k(\Omega ,C^{0,1})}&= \epsilon \left( \int _\Omega \Vert \kappa (\cdot ,\omega ) {\varvec{n}}^0\Vert _{C^{0,1}(\Gamma ^0)}^k \, d {\mathbb {P}}(\omega ) \right) ^{\frac{1}{k}}\\&\le {2}\epsilon \Vert \kappa \Vert _{L^k(\Omega ,{C^{0,1}(\Gamma ^0)})} \Vert {\varvec{n}}^0\Vert _{C^{0,1}(\Gamma ^0)}. \end{aligned} \end{aligned}$$

(2.8)

This implies that for almost all $\omega \in \Omega$ and a sufficiently small $\epsilon \ge 0$ the surface $\Gamma ^\epsilon (\omega )$ is a Lipschitz continuous closed surface. From (2.7) we observe that the mean random surface is defined by

$$\begin{aligned} {\mathbb {E}}[\Gamma ^\epsilon ] = \big \{ {\varvec{x}}+ \epsilon {{\mathbb {E}}[\kappa ({\varvec{x}},\cdot )] \varvec{n}^0}({\varvec{x}}), \ {\varvec{x}}\in \Gamma ^0 \big \}. \end{aligned}$$

Without loss of generality, we may assume that the random perturbation amplitude $\kappa ({\varvec{x}},\omega )$ is centered, i.e.,

$$\begin{aligned} {{\mathbb {E}}[\kappa ({\varvec{x}},\cdot )]} = 0 \qquad \forall {\varvec{x}}\in \Gamma ^0. \end{aligned}$$

(2.9)

In this case,

$$\begin{aligned} {\mathbb {E}}[\Gamma ^\epsilon ] = \Gamma ^0 \qquad \text {and} \qquad \mathrm{{Covar}}[\kappa ]({\varvec{x}},{\varvec{y}}) = \mathrm{{Cor}}[\kappa ]({\varvec{x}},{\varvec{y}}). \end{aligned}$$

2.3 Sobolev Spaces

In this section, we introduce the function spaces needed for the forthcoming analysis. Let U be a bounded domain in ${{\mathbb {R}}}^3$ with boundary $\partial U$. The Sobolev space $H^1(U)$ is defined, as usual, as the space of all distributions which together with their first-order partial derivatives are square integrable. The Sobolev space $H^{1/2}(\partial U)$ is defined by

$$\begin{aligned} H^{1/2}(\partial U) = \{ g: \partial U\rightarrow {{\mathbb {R}}}\ |\ g=v \ \text {on}\ {\partial U} \ (\text {in the trace sense})\ \text {for some}\ v\in H^1(U) \} \end{aligned}$$

and equipped with the following norm

$$\begin{aligned} \left\| {g}\right\| _{H^{1/2}(\partial U)} := \inf \{ \left\| {v}\right\| _{H^1(U)}: g = v|_{\partial U} \}. \end{aligned}$$

The dual of $H^{1/2}(\partial U)$ is denoted by $H^{-1/2}(\partial U)$.

Sobolev spaces defined on unbounded domains require a special care. Following [18], for an unbounded domain $U \subset {{\mathbb {R}}}^3$ we introduce the weighted space

$$\begin{aligned} H_w^1(U) := \left\{ v\in \mathcal{D}'(U) : \left\| {v}\right\| _{H_w^1(U)} = \left( \int _{U}\Big ( \left| {\nabla v}\right| _{}^2 + \frac{\left| {v({\varvec{x}})}\right| _{}^2}{1+\left\| {{\varvec{x}}}\right\| _{}^2 }\Big )\,d{\varvec{x}}\right) ^{1/2} < +\infty \right\} \nonumber \\ \end{aligned}$$

(2.10)

with the corresponding norm and seminorm

$$\begin{aligned}&\left\| {v}\right\| _{H_w^1(U)} := \left( \int _{U}\Big ( \left| {\nabla v}\right| _{}^2 + \frac{\left| {v({\varvec{x}})}\right| _{}^2}{1+\left\| {{\varvec{x}}}\right\| _{}^2}\Big )\,d{\varvec{x}}\right) ^{1/2} \quad \text {and} \nonumber \\&\quad \left| {v}\right| _{H^1(U)} := \left( \int _{U} \left| {\nabla v}\right| _{}^2\, d{\varvec{x}}\right) ^{1/2}. \end{aligned}$$

(2.11)

For the Dirichlet problem, we also need the following Sobolev space of functions whose traces on the boundary of U vanish:

$$\begin{aligned} \mathring{H}_w^1(U) := \{ v\in H_w^1(U): v = 0 \ \text {on} \ \partial U \} \end{aligned}$$

(2.12)

with the seminorm and norm defined in (2.11).

The following lemma, proved in [18, Theorem 2.10.10], states the equivalence between the norm $\left\| {\cdot }\right\| _{H_w^1(U)}$ and seminorm $\left| {\cdot }\right| _{H^1(U)}$.

Lemma 2.1

The seminorm $\left| {\cdot }\right| _{H^1(U)}$ is also a norm in $H_w^1(U)$ and $\mathring{H}_w^1(U)$ which is equivalent to $\left\| {\cdot }\right\| _{H_w^1(U)}$.

Remark 2.2

In this paper, this lemma is used for $U = D^\epsilon (\omega )$. If there exist two fixed bounded domains $U_1$ and $U_2$ (independent of $\epsilon$ and $\omega$) satisfying $U_1 \subset [D^\epsilon (\omega )]^c \subset U_2$, then the equivalence constants are independent of $\epsilon$ and $\omega$; see the proof of [18, Theorem 2.10.10].

Besides the above Sobolev spaces, we also need the tensor product space $H_{\mathrm{mix}}^{1/2}(\partial U^k)$ which is the k-time tensor product of $H^{1/2}(\partial U)$, namely

$$\begin{aligned} H_{\text { mix}}^{1/2}(\partial U^k) = \underbrace{H^{1/2}(\partial U)\otimes \cdots \otimes H^{1/2}(\partial U)}_{k\ \text {times}}. \end{aligned}$$

3 Shape Calculus

In this section, we temporarily stay away from randomness and consider only deterministic perturbed surfaces. We will prove the existence of shape derivatives of the solutions of the exterior (unbounded domain) Dirichlet and Neumann problems. Shape derivatives of solutions to the Laplacian equation in bounded domains with homogeneous Dirichlet and Neumann boundary conditions were investigated in [4]. In this paper, we adopt a mapping $T^\epsilon :{{\mathbb {R}}}^3\rightarrow {{\mathbb {R}}}^3$ which transforms reference boundary, exterior and interior domains into perturbed ones so that the mapping is an identity at points which are sufficiently far away from the origin. Standard variational formulation techniques for unbounded domains will be frequently used.

3.1 Perturbation of Deterministic Surfaces

In this subsection, we collect several properties of perturbed surfaces which are important for the subsequent analysis. Assume in this and the next two subsections that the perturbation function $\kappa$ is a fixed deterministic function in $W^{1,\infty }(\Gamma ^0)$. Then $\Gamma ^\epsilon$ is defined by

$$\begin{aligned} \Gamma ^{\epsilon } := \{ {\varvec{x}}+ \epsilon \kappa ({\varvec{x}}){\varvec{n}}^0({\varvec{x}}) : {\varvec{x}}\in \Gamma ^0 \}, \quad \epsilon > 0. \end{aligned}$$

(3.1)

As already noticed in Sect. 2.2, $\Gamma ^\epsilon$ is a closed Lipschitz surface in ${{\mathbb {R}}}^3$ provided $0 \le \epsilon \le \epsilon _0$ and $\epsilon _0$ is sufficiently small. The unbounded domain exterior to $\Gamma ^\epsilon$ is denoted by $D^\epsilon$, for $\epsilon \ge 0$. We assume that there exist two unbounded domains ${\underline{D}}$ and ${\overline{D}}$ satisfying

$$\begin{aligned} {\underline{D}} \subsetneqq D^\epsilon \subsetneqq {\overline{D}} \quad \forall \epsilon \le \epsilon _0. \end{aligned}$$

(3.2)

Following [19], we define a mapping $T^\epsilon :{{\mathbb {R}}}^3\rightarrow {{\mathbb {R}}}^3$ which transforms $\Gamma ^0$ and $D^0$ into $\Gamma ^\epsilon$ and $D^\epsilon$, respectively, by

$$\begin{aligned} T^\epsilon ({\varvec{x}}) := {\varvec{x}}+ \epsilon {\tilde{\kappa }}({\varvec{x}}) {\tilde{{\varvec{n}}}}{}^0({\varvec{x}}), \quad {\varvec{x}}\in {{\mathbb {R}}}^3, \end{aligned}$$

(3.3)

where ${\tilde{\kappa }}$ and ${\tilde{{\varvec{n}}}}{}^0$ are any smoothness-preserving extensions of $\kappa$ and ${\varvec{n}}^0$ into ${{\mathbb {R}}}^3$. We require in particular that ${{\tilde{\kappa }}}\in W^{1,\infty }({\overline{D}})$. Similar to the case of half-plane (see [7]), we may assume without loss of generality that the extension ${{\tilde{\kappa }}}$ vanishes outside a sufficiently large ball $B_{R} := \{{\varvec{x}}\in {{\mathbb {R}}}^3 : |{\varvec{x}}|<R\}$ containing $\Gamma ^\epsilon$ for all $\epsilon$ satisfying $0 \le \epsilon \le \epsilon _0$. This implies that the perturbation mapping $T^{\epsilon }$ is an identity in the complement $B_R^c := {{\mathbb {R}}}^3 {\setminus } \overline{B_R}$, i.e.,

$$\begin{aligned} T^{\epsilon }({\varvec{x}}) = {\varvec{x}}\qquad \forall {\varvec{x}}\in {B_{R}^c}. \end{aligned}$$

(3.4)

For ease of notation, we abbreviate

$$\begin{aligned} V({\varvec{x}}) := {{\tilde{\kappa }}}({\varvec{x}}){{\tilde{{\varvec{n}}}}}{}^0({\varvec{x}}), \qquad {\varvec{x}}\in {{\mathbb {R}}}^3. \end{aligned}$$

(3.5)

In [19], V is called the velocity field of the mapping $T^\epsilon$.

In the subsequent analysis, a surface integral over $\Gamma ^\epsilon$ is computed via a surface integral over $\Gamma ^0$ as follows

$$\begin{aligned} \int _{\Gamma ^\epsilon } v({\varvec{y}})\,d\sigma _{{\varvec{y}}} = \int _{\Gamma ^0} v\circ T^\epsilon ({\varvec{x}}) \beta (\epsilon ,{\varvec{x}}) \,d\sigma _{{\varvec{x}}}, \end{aligned}$$

where $\beta (\epsilon ,{\varvec{x}}) = \left\| {M(T^\epsilon ({\varvec{x}}))\cdot {\varvec{n}}^0({\varvec{x}})}\right\| _{}$ with $M(T^\epsilon )$ being the cofactor matrix of the Jacobian matrix $J_{T^\epsilon }$.

Recall that the surface divergence of V is defined by

$$\begin{aligned} {{\,\mathrm{{div }}\,}}_{\Gamma ^0} V = {{\,\mathrm{{div }}\,}}V - (J_V^\top {\varvec{n}}^0)\cdot {\varvec{n}}^0, \end{aligned}$$

where $J_V$ is the Jacobian matrix of V and $J_V^\top$ is the transpose matrix. We now prove a technical lemma which involves the derivative with respect to $\epsilon$ of $\beta (\epsilon ,\cdot )(v\circ T^\epsilon )$ for any function $v\in H^1(U)$, where U is a bounded domain containing all $\Gamma ^\epsilon$.

Lemma 3.1

For any $v\in H^1(U)$, where U is an open bounded domain which contains all $\Gamma ^\epsilon$, there holds

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \left\| {\frac{ \beta (\epsilon ,\cdot )(v\circ T^\epsilon ) - v }{\epsilon } - v{{\,\mathrm{{div }}\,}}_{\Gamma ^0} V - V\cdot \nabla v }\right\| _{L^2(\Gamma ^0)} = 0. \end{aligned}$$

(3.6)

Proof

The triangle inequality gives

$$\begin{aligned}&\left\| {\frac{ \beta (\epsilon ,\cdot )(v\circ T^\epsilon ) - v }{\epsilon } - v{{\,\mathrm{{div }}\,}}_{\Gamma ^0} V - V\cdot \nabla v }\right\| _{L^2(\Gamma ^0)} \nonumber \\&\quad \le \left\| {\frac{ \beta (\epsilon ,\cdot )-1 }{\epsilon }(v\circ T^\epsilon ) - v{{\,\mathrm{{div }}\,}}_{\Gamma ^0} V }\right\| _{L^2(\Gamma ^0)} + \left\| {\frac{ (v\circ T^\epsilon ) - v }{\epsilon } - V\cdot \nabla v }\right\| _{L^2(\Gamma ^0)}. \end{aligned}$$

(3.7)

Elementary calculations give

$$\begin{aligned} \beta (\epsilon ,{\varvec{x}})&= \left\| {M(T^\epsilon ({\varvec{x}}))\cdot {\varvec{n}}^0({\varvec{x}})}\right\| _{} = \left( 1 + 2\epsilon {{\,\mathrm{{div }}\,}}_{\Gamma ^0}V({\varvec{x}}) + \epsilon ^2 q_1({\varvec{x}}) \right) ^{1/2}, \end{aligned}$$

(3.8)

where $q_1$ is a polynomial of partial derivatives of V which plays no important role in the following calculations (hence, an explicit form is not given). From (3.8), we deduce

$$\begin{aligned} \beta _\epsilon '(0,\cdot )&= {{\,\mathrm{{div }}\,}}_{\Gamma ^0}V. \end{aligned}$$

Hence, it can be shown that

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \left\| {\frac{\beta (\epsilon ,\cdot ) - 1}{\epsilon } - {{\,\mathrm{{div }}\,}}_{\Gamma ^0}V}\right\| _{L^\infty (U)} = 0. \end{aligned}$$

This together with the fact that $\lim _{\epsilon \rightarrow 0} \left\| {v\circ T^\epsilon -v}\right\| _{L^2(\Gamma ^0)} = 0$ yields

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \left\| {\frac{ \beta (\epsilon ,\cdot )-1 }{\epsilon }(v\circ T^\epsilon ) - v{{\,\mathrm{{div }}\,}}_{\Gamma ^0} V }\right\| _{L^2(\Gamma ^0)} = 0. \end{aligned}$$

(3.9)

On the other hand, by using a density argument, it can be shown that

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \left\| {\frac{ v\circ T^\epsilon - v }{\epsilon } - V\cdot \nabla v }\right\| _{L^2(\Gamma ^0)} = 0, \end{aligned}$$

noting that $V = \partial T^\epsilon / \partial \epsilon$ at $\epsilon =0$. This together with (3.7) and (3.9) completes the proof of the lemma. $\square$

We finish this subsection by introducing the definitions of material and shape derivatives; see, e.g., [9, 19].

Definition 3.2

For any sufficiently small $\epsilon$, let $v^\epsilon$ be an element in $H_w^1(D^\epsilon )$ or $H^{1/2}(\Gamma ^\epsilon )$.

• (i) The material derivative of $v^\epsilon$, denoted by $\dot{v}$, is defined by

  $$\begin{aligned} \dot{v} := \lim _{\epsilon \rightarrow 0} \frac{v^\epsilon \circ T^\epsilon - v^0}{\epsilon }, \end{aligned}$$

  (3.10)

  if the limit exists in the corresponding space $H_w^1(D^0)$ or $H^{1/2}(\Gamma ^0)$.

• (ii) The shape derivative of $v^\epsilon$ is defined by

  $$\begin{aligned} v' = {\left\{ \begin{array}{ll} \dot{v} - \nabla v^0\cdot V &{} \text {if}\ v^\epsilon \in H_w^1(D^\epsilon ), \\ \dot{v} - \nabla _{\Gamma ^0} v^0\cdot V &{} \text {if}\ v^\epsilon \in H^{1/2}(\Gamma ^\epsilon ), \end{array}\right. } \end{aligned}$$

  (3.11)

  where $\nabla _{\Gamma ^0}$ denotes the surface gradient.

The following lemma which will frequently be used in the remainder of this paper is proved in [9, Lemma 3.8].

Lemma 3.3

Let $\dot{v}$, $\dot{w}$ be material derivatives, and $v'$, $w'$ be shape derivatives of $v^\epsilon$, $w^\epsilon$ in $H_w^1(D^\epsilon )$, $\epsilon \ge 0$, respectively. Then the following statements are true.

• (i) The material and shape derivatives of the product $v^\epsilon w^\epsilon$ are $\dot{v} w^0 + v^0\dot{w}$ and $v' w^0 + v^0 w'$, respectively.

• (ii) If $v^\epsilon = v$ for all $\epsilon \ge 0$, then $\dot{v} = \nabla v^0\cdot V = \nabla v\cdot V$ and $v' = 0$.

• (iii) For $i=1,2$, if

  $$\begin{aligned} {\mathcal {J}}_1(D^\epsilon ):= & {} \displaystyle \int _{D^\epsilon } v^\epsilon \,d{\varvec{x}}, \quad {\mathcal {J}}_2(D^\epsilon ) \\:= & {} \displaystyle \int _{\Gamma ^\epsilon } v^\epsilon \,d\sigma , \quad \text {and} \quad d{\mathcal {J}}_i(D^\epsilon )|_{\epsilon =0} := \lim _{\epsilon \rightarrow 0} \frac{{\mathcal {J}}_i(D^\epsilon )-{\mathcal {J}}_i(D^0)}{ \epsilon }, \end{aligned}$$

  then

  $$\begin{aligned} d{\mathcal {J}}_1(D^\epsilon )|_{\epsilon =0} = \int _{D^0} v'\,d{\varvec{x}}+ \int _{\Gamma ^0} v^0 \left\langle {V},{{{\varvec{n}}^0}}\right\rangle \,d\sigma \end{aligned}$$

  and

  $$\begin{aligned} d{\mathcal {J}}_2(D^\epsilon )|_{\epsilon =0} = \int _{\Gamma ^0} v'\,d\sigma + \int _{\Gamma ^0} \left( \frac{\partial v^0}{\partial n} + {{\,\mathrm{{div }}\,}}_{\Gamma ^0}({\varvec{n}}^0)\, v^0 \right) \left\langle {V},{{{\varvec{n}}^0}}\right\rangle \,d\sigma . \end{aligned}$$

We note here that the surface divergence of the normal vector field ${{\,\mathrm{{div }}\,}}_{\Gamma ^0}({\varvec{n}}^0)$ is the mean curvature (with the minus sign), i.e., ${{\,\mathrm{{div }}\,}}_{\Gamma ^0}({\varvec{n}}^0) = -H$ where H is the mean curvature of $\Gamma ^0$, see, e.g., [12, page 24].

3.2 Shape Derivative of Solutions to the Exterior Neumann Problem

In this subsection, we shall discuss the existence of material and shape derivatives of the solutions of exterior Neumann problems on perturbed surfaces. Consider a deterministic problem with respect to the reference surface $\Gamma ^0$:

$$\begin{aligned} - \Delta u^0&= f \quad \text {in } D^0, \end{aligned}$$

(3.12a)

$$\begin{aligned} \frac{\partial u^0}{\partial \varvec{n}^0}&= g_N \quad \text {on } \Gamma ^0, \end{aligned}$$

(3.12b)

$$\begin{aligned} u^0({{\varvec{x}}}{})&= {\mathcal {O}}(\left\| {{\varvec{x}}}\right\| _{}^{-1}) \quad \text {when } \left\| {{\varvec{x}}}\right\| _{}\rightarrow \infty . \end{aligned}$$

(3.12c)

The perturbed problem on the perturbed surface $\Gamma ^{\epsilon }$, see (3.1), is given by

$$\begin{aligned} -\Delta u^\epsilon&= f \quad \text {in } D^{\epsilon }, \end{aligned}$$

(3.13a)

$$\begin{aligned} \frac{\partial u^\epsilon }{\partial \varvec{n}^\epsilon }&= g_N \quad \text {on } \Gamma ^{\epsilon }, \end{aligned}$$

(3.13b)

$$\begin{aligned} u^\epsilon ({{\varvec{x}}}{})&= {\mathcal {O}}(\left\| {{\varvec{x}}}\right\| _{}^{-1}) \quad \text {when } \left\| {{\varvec{x}}}\right\| _{}\rightarrow \infty . \end{aligned}$$

(3.13c)

Lemma 3.4

Suppose $f\in L^2({\overline{D}})\cap {\big (H_w^1(D^0)\big )^*}$ and $\kappa \in C^1(\Gamma ^0)$ where $\big (H_w^1(D^0)\big )^*$ denotes the dual space of $H_w^1(D^0)$ with respect to the $L^2$-inner product. Suppose further that $g_N\in H^{s}({\mathcal {U}})$ for some $s>1/2$ where ${\mathcal {U}} = {\overline{D}}{\setminus } \mathrm{{cl}}({\underline{D}})$ contains all $\Gamma ^\epsilon$; see (3.2). Here $\mathrm{{cl}}({\underline{D}})$ denotes the closure of ${\underline{D}}$. If $u^0$ and $u^\epsilon$ are solutions of (3.12) and (3.13), respectively, then

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \left\| {u^\epsilon \circ T^\epsilon -u^0}\right\| _{{H_w^1(D^0)}} = 0. \end{aligned}$$

(3.14)

Proof

Multiplying both sides of (3.13a) by an arbitrary function $v\in C_0^\infty ({\overline{D}})$, integrating over $D^\epsilon$ and using Green’s identity, we obtain, noting (3.13b),

$$\begin{aligned} \int _{D^\epsilon } \nabla u^\epsilon ({\varvec{x}})\cdot \nabla v({\varvec{x}}) + \int _{\Gamma ^\epsilon } g_N({\varvec{x}}) v({\varvec{x}})\,d\sigma _{{\varvec{x}}} = \int _{D^\epsilon } fv\,d{\varvec{x}}\quad \forall v\in C_0^\infty ({\overline{D}}).\nonumber \\ \end{aligned}$$

(3.15)

Since the space $\{v|_{D^\epsilon }: v\in C_0^\infty ({\overline{D}})\}$ is dense in $H_w^1(D^\epsilon )$ (see [18, Remark 2.9.3]), there holds

$$\begin{aligned} \int _{D^\epsilon } \nabla u^\epsilon ({\varvec{x}})\cdot \nabla v({\varvec{x}}) = \int _{D^\epsilon } fv\,d{\varvec{x}}- \int _{\Gamma ^\epsilon } g_N({\varvec{x}})v({\varvec{x}})\,d\sigma _{{\varvec{x}}} \quad \forall v\in H_w^1(D^\epsilon ).\nonumber \\\ \end{aligned}$$

(3.16)

Choosing $v = u^\epsilon$ gives

$$\begin{aligned} \left| {u^\epsilon }\right| _{{H^1(D^\epsilon )}}^2&\lesssim \left\| {f}\right\| _{{[H_w^1(D^\epsilon )]^*}} \left\| {u^\epsilon }\right\| _{H_w^1(D^\epsilon )} + \left\| {g_N|_{\Gamma ^\epsilon }}\right\| _{H^{-1/2}(\Gamma ^\epsilon )} \left\| {u^\epsilon }\right\| _{H^{1/2}(\Gamma ^\epsilon )} \\&\lesssim \big (\left\| {f}\right\| _{{[H_w^1(D^\epsilon )]^*}} + \left\| {g_N}\right\| _{H^{s}({\mathcal {U}})}\big ) \left\| {u^\epsilon }\right\| _{H_w^1(D^\epsilon )}. \end{aligned}$$

It follows from Lemma 2.1 that

$$\begin{aligned} \left\| {u^\epsilon }\right\| _{H_w^1(D^\epsilon )} \lesssim \left\| {f}\right\| _{{[H_w^1(D^\epsilon )]^*}} + \left\| {g_N}\right\| _{H^{s}({\mathcal {U}})} \lesssim \left\| {f}\right\| _{{[H_w^1({\underline{D}})]^*}} + \left\| {g_N}\right\| _{H^{s}({\mathcal {U}})}.\nonumber \\ \end{aligned}$$

(3.17)

Here, we have used the fact that if $X_1$ and $X_2$ are two Hilbert spaces such that $X_1\hookrightarrow X_2$, where $\hookrightarrow$ denotes a continuous embedding, then the dual spaces satisfy $X_2^*\hookrightarrow X_1^*$.

On the other hand, using the change of variables ${\varvec{x}}= T^\epsilon ({\varvec{y}})$ in (3.16), we have

$$\begin{aligned}&\int _{D^0} (\nabla w({\varvec{y}}))^{\top } \,A(\epsilon ,{\varvec{y}})\, \nabla (u^\epsilon \circ T^\epsilon )({\varvec{y}}) \,d{\varvec{y}}\nonumber \\&\quad = \int _{D^0} f(T^\epsilon ({\varvec{y}}))\, w({\varvec{y}}) \gamma (\epsilon ,{\varvec{y}}) \,d{\varvec{y}}- \int _{\Gamma ^0} g_N(T^\epsilon ({\varvec{y}}))w({\varvec{y}})\beta (\epsilon ,{\varvec{y}})\, d\sigma _{{\varvec{y}}},\nonumber \\ \end{aligned}$$

(3.18)

for any $w\in H_w^1(D^0)$. Here,

$$\begin{aligned} A(\epsilon ,\cdot ) := \gamma (\epsilon ,\cdot ) J_{T^\epsilon }^{-1} J_{T^\epsilon }^{-\top } \quad \text {and} \quad \gamma (\epsilon ,\cdot ) = {{\,\mathrm{{det }}\,}}(J_{T^\epsilon }) \end{aligned}$$

(3.19)

with $J_{T^\epsilon }$ being the Jacobian matrix of the mapping $T^\epsilon$. We also obtain from problem (3.12)

$$\begin{aligned} \int _{D^0} (\nabla w({\varvec{y}}))^{\top }\, \nabla u^0({\varvec{y}})\, d{\varvec{y}}= \int _{D^0} f({\varvec{y}})\, w({\varvec{y}}) \,d{\varvec{y}}- \int _{\Gamma ^0} g_N({\varvec{y}})w({\varvec{y}})\,d\sigma _{{\varvec{y}}},\qquad \end{aligned}$$

(3.20)

for any $w\in H_w^1(D^0)$. Subtracting (3.20) from (3.18), we deduce

$$\begin{aligned}&\int _{D^0} \nabla w({\varvec{y}})^{\top } \, \nabla \Big ((u^\epsilon \circ T^\epsilon )({\varvec{y}}) - u^0({\varvec{y}}) \Big ) \,d{\varvec{y}}\nonumber \\&\quad = -\int _{D^0} \big (\nabla w({\varvec{y}})\big )^{\top } \, \Big ( A(\epsilon ,{\varvec{y}}) -I \Big ) \, \nabla (u^\epsilon \circ T^\epsilon )({\varvec{y}}) \,d{\varvec{y}}\nonumber \\&\qquad + \int _{D^0} \Big ( \gamma (\epsilon ,{\varvec{y}})f(T^\epsilon ({\varvec{y}})) - f({\varvec{y}}) \Big ) w({\varvec{y}}) \,d{\varvec{y}}\nonumber \\&\qquad - \int _{\Gamma ^0} \Big (g_N(T^\epsilon ({\varvec{y}}))\beta (\epsilon ,{\varvec{y}}) - g_N({\varvec{y}})\Big ) w({\varvec{y}})\,d\sigma _{{\varvec{y}}} \qquad \forall w\in H_w^1(D^0). \end{aligned}$$

(3.21)

Choosing $w = u^\epsilon \circ T^\epsilon - u^0$ gives

$$\begin{aligned} \int _{D^0}&\left| \nabla \Big ((u^\epsilon \circ T^\epsilon )({\varvec{y}}) - u^0({\varvec{y}}) \Big ) \right| ^2 \,d{\varvec{y}}\nonumber \\&= -\int _{D^0} \left( \nabla \Big ((u^\epsilon \circ T^\epsilon )({\varvec{y}}) - u^0({\varvec{y}}) \Big )\right) ^{\top } \Big ( A(\epsilon ,{\varvec{y}}) -I \Big ) \nabla (u^\epsilon \circ T^\epsilon )({\varvec{y}}) \,d{\varvec{y}}\nonumber \\&\quad + \int _{D^0} \sqrt{1 + \left| {{\varvec{y}}}\right| _{}^2} \Big ( \gamma (\epsilon ,{\varvec{y}})f(T^\epsilon ({\varvec{y}})) - f({\varvec{y}}) \Big ) \frac{(u^\epsilon \circ T^\epsilon )({\varvec{y}}) - u^0({\varvec{y}})}{\sqrt{1 + \left| {{\varvec{y}}}\right| _{}^2}} \,d{\varvec{y}}\nonumber \\&\quad - \int _{\Gamma ^0} \Big ( g_N(T^\epsilon ({\varvec{y}}))\beta (\epsilon ,{\varvec{y}}) - g_N({\varvec{y}}) \Big ) \Big ( (u^\epsilon \circ T^\epsilon )({\varvec{y}}) - u^0({\varvec{y}}) \Big )\,d\sigma _{{\varvec{y}}}\nonumber \\&\lesssim \left\| {\big ( A(\epsilon ,\cdot ) - I \big )}\right\| _{L^\infty ({\overline{D}})} \left\| {\nabla (u^\epsilon \circ T^\epsilon )}\right\| _{L^2({\overline{D}})} \left\| {\nabla \big (u^\epsilon \circ T^\epsilon - u^0 \big )}\right\| _{L^2({\overline{D}})} \nonumber \\&\quad + \left\| {\sqrt{1 + \left| {\cdot }\right| _{}^2}\big (\gamma (\epsilon ,\cdot ) f\circ T^\epsilon - f\big )}\right\| _{L^2({\overline{D}})} \, \Big \Vert {\frac{u^\epsilon \circ T^\epsilon - u^0}{\sqrt{1 + \left| {\cdot }\right| _{}^2}}}\Big \Vert _{L^2({\overline{D}})} \nonumber \\&\quad + \left\| {(g_N\circ T^\epsilon ))\beta (\epsilon ,\cdot ) - g_N}\right\| _{H^{-1/2}(\Gamma ^0)} \left\| {u^\epsilon \circ T^\epsilon - u^0}\right\| _{H^{1/2}(\Gamma ^0)}, \end{aligned}$$

(3.22)

implying

$$\begin{aligned} \left\| {u^\epsilon \circ T^\epsilon - u^0}\right\| _{H_w^1(D^0)}&\lesssim \left\| { A(\epsilon ,\cdot ) - I }\right\| _{L^\infty ({\overline{D}})} \left\| {\nabla (u^\epsilon \circ T^\epsilon )}\right\| _{L^2({\overline{D}})} \\&+ \left\| {\sqrt{1 + \left| {\cdot }\right| _{}^2}\big ( \gamma (\epsilon ,\cdot )f\circ T^\epsilon - f\big )}\right\| _{L^2({\overline{D}})}\\&+ \left\| {(g_N\circ T^\epsilon )\beta (\epsilon ,\cdot ) - g_N}\right\| _{L^2(\Gamma ^0)}. \end{aligned}$$

It is shown in [9, Lemmas 3.2 and 3.3], respectively, that

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \left\| { A(\epsilon ,\cdot ) - I }\right\| _{L^\infty ({\overline{D}})} = 0 \quad \text {and} \quad \lim _{\epsilon \rightarrow 0} \left\| {\sqrt{1 + \left| {\cdot }\right| _{}^2}\big ( \gamma (\epsilon ,\cdot )f\circ T^\epsilon - f\big )}\right\| _{L^2({\overline{D}})} = 0. \end{aligned}$$

Moreover, by noting

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \left\| { \beta (\epsilon ,\cdot ) - 1 }\right\| _{L^\infty (\Gamma ^0)} = 0 \quad \text {and} \quad \lim _{\epsilon \rightarrow 0} \left\| {g_N\circ T^\epsilon - g_N}\right\| _{L^2(\Gamma ^0)} = 0, \end{aligned}$$

one can prove that

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \left\| {(g_N\circ T^\epsilon )\beta (\epsilon ,\cdot ) - g_N}\right\| _{L^2(\Gamma ^0)} = 0. \end{aligned}$$

Hence,

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \left\| {u^\epsilon \circ T^\epsilon - u^0}\right\| _{H_w^1(D^0)} = 0, \end{aligned}$$

finishing the proof of this lemma. $\square$

Lemma 3.5

Under the assumptions of Lemma 3.4, there holds

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \left\| {\nabla (u^\epsilon \circ T^\epsilon )\,\frac{A(\epsilon ,\cdot )-I}{\epsilon } -\nabla u^0 A'(0,\cdot )}\right\| _{L^2(D^0)} = 0. \end{aligned}$$

Here, $A'(0,\cdot )$ is the Gâteaux derivative at $\epsilon =0$ of $A(\epsilon ,\cdot )$ defined in (3.19), namely

$$\begin{aligned} A'(0,{\varvec{x}}) = \lim _{\epsilon \rightarrow 0} \frac{A(\epsilon ,{\varvec{x}}) - I({\varvec{x}})}{\epsilon }, \quad {\varvec{x}}\in {\overline{D}}. \end{aligned}$$

(3.23)

Proof

Let

$$\begin{aligned} J(\epsilon ) := \left\| {\nabla (u^\epsilon \circ T^\epsilon )\,\frac{A(\epsilon ,\cdot )-I}{\epsilon } -\nabla u^0 A'(0,\cdot )}\right\| _{L^2(D^0)}. \end{aligned}$$

The triangle inequality gives

$$\begin{aligned} J(\epsilon )&\le \left\| { \nabla (u^\epsilon \circ T^\epsilon ) \left( \frac{A(\epsilon ,\cdot )-I}{\epsilon } - A'(0,\cdot )\right) }\right\| _{L^2(D^0)}\\&\quad + \left\| { \big (\nabla (u^\epsilon \circ T^\epsilon ) - \nabla u^0\big ) A'(0,\cdot ) }\right\| _{L^2(D^0)}\\&\le \left\| {\frac{A(\epsilon ,\cdot )-I}{\epsilon } - A'(0,\cdot )}\right\| _{L^\infty (D^0)} \left\| {u^\epsilon \circ T^\epsilon }\right\| _{H_w^1(D^0)}\\&\quad + \left\| {A'(0,\cdot )}\right\| _{L^\infty (D^0)} \left\| {u^\epsilon \circ T^\epsilon - u^0}\right\| _{H_w^1(D^0)}. \end{aligned}$$

It is shown in [9, Lemma 3.2] that

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \left\| {\dfrac{A(\epsilon ,\cdot ) - I}{\epsilon } - A'(0,\cdot )}\right\| _{L^\infty (D^0)} = 0. \end{aligned}$$

This together with (3.14) gives the desired result. $\square$

Lemma 3.6

Assume that $f\in H^1({\overline{D}})\cap {H_w^1(D^0)^*}$ and $\kappa \in C^1(\Gamma ^0)$. Assume further that $g_N\in H^s({\mathcal {U}})$ for some $s> 3/2$. Then, $u^\epsilon$ has a material derivative belonging to $H_w^1(D^0)$ which is the solution to the following equation with unknown z:

$$\begin{aligned} \int _{D^0} \, \nabla z({\varvec{y}}) \cdot \nabla w({\varvec{y}}) \,d{\varvec{y}}&= -\int _{D^0} \nabla u^0({\varvec{y}}) A'(0,{\varvec{y}}) \big (\nabla w({\varvec{y}})\big )^{\top } \,d{\varvec{y}}\nonumber \\&\quad + \int _{D^0} {{\,\mathrm{{div }}\,}}\left( V({\varvec{y}})f({\varvec{y}})\right) w({\varvec{y}}) \,d{\varvec{y}}\nonumber \\&\quad - \int _{\Gamma ^0} \Big ( g_N({\varvec{y}}){{\,\mathrm{{div }}\,}}_{\Gamma ^0} V({\varvec{y}}) + V({\varvec{y}})\cdot \nabla g_N({\varvec{y}}) \Big ) w({\varvec{y}})\,d\sigma _{{\varvec{y}}} \end{aligned}$$

(3.24)

for all $w\in {H_w^1(D^0)}$. Here, $A'(0,\cdot )$ is the Gâteaux derivative at $\epsilon =0$ of $A(\epsilon ,\cdot )$ defined in (3.23).

Proof

The unique existence of the solution $z\in {H_w^1(D^0)}$ to the above equation is confirmed by [18, Theorem 2.10.12]. Let $z^\epsilon := (u^\epsilon \circ T^\epsilon - u^0)/\epsilon$; see (3.10). Our task is to prove that $\displaystyle \lim _{\epsilon \rightarrow 0} \left\| {z^\epsilon -z}\right\| _{{H_w^1(D^0)}}= 0$. Dividing (3.21) by $\epsilon$, we obtain

$$\begin{aligned} \int _{D^0} \, \nabla z^\epsilon ({\varvec{y}}) \cdot \nabla w({\varvec{y}}) \,d{\varvec{y}}&= -\int _{D^0} \nabla (u^\epsilon \circ T^\epsilon )({\varvec{y}}) \frac{ A(\epsilon ,{\varvec{y}}) -I }{\epsilon } \big (\nabla w({\varvec{y}})\big )^{\top } \,d{\varvec{y}}\nonumber \\&\quad + \int _{D^0} \frac{ \gamma (\epsilon ,{\varvec{y}})f(T^\epsilon ({\varvec{y}})) - f({\varvec{y}}) }{\epsilon } w({\varvec{y}}) \,d{\varvec{y}}\nonumber \\&\quad - \int _{\Gamma ^0} \frac{ \beta (\epsilon ,{\varvec{y}})g_N(T^\epsilon ({\varvec{y}})) - g_N({\varvec{y}}) }{\epsilon } w({\varvec{y}}) \,d\sigma _{{\varvec{y}}} \qquad \forall {w\in {H_w^1(D^0)}}. \end{aligned}$$

(3.25)

Subtracting (3.24) from (3.25) yields

$$\begin{aligned} \int _{D^0}&\, \nabla \left( z^\epsilon ({\varvec{y}})-z({\varvec{y}})\right) \cdot \nabla w({\varvec{y}}) \,d{\varvec{y}}\nonumber \\&= - \int _{D^0} \left( \nabla (u^\epsilon \circ T^\epsilon )({\varvec{y}})\,\frac{A(\epsilon ,{\varvec{y}})-I}{\epsilon } -\nabla u^0({\varvec{y}}) A'(0,{\varvec{y}})\right) \cdot \nabla w({\varvec{y}})\,d{\varvec{y}}\nonumber \\&\quad + \int _{D^0} \left( \frac{\gamma (\epsilon ,{\varvec{y}}) f(T^\epsilon ({\varvec{y}})) - f({\varvec{y}})}{\epsilon } - {{\,\mathrm{{div }}\,}}\Big ( V({\varvec{y}})f({\varvec{y}}) \Big ) \right) w({\varvec{y}})\,d{\varvec{y}}\nonumber \\&\quad - \int _{\Gamma ^0} \Big ( \frac{ \beta (\epsilon ,{\varvec{y}})g_N(T^\epsilon ({\varvec{y}})) - g_N({\varvec{y}}) }{\epsilon } - g_N({\varvec{y}}){{\,\mathrm{{div }}\,}}_{\Gamma ^0} V({\varvec{y}}) - V({\varvec{y}})\cdot \nabla g_N({\varvec{y}}) \Big ) w({\varvec{y}}) \,d\sigma _{{\varvec{y}}}. \end{aligned}$$

(3.26)

Denoting the last integral on the right-hand side of (3.26) by J, we note that

$$\begin{aligned} \left| J\right|&\le \left\| { \frac{ \beta (\epsilon ,\cdot ) (g_N\circ T) - g_N }{\epsilon } - g_N{{\,\mathrm{{div }}\,}}_{\Gamma ^0} V - V\cdot \nabla g_N }\right\| _{L^2(\Gamma ^0)} \left\| {w}\right\| _{L^2(\Gamma ^0)}. \end{aligned}$$

The trace theorem gives $\left\| {w}\right\| _{L^2(\Gamma ^0)}\lesssim \left\| {w}\right\| _{H^1(D^0{\setminus } \mathrm {cl}({\underline{D}}))}\lesssim \left\| {w}\right\| _{H_w^1(D^0)}$. Therefore, by letting $w = z^\epsilon - z$ in (3.26), using Lemma 2.1 and canceling the common term, we obtain

$$\begin{aligned} \left\| {z^\epsilon -z}\right\| _{H_w^1(D^0)}&\le \left\| {\nabla (u^\epsilon \circ T^\epsilon )\,\frac{A(\epsilon ,\cdot )-I}{\epsilon } -\nabla u^0 A'(0,\cdot )}\right\| _{L^2(D^0)}\\&+ \left\| {\sqrt{1 + \left| \cdot \right| ^2} \left( \frac{\gamma (\epsilon ,\cdot ) (f\circ T^\epsilon ) - f}{\epsilon } - {{\,\mathrm{{div }}\,}}\big ( Vf \big ) \right) }\right\| _{L^2(D^0)}\\&+ \left\| { \frac{ \beta (\epsilon ,\cdot ) (g_N\circ T) - g_N }{\epsilon } - g_N{{\,\mathrm{{div }}\,}}_{\Gamma ^0} V - V\cdot \nabla g_N }\right\| _{L^2(\Gamma ^0)}. \end{aligned}$$

The fact that the terms on the right-hand side go to zero when $\epsilon$ goes to zero are proved in Lemma 3.5, [9, Lemma 3.4] and Lemma 3.1, respectively. This finishes the proof of this lemma. $\square$

Hence, we have shown that the solution of the exterior Neumann problem (3.13) has a material derivative, and thus a shape derivative. The latter turns out to be the solution of another homogeneous exterior Neumann problem on the nominal surface $\Gamma ^0$. We state and prove that result in the following theorem.

Theorem 3.7

Under the assumption of Lemma 3.6, the shape derivative $u'$ of $u^\epsilon$ exists and is the weak solution of the Neumann problem

$$\begin{aligned} \begin{aligned} \Delta u'&= 0 \quad \text {in } D^0\\ \dfrac{\partial u'}{\partial \varvec{n}^0}&= G_N \quad \text {on } \Gamma ^0\\ \left| {u'({\varvec{x}})}\right| _{}&= {\mathcal {O}}\left( {\left\| {{\varvec{x}}}\right\| _{}^{-1}}\right) \quad \text {as } \left\| {{\varvec{x}}}\right\| _{}\rightarrow \infty , \end{aligned} \end{aligned}$$

(3.27)

where

$$\begin{aligned} G_N := \nabla _{\Gamma ^0}\cdot \Big ( \left\langle {V},{{\varvec{n}}^0}\right\rangle \nabla _{\Gamma ^0}u^0 \Big ) + \left\langle {V},{{\varvec{n}}^0}\right\rangle \left( \frac{\partial g_N}{\partial {\varvec{n}}^0} - H g_N + f \right) . \end{aligned}$$

(3.28)

Proof

The existence of $u'$ is confirmed by Lemma 3.6. From (3.15) and Green’s identity, we deduce

$$\begin{aligned} - \int _{D^\epsilon } u^\epsilon \Delta v\,d{\varvec{x}}- \int _{\Gamma ^\epsilon } u^\epsilon \frac{\partial v}{\partial n^\epsilon } \,d\sigma + \int _{\Gamma ^\epsilon } g_Nv\,d\sigma = \int _{D^\epsilon } fv\,d{\varvec{x}}\qquad \forall v\in C_0^\infty ({\overline{D}}).\nonumber \\ \end{aligned}$$

(3.29)

It follows from Lemma 3.3 (i) and (ii) that the shape derivative of $u^\epsilon \Delta v$ is $u'\Delta v$. Furthermore, it is shown in [9, Lemma 3.9] that the shape derivative of ${\varvec{n}}^\epsilon$ is $-\nabla _{\Gamma ^0}\kappa$. Thus, the shape derivative of $\dfrac{\partial v}{\partial {\varvec{n}}^\epsilon } =\nabla v\cdot {\varvec{n}}^\epsilon$ is $-\nabla _{\Gamma ^0} v\cdot \nabla _{\Gamma ^0}\left\langle {V},{{\varvec{n}}^0}\right\rangle$, so that the shape derivative of $\displaystyle u^\epsilon \dfrac{\partial v}{\partial {\varvec{n}}^\epsilon }$ is $\displaystyle u' \dfrac{\partial v}{\partial {\varvec{n}}^0} - u^0\Big (\nabla _{\Gamma ^0} v\cdot \nabla _{\Gamma ^0}\left\langle {V},{{\varvec{n}}^0}\right\rangle \Big )$. Using Lemma 3.3 (iii), we deduce

$$\begin{aligned}&- \int _{D^0} u' \Delta v\,d{\varvec{x}}+ \int _{\Gamma ^0} u^0 \Delta v \left\langle {V},{{\varvec{n}}^0}\right\rangle \,d\sigma - \int _{\Gamma ^0} u'\frac{\partial v}{\partial {\varvec{n}}^0} \nonumber \\&\quad - \int _{\Gamma ^0} u^0 \big (\nabla _{\Gamma ^0} v\cdot \nabla _{\Gamma ^0} \left\langle {V},{{\varvec{n}}^0}\right\rangle \big )\,d\sigma \nonumber \\&- \int _{\Gamma ^0} \frac{\partial }{\partial {\varvec{n}}^0} \Big ( u^0\frac{\partial v}{\partial {\varvec{n}}^0} \Big ) \left\langle {V},{{\varvec{n}}^0}\right\rangle \,d\sigma - \int _{\Gamma ^0} {{\,\mathrm{{div }}\,}}_{\Gamma ^0}({\varvec{n}}^0) u^0\frac{\partial v}{\partial {\varvec{n}}^0} \left\langle {V},{{\varvec{n}}^0}\right\rangle \,d\sigma \nonumber \\&+ \int _{\Gamma ^0} \frac{\partial }{\partial {\varvec{n}}^0}(g_Nv) \left\langle {V},{{\varvec{n}}^0}\right\rangle \,d\sigma + \int _{\Gamma ^0} {{\,\mathrm{{div }}\,}}_{\Gamma ^0}({\varvec{n}}^0) g_Nv\left\langle {V},{{\varvec{n}}^0}\right\rangle \,d\sigma \nonumber \\&\quad = - \int _{\Gamma ^0} fv\left\langle {V},{{\varvec{n}}^0}\right\rangle \,d\sigma . \end{aligned}$$

(3.30)

Simple but lengthy calculations give

$$\begin{aligned} \Delta v = \Delta _{\Gamma ^0} v + {{\,\mathrm{{div }}\,}}_{\Gamma ^0}({\varvec{n}}^0) \partial v/\partial {\varvec{n}}+ \partial ^2v/\partial {\varvec{n}}^2. \end{aligned}$$

(3.31)

Applying Green’s identity for the first and the third integrals in (3.30), using (3.31) for the second integral, and noting the boundary condition (3.12b), we obtain

$$\begin{aligned}&\int _{D^0} \nabla u'\cdot \nabla v \,d{\varvec{x}}+ \int _{\Gamma ^0} u^0\left\langle {V},{{\varvec{n}}^0}\right\rangle \Delta _{\Gamma ^0} v\,d\sigma - \int _{\Gamma ^0} u^0 \Big ( \nabla _{\Gamma ^0} v\cdot \nabla _{\Gamma ^0}\left\langle {V},{{\varvec{n}}^0}\right\rangle \Big )\,d\sigma \\&\qquad + \int _{\Gamma ^0} \frac{\partial g_N}{\partial {\varvec{n}}^0} v\left\langle {V},{{\varvec{n}}^0}\right\rangle \,d\sigma + \int _{\Gamma ^0} {{\,\mathrm{{div }}\,}}_{\Gamma ^0}({\varvec{n}}^0) g_Nv\left\langle {V},{{\varvec{n}}^0}\right\rangle \,d\sigma \\&\quad = - \int _{\Gamma ^0} fv\left\langle {V},{{\varvec{n}}^0}\right\rangle \,d\sigma \quad \forall v\in C_0^\infty ({\overline{D}}). \end{aligned}$$

The tangential Green identity gives

$$\begin{aligned}&\int _{\Gamma ^0} u^0\left\langle {V},{{\varvec{n}}^0}\right\rangle \Delta _{\Gamma ^0} v\,d\sigma - \int _{\Gamma ^0} u^0 \Big ( \nabla _{\Gamma ^0} v\cdot \nabla _{\Gamma ^0}\left\langle {V},{{\varvec{n}}^0}\right\rangle \Big )\,d\sigma \\&\quad = \int _{\Gamma ^0} \Big ( \nabla _{\Gamma ^0}u^0\cdot \nabla _{\Gamma ^0} v \Big ) \left\langle {V},{{\varvec{n}}^0}\right\rangle \,d\sigma . \end{aligned}$$

Hence,

$$\begin{aligned}&\int _{D^0} \nabla u'\cdot \nabla v \,d{\varvec{x}}+ \int _{\Gamma ^0} \Big ( \nabla _{\Gamma ^0}u^0\cdot \nabla _{\Gamma ^0} v \Big ) \left\langle {V},{{\varvec{n}}^0}\right\rangle \,d\sigma + \int _{\Gamma ^0} \frac{\partial g}{\partial {\varvec{n}}^0} v\left\langle {V},{{\varvec{n}}^0}\right\rangle \,d\sigma \nonumber \\&\quad + \int _{\Gamma ^0} {{\,\mathrm{{div }}\,}}_{\Gamma ^0}({\varvec{n}}^0) gv\left\langle {V},{{\varvec{n}}^0}\right\rangle \,d\sigma = - \int _{\Gamma ^0} fv\left\langle {V},{{\varvec{n}}^0}\right\rangle \,d\sigma \quad \forall v\in C_0^\infty ({\overline{D}}). \end{aligned}$$

(3.32)

Choosing $v\in C_0^\infty (D^0)$ so that $v= 0$ and $\nabla v = 0$ on $\Gamma ^0$, we obtain

$$\begin{aligned} \int _{D^0}\nabla u'\cdot \nabla v = 0 \quad \forall v\in C_0^\infty (D^0). \end{aligned}$$

This implies

$$\begin{aligned} \Delta u' = 0 \quad \text {a.e. in } D^0. \end{aligned}$$

(3.33)

We now choose $v\in C_0^\infty ({\overline{D}})$ and applying Green’s identity and the tangential Green identity to the first and the second integrals in (3.32), respectively, noting (3.33) and obtain

$$\begin{aligned}&- \int _{\Gamma ^0} \frac{\partial u'}{\partial {\varvec{n}}^0} v\,d\sigma + \int _{\Gamma ^0} v \nabla _{\Gamma ^0}\cdot \Big ( \left\langle {V},{{\varvec{n}}^0}\right\rangle \nabla _{\Gamma ^0}u^0 \Big ) \,d\sigma + \int _{\Gamma ^0} \frac{\partial g}{\partial {\varvec{n}}^0} v\left\langle {V},{{\varvec{n}}^0}\right\rangle \,d\sigma \\&\quad + \int _{\Gamma ^0} H {{\,\mathrm{{div }}\,}}_{\Gamma ^0}({\varvec{n}}^0) gv\left\langle {V},{{\varvec{n}}^0}\right\rangle \,d\sigma = - \int _{\Gamma ^0} fv\left\langle {V},{{\varvec{n}}^0}\right\rangle \,d\sigma \quad \forall v\in C_0^\infty ({\overline{D}}), \end{aligned}$$

implying

$$\begin{aligned} \frac{\partial u'}{\partial {\varvec{n}}^0}&= \nabla _{\Gamma ^0}\cdot \Big ( \left\langle {V},{{\varvec{n}}^0}\right\rangle \nabla _{\Gamma ^0}u^0 \Big ) + \frac{\partial g}{\partial {\varvec{n}}^0} \left\langle {V},{{\varvec{n}}^0}\right\rangle \\&\quad + {{\,\mathrm{{div }}\,}}_{\Gamma ^0}({\varvec{n}}^0) g\left\langle {V},{{\varvec{n}}^0}\right\rangle + f\left\langle {V},{{\varvec{n}}^0}\right\rangle \quad \text {on} \quad \Gamma ^0. \end{aligned}$$

Hence, the shape derivative $u'$ is the weak solution of the Neumann problem (3.27). $\square$

3.3 Shape Derivative of Solutions of Exterior Dirichlet Problem

In this subsection, we shall discuss the existence of material and shape derivatives of the solutions of exterior Dirichlet problems on perturbed surfaces. Consider a deterministic problem with respect to the reference surface $\Gamma ^0$:

$$\begin{aligned} - \Delta u^0&= f \quad \text {in } D^0, \end{aligned}$$

(3.34a)

$$\begin{aligned} u^0&= g_D \quad \text {on } \Gamma ^0, \end{aligned}$$

(3.34b)

$$\begin{aligned} u^0({{\varvec{x}}}{})&= {\mathcal {O}}(\left\| {{\varvec{x}}}\right\| _{}^{-1}) \quad \text {when } \left\| {{\varvec{x}}}\right\| _{}\rightarrow \infty , \end{aligned}$$

(3.34c)

and the perturbed problem

$$\begin{aligned} -\Delta u^\epsilon&= f \quad \text {in } D^{\epsilon }, \end{aligned}$$

(3.35a)

$$\begin{aligned} u^\epsilon&= g_D \quad \text {on } \Gamma ^{\epsilon }, \end{aligned}$$

(3.35b)

$$\begin{aligned} u^\epsilon ({{\varvec{x}}}{})&= {\mathcal {O}}(\left\| {{\varvec{x}}}\right\| _{}^{-1}) \quad \text {when } \left\| {{\varvec{x}}}\right\| _{}\rightarrow \infty . \end{aligned}$$

(3.35c)

With the assumptions that $f\in [H_w^1(D^\epsilon )]^*$ and $g_D\in H^{1/2}(\Gamma ^\epsilon )$, the problem (3.35) has a unique weak solution $u^\epsilon \in H_w^1(D^\epsilon )$ for all $\epsilon \ge 0$. The following lemma shows that $u^0$ is the limit of $u^\epsilon \circ T^\epsilon$ in the weighted Sobolev space $H_w^1(D^0)$.

Lemma 3.8

Suppose $f\in L^2({\overline{D}})\cap {\big (H_w^1(D^0)\big )^*}$ and $\kappa \in C^1(\Gamma ^0)$ where $\big (H_w^1(D^0)\big )^*$ denotes the dual space of $H_w^1(D^0)$ with respect to the $L^2$-inner product. Suppose further that $g_D\in H^{s}({\mathcal {U}})$ for some $s>3/2$. If $u^0$ and $u^\epsilon$ are solutions of (3.34) and (3.35), respectively, then

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \left\| {u^\epsilon \circ T^\epsilon -u^0}\right\| _{{H_w^1(D^0)}} = 0. \end{aligned}$$

(3.36)

Proof

In this proof only, for ease of notation we denote also by $g_D$ a smoothness-preserving extension of $g_D$ into ${\overline{D}}$. Furthermore, we can assume, without loss of generality, that the extension vanishes outside a bounded domain ${\mathcal {U}}^*$ containing ${\mathcal {U}}$. The weak solution $u^\epsilon$ of (3.35) can then be written as

$$\begin{aligned} u^\epsilon = u_*^\epsilon + g_D \end{aligned}$$

(3.37)

where $u_*^\epsilon \in \mathring{H}_w^1(D^\epsilon )$ satisfies

$$\begin{aligned} \int _{D^\epsilon } \nabla u_*^\epsilon ({\varvec{x}}) \cdot \nabla w({\varvec{x}})\,d{\varvec{x}}= & {} \int _{D^\epsilon } f({\varvec{x}}) w({\varvec{x}}) \,d{\varvec{x}}\nonumber \\&- \int _{D^\epsilon } \nabla g_D({\varvec{x}}) \cdot \nabla w({\varvec{x}})\,d{\varvec{x}}\quad \forall w\in \mathring{H}_w^1(D^\epsilon ).\qquad \end{aligned}$$

(3.38)

In particular, $u^0 = u_*^0 + g_D$ where $u_*^0\in \mathring{H}_w^1(D^0)$ satisfies

$$\begin{aligned} \int _{D^0} \nabla u_*^0({\varvec{x}}) \cdot \nabla w({\varvec{x}})\,d{\varvec{x}}= & {} \int _{D^0} f({\varvec{x}}) w({\varvec{x}}) \,d{\varvec{x}}\nonumber \\&- \int _{D^0} \nabla g_D({\varvec{x}}) \cdot \nabla w({\varvec{x}})\,d{\varvec{x}}\quad \forall w\in \mathring{H}_w^1(D^0).\qquad \end{aligned}$$

(3.39)

Choosing $w = u_*^\epsilon$ in (3.38), noting that the seminorm $\left| {\cdot }\right| _{H_w^1(D^\epsilon )}$ is a norm in $\mathring{H}_w^1(D^\epsilon )$ and applying the duality argument, we obtain

$$\begin{aligned} \left\| {u_*^\epsilon }\right\| _{H_w^1(D^\epsilon )} \lesssim \left\| {f}\right\| _{{[H_w^1(D^\epsilon )]^*}} + \left\| {g_D}\right\| _{H^1({\mathcal {U}}^*)} \lesssim \left\| {f}\right\| _{[H_w^1({\underline{D}})]^*} + \left\| {g_D}\right\| _{H^1({\mathcal {U}}^*)}. \end{aligned}$$

(3.40)

Similarly as in the proof of Lemma 3.4, the change of variables ${\varvec{x}}= T^\epsilon ({\varvec{y}})$ in (3.38) gives

$$\begin{aligned}&\int _{D^0} (\nabla w({\varvec{y}}))^{\top } \,A(\epsilon ,{\varvec{y}})\, \nabla (u_*^\epsilon \circ T^\epsilon )({\varvec{y}}) \,d{\varvec{y}}\nonumber \\&\quad = \int _{D^0} f(T^\epsilon ({\varvec{y}}))\, w({\varvec{y}}) \gamma (\epsilon ,{\varvec{y}}) \,d{\varvec{y}}\nonumber \\&\qquad - \int _{D^0} (\nabla w({\varvec{y}}))^{\top } \,A(\epsilon ,{\varvec{y}})\, \nabla g_D({\varvec{y}}) \,d{\varvec{y}}\quad \forall w\in \mathring{H}_w^1(D^0), \end{aligned}$$

(3.41)

where $A(\epsilon ,\cdot )$ and $\gamma (\epsilon ,\cdot )$ are given by (3.19). Subtracting (3.39) from (3.41), we deduce

$$\begin{aligned}&\int _{D^0} \nabla (w({\varvec{y}}))^{\top } \, \nabla \Big ((u_*^\epsilon \circ T^\epsilon )({\varvec{y}}) - u_*^0({\varvec{y}}) \Big ) \,d{\varvec{y}}\nonumber \\&\quad = -\int _{D^0} \big (\nabla w({\varvec{y}})\big )^{\top } \, \Big ( A(\epsilon ,{\varvec{y}}) -I \Big ) \, \nabla (u_*^\epsilon \circ T^\epsilon )({\varvec{y}}) \,d{\varvec{y}}\nonumber \\&\qquad + \int _{D^0} \Big ( \gamma (\epsilon ,{\varvec{y}})f(T^\epsilon ({\varvec{y}})) - f({\varvec{y}}) \Big ) w({\varvec{y}}) \,d{\varvec{y}}\nonumber \\&\qquad - \int _{D^0} (\nabla w({\varvec{y}}))^{\top } \big (A(\epsilon ,{\varvec{y}}) - I\big ) \nabla g_D({\varvec{y}}) \,d\sigma _{{\varvec{y}}}, \qquad \forall w\in H_w^1(D^0). \end{aligned}$$

(3.42)

Choosing in (3.42) $w = u_*^\epsilon \circ T^\epsilon - u_*^0$ and using the duality inequality, we deduce

$$\begin{aligned} \left\| { u_*^\epsilon \circ T^\epsilon - u_*^0}\right\| _{H_w^1(D^0)}&\lesssim \left\| {\big ( A(\epsilon ,\cdot ) - I \big )}\right\| _{L^\infty (D^0)} \Big ( \left\| {\nabla (u_*^\epsilon \circ T^\epsilon )}\right\| _{L^2(D^0)} + \left\| {g_D}\right\| _{H^1({\mathcal {U}}^*)} \Big ) \\&\quad + \left\| {\sqrt{1 + \left| {\cdot }\right| _{}^2}\big (\gamma (\epsilon ,\cdot ) f\circ T^\epsilon - f\big )}\right\| _{L^2({\overline{D}})}. \end{aligned}$$

Applying [9, Lemmas 3.2 and 3.3] yields

$$\begin{aligned} \lim _{\epsilon \rightarrow 0} \left\| {u^\epsilon \circ T^\epsilon - u^0}\right\| _{H_w^1(D^0)} = \lim _{\epsilon \rightarrow 0} \left\| {u_*^\epsilon \circ T^\epsilon - u_*^0}\right\| _{H_w^1(D^0)} = 0, \end{aligned}$$

finishing the proof of this lemma. $\square$

Lemma 3.9

Assume that $f\in H^1({\overline{D}})\cap {H_w^1(D^0)^*}$ and $\kappa \in C^1(\Gamma ^0)$. Assume further that $g_D\in H^s({\mathcal {U}})$ for some $s> 3/2$. Then, $u_*^\epsilon$ has a material derivative belonging to $H_w^1(D^0)$ which is the solution to the following equation with unknown z:

$$\begin{aligned} \int _{D^0} \, \nabla z({\varvec{y}}) \cdot \nabla w({\varvec{y}}) \,d{\varvec{y}}&= -\int _{D^0} \big (\nabla w({\varvec{y}})\big )^{\top } A'(0,{\varvec{y}}) \nabla \big ( g_D({\varvec{y}}) + u_*^0({\varvec{y}}) \big ) \,d{\varvec{y}}\nonumber \\&\quad + \int _{D^0} {{\,\mathrm{{div }}\,}}\left( V({\varvec{y}})f({\varvec{y}})\right) w({\varvec{y}}) \,d{\varvec{y}}, \end{aligned}$$

(3.43)

for all $w\in {H_w^1(D^0)}$.

Proof

The unique existence of the solution $z\in {H_w^1(D^0)}$ to the above equation is confirmed by [18, Theorem 2.10.12]. Let $z^\epsilon := (u_*^\epsilon \circ T^\epsilon - u_*^0)/\epsilon$; see (3.10). Our task is to prove that $\displaystyle \lim _{\epsilon \rightarrow 0} \left\| {z^\epsilon -z}\right\| _{{H_w^1(D^0)}}= 0$. Dividing (3.42) by $\epsilon$, we obtain

$$\begin{aligned} \int _{D^0} \, \nabla z^\epsilon ({\varvec{y}}) \cdot \nabla w({\varvec{y}}) \,d{\varvec{y}}&= - \int _{D^0} (\nabla w({\varvec{y}}))^{\top } \frac{A(\epsilon ,{\varvec{y}}) - I}{\epsilon } \big (\nabla g_D({\varvec{y}}) + \nabla (u_*^\epsilon \circ T^\epsilon )({\varvec{y}}) \big ) \,d\sigma _{{\varvec{y}}} \nonumber \\&\quad + \int _{D^0} \frac{ \gamma (\epsilon ,{\varvec{y}})f(T^\epsilon ({\varvec{y}})) - f({\varvec{y}}) }{\epsilon } w({\varvec{y}}) \,d{\varvec{y}}\quad \forall {w\in {H_w^1(D^0)}}. \end{aligned}$$

(3.44)

Subtracting (3.43) from (3.44) yields

$$\begin{aligned} \int _{D^0}&\, \nabla \left( z^\epsilon ({\varvec{y}})-z({\varvec{y}})\right) \cdot \nabla w({\varvec{y}}) \,d{\varvec{y}}\\&= - \int _{D^0} \left( \nabla (u^\epsilon \circ T^\epsilon )({\varvec{y}})\,\frac{A(\epsilon ,{\varvec{y}})-I}{\epsilon } -\nabla u^0({\varvec{y}}) A'(0,{\varvec{y}})\right) \cdot \nabla w({\varvec{y}})\,d{\varvec{y}}\\&\quad - \int _{D^0} (\nabla w({\varvec{y}}))^\top \Big ( \frac{ A(\epsilon ,{\varvec{y}}) - I }{\epsilon } - A'(0,{\varvec{y}}) \Big ) \nabla g_D({\varvec{y}}) \,d\sigma _{{\varvec{y}}} \\&\quad + \int _{D^0} \left( \frac{\gamma (\epsilon ,{\varvec{y}}) f(T^\epsilon ({\varvec{y}})) - f({\varvec{y}})}{\epsilon } - {{\,\mathrm{{div }}\,}}\Big ( V({\varvec{y}})f({\varvec{y}}) \Big ) \right) w({\varvec{y}})\,d{\varvec{y}}. \end{aligned}$$

Letting $w = z^\epsilon - z$ and using similar argument as in the proof of Lemma 3.6, we obtain

$$\begin{aligned} \left\| {z^\epsilon -z}\right\| _{H_w^1(D^0)}&\le \left\| {\nabla (u^\epsilon \circ T^\epsilon )\,\frac{A(\epsilon ,\cdot )-I}{\epsilon } -\nabla u^0 A'(0,\cdot )}\right\| _{L^2(D^0)} \\&+ \left\| {\Big ( \frac{ A(\epsilon ,\cdot ) - I }{\epsilon } - A'(0,\cdot ) \Big ) \nabla g_D }\right\| _{L^2(D^0)} \\&+ \left\| {\sqrt{1 + \left| \cdot \right| ^2} \left( \frac{\gamma (\epsilon ,\cdot ) (f\circ T^\epsilon ) - f}{\epsilon } - {{\,\mathrm{{div }}\,}}\big ( Vf \big ) \right) }\right\| _{L^2(D^0)}. \end{aligned}$$

The proof that the first and the third terms tend to zero when $\epsilon$ goes to zero has been done in the proof of Lemma 3.6. The second term is bounded by

$$\begin{aligned} \left\| {\Big ( \frac{ A(\epsilon ,\cdot ) - I }{\epsilon } - A'(0,\cdot ) \Big ) \nabla g_D }\right\| _{L^2(D^0)} \le \left\| { \frac{A(\epsilon ,\cdot ) - I}{\epsilon } - A'(0,\cdot ) }\right\| _{L^\infty (D^0)} \left\| {\nabla g_D}\right\| _{L^2(D^0)}, \end{aligned}$$

and it tends to zero since the first factor on the right-hand side of the above inequality goes to zero when $\epsilon$ goes to zero; see [9, Lemma 3.2]. Hence, we have shown that $\lim _{\epsilon \rightarrow 0} \left\| {z^\epsilon -z}\right\| _{H_w^1(D^0)} = 0$, finishing the proof of this lemma. $\square$

Lemma 3.9 has shown the unique existence of the material derivative of $u_*^\epsilon$ and thus the existence of the shape derivative of $u_*^\epsilon$. Recalling that $u^\epsilon = u_*^\epsilon + g_D$ (see (3.37)) and applying Lemma 3.3 (ii), we deduce the existence of the shape derivative of $u^\epsilon$ and

$$\begin{aligned} u' = u_*' + g_D' = u_*'. \end{aligned}$$

(3.45)

Theorem 3.10

Under the assumption of Lemma 3.9, the shape derivative $u'$ of $u^\epsilon$ exists and is the weak solution of the Dirichlet problem

$$\begin{aligned} \begin{aligned} \Delta u'&= 0 \quad \text {in } D^0 \\ u'&= G_D \quad \text {on } \Gamma ^0 \\ \left| {u'({\varvec{x}})}\right| _{}&= {\mathcal {O}}\left( {\left\| {{\varvec{x}}}\right\| _{}^{-1}}\right) \quad \text {as } \left\| {{\varvec{x}}}\right\| _{}\rightarrow \infty , \end{aligned} \end{aligned}$$

(3.46)

where

$$\begin{aligned} G_D := \left( - \frac{\partial u^0}{\partial {\varvec{n}}^0} + \frac{\partial g_D}{\partial {\varvec{n}}^0} \right) \left\langle {V},{{\varvec{n}}^0}\right\rangle . \end{aligned}$$

(3.47)

Proof

The existence of $u'$ is confirmed by Lemma 3.9. From (3.38) and Green’s identity, we deduce

$$\begin{aligned}&- \int _{D^\epsilon } u_*^\epsilon \Delta v\,d{\varvec{x}}- \int _{\Gamma ^\epsilon } u_*^\epsilon \frac{\partial v}{\partial n^\epsilon } \,d\sigma + \int _{D^\epsilon } \nabla g_D({\varvec{x}})\cdot \nabla v \,d\sigma \nonumber \\&\quad = \int _{D^\epsilon } fv\,d{\varvec{x}}\qquad \forall v\in C_0^\infty ({\overline{D}}). \end{aligned}$$

(3.48)

Similar argument as in the proof of Theorem 3.7 gives

$$\begin{aligned}&- \int _{D^0} u_*' \Delta v\,d{\varvec{x}}+ \int _{\Gamma ^0} u_*^0 \Delta v \left\langle {V},{{\varvec{n}}^0}\right\rangle \,d\sigma - \int _{\Gamma ^0} u_*'\frac{\partial v}{\partial {\varvec{n}}^0} - \int _{\Gamma ^0} u_*^0 \big (\nabla _{\Gamma ^0} v\cdot \nabla _{\Gamma ^0} \left\langle {V},{{\varvec{n}}^0}\right\rangle \big )\,d\sigma \nonumber \\&\quad - \int _{\Gamma ^0} \frac{\partial }{\partial {\varvec{n}}^0} \Big ( u_*^0\frac{\partial v}{\partial {\varvec{n}}^0} \Big ) \left\langle {V},{{\varvec{n}}^0}\right\rangle \,d\sigma - \int _{\Gamma ^0} {{\,\mathrm{{div }}\,}}_{\Gamma ^0}({\varvec{n}}^0) u_*^0\frac{\partial v}{\partial {\varvec{n}}^0} \left\langle {V},{{\varvec{n}}^0}\right\rangle \,d\sigma \nonumber \\&\quad + \int _{\Gamma ^0} \nabla g_D\cdot \nabla v \left\langle {V},{{\varvec{n}}^0}\right\rangle \,d\sigma = - \int _{\Gamma ^0} fv\left\langle {V},{{\varvec{n}}^0}\right\rangle \,d\sigma . \end{aligned}$$

(3.49)

Applying Green’s identity for the first and the third integrals in (3.49), using (3.31) for the second integral, and noting (3.12b) and (3.45), we obtain

$$\begin{aligned}&\int _{D^0} \nabla u'\cdot \nabla v \,d{\varvec{x}}+ \int _{\Gamma ^0} u^0\left\langle {V},{{\varvec{n}}^0}\right\rangle \Delta _{\Gamma ^0} v\,d\sigma - \int _{\Gamma ^0} u^0 \Big ( \nabla _{\Gamma ^0} v\cdot \nabla _{\Gamma ^0}\left\langle {V},{{\varvec{n}}^0}\right\rangle \Big )\,d\sigma \\&\quad + \int _{\Gamma ^0} \frac{\partial g_D}{\partial {\varvec{n}}^0} v\left\langle {V},{{\varvec{n}}^0}\right\rangle \,d\sigma + \int _{\Gamma ^0} {{\,\mathrm{{div }}\,}}_{\Gamma ^0}({\varvec{n}}^0) g_Dv\left\langle {V},{{\varvec{n}}^0}\right\rangle \,d\sigma \\&\quad = - \int _{\Gamma ^0} fv\left\langle {V},{{\varvec{n}}^0}\right\rangle \,d\sigma \quad \forall v\in C_0^\infty ({\overline{D}}). \end{aligned}$$

The tangential Green identity gives

$$\begin{aligned}&\int _{\Gamma ^0} u^0\left\langle {V},{{\varvec{n}}^0}\right\rangle \Delta _{\Gamma ^0} v\,d\sigma - \int _{\Gamma ^0} u^0 \Big ( \nabla _{\Gamma ^0} v\cdot \nabla _{\Gamma ^0}\left\langle {V},{{\varvec{n}}^0}\right\rangle \Big )\,d\sigma \\&\quad = \int _{\Gamma ^0} \Big ( \nabla _{\Gamma ^0}u^0\cdot \nabla _{\Gamma ^0} v \Big ) \left\langle {V},{{\varvec{n}}^0}\right\rangle \,d\sigma . \end{aligned}$$

Hence,

$$\begin{aligned}&\int _{D^0} \nabla u'\cdot \nabla v \,d{\varvec{x}}+ \int _{\Gamma ^0} \Big ( \nabla _{\Gamma ^0}u^0\cdot \nabla _{\Gamma ^0} v \Big ) \left\langle {V},{{\varvec{n}}^0}\right\rangle \,d\sigma + \int _{\Gamma ^0} \frac{\partial g_D}{\partial {\varvec{n}}^0} v\left\langle {V},{{\varvec{n}}^0}\right\rangle \,d\sigma \nonumber \\&+ \int _{\Gamma ^0} {{\,\mathrm{{div }}\,}}_{\Gamma ^0}({\varvec{n}}^0) g_D\,v\left\langle {V},{{\varvec{n}}^0}\right\rangle \,d\sigma = - \int _{\Gamma ^0} fv\left\langle {V},{{\varvec{n}}^0}\right\rangle \,d\sigma \quad \forall v\in C_0^\infty ({\overline{D}}). \end{aligned}$$

(3.50)

Choosing $v\in C_0^\infty (D^0)$ so that $v= 0$ and $\nabla v = 0$ on $\Gamma ^0$, we obtain

$$\begin{aligned} \int _{D^0}\nabla u'\cdot \nabla v = 0 \quad \forall v\in C_0^\infty (D^0). \end{aligned}$$

This implies

$$\begin{aligned} \Delta u' = 0 \quad \text {a.e. in } D^0. \end{aligned}$$

(3.51)

Choosing $v\in C^\infty ({\overline{D}})$ and noting (3.35b), we deduce

$$\begin{aligned} \int _{\Gamma ^\epsilon } u^\epsilon v\,d\sigma = \int _{\Gamma ^\epsilon } g_D v\,d\sigma . \end{aligned}$$

Applying Lemma 3.3 (iii) yields

$$\begin{aligned}&\int _{\Gamma ^0} u' v\,d\sigma + \int _{\Gamma ^0} \Big ( \frac{\partial (u^0 v)}{\partial {\varvec{n}}^0} + {{\,\mathrm{{div }}\,}}_{\Gamma ^0} ({\varvec{n}}^0) u^0 v \Big ) \left\langle {V},{{\varvec{n}}^0}\right\rangle d\sigma \nonumber \\&\quad = \int _{\Gamma ^0} \Big ( \frac{\partial (g_D v)}{\partial {\varvec{n}}^0} + {{\,\mathrm{{div }}\,}}_{\Gamma ^0} ({\varvec{n}}^0) g_D v \Big ) \left\langle {V},{{\varvec{n}}^0}\right\rangle d\sigma . \end{aligned}$$

(3.52)

Noting that $u^0 = g_D$ on $\Gamma ^0$, we have

$$\begin{aligned} \int _{\Gamma ^0} u' v\,d\sigma + \int _{\Gamma ^0} \frac{\partial u^0}{\partial {\varvec{n}}^0} v \left\langle {V},{{\varvec{n}}^0}\right\rangle \,d\sigma = \int _{\Gamma ^0} \frac{\partial g_D}{\partial {\varvec{n}}^0} v \left\langle {V},{{\varvec{n}}^0}\right\rangle \,d\sigma \quad \forall v\in C^\infty ({\overline{D}}). \end{aligned}$$

This implies that

$$\begin{aligned} u' = \left( - \frac{\partial u^0}{\partial {\varvec{n}}^0} + \frac{\partial g_D}{\partial {\varvec{n}}^0} \right) \left\langle {V},{{\varvec{n}}^0}\right\rangle \ \text {on} \ \Gamma ^0, \end{aligned}$$

completing the proof of this theorem. $\square$

Theorems 3.7 and 3.10 show the existence of shape derivatives of the solutions of the Neumann and Dirichlet problems on deterministic boundary surfaces $\Gamma ^\epsilon$, given by (3.1). Furthermore, they are solutions to homogeneous boundary value problems (3.27) and (3.46). In the next section, we consider shape derivatives of the solutions of these problems on random boundary surfaces $\Gamma ^\epsilon (\omega )$ given by (2.7).

4 Random Domains

In Sects. 3.2 and 3.3, we have defined shape derivatives of the solutions of the exterior Neumann and Dirichlet problems in which the quantity $\kappa ({\varvec{x}})$ does not contain uncertainty. Since the Laplacian equation (1.1) with the Dirichlet and Neumann conditions, (1.2) and (1.3), respectively, is posed on a domain with a random interface [see (2.7)], the shape derivatives also depend on $\omega$, and it is necessary to compute the statistical moments, such as the mean and the covariance fields, of the random solutions. The following theorem, which was proved in [9, Theorem 3.13], shows that the statistical moments of the solutions can be approximated by those of the corresponding shape derivatives.

Theorem 4.1

Let $u^\epsilon (\omega )$ be the solution of the Laplacian equation (1.1) in the exterior domain of the random surface $\Gamma ^\epsilon (\omega )$ given by (2.7), with either the Dirichlet or the Neumann boundary conditions (1.2) and (1.3), respectively. Let $u^0$ denote the corresponding solutions of the Dirichlet or Neumann problems with the reference boundary $\Gamma ^0$. Assume that the perturbation function $\kappa$ belongs to $L^k(\Omega , {C^1(\Gamma ^0)})$ for an integer k and $f\in H^1({\overline{D}})\cap {H_w^1(D^0)^*}$. Then, for any compact subset $K\subset \subset D^0$, the expectation and the k-th-order central moments of the solution $u^\epsilon (\omega )$ can be approximated, respectively, by

$$\begin{aligned} {{\mathbb {E}}[u^\epsilon ]} = u^0 + o(\epsilon ) \quad \text {in} \quad {H^1(K)} \end{aligned}$$

(4.1)

and

$$\begin{aligned} \begin{aligned} {{\mathcal {M}}^k[u^\epsilon - {\mathbb {E}}[u^\epsilon ]] = \epsilon ^k{\mathcal {M}}^k[u'] + o(\epsilon ^k) \quad \text {in} \quad H^1_{\mathrm{mix}}(K^k).} \end{aligned} \end{aligned}$$

(4.2)

Moreover,

$$\begin{aligned} \begin{aligned} {{\mathcal {M}}^k[u^\epsilon - u^0] = \epsilon ^k{\mathcal {M}}^k[u'] + o(\epsilon ^k) \quad \text {in} \quad H^1_{\mathrm{mix}}(K^k).} \end{aligned} \end{aligned}$$

(4.3)

Here, $H^1_{\mathrm{mix}}(K^k)$ is the tensor product $H^1(K)^{(k)}$.

We now briefly recall the boundary integral equation methods to find the shape derivative $u'$ of the solutions to the exterior Dirichlet and Neumann problems in random domains. For boundary integral equation methods to find shape derivative of the solutions to interior Dirichlet problems in random bounded domains, please refer to the work by Harbrecht et al. [15]. Note here that if $u^\epsilon (\omega )$ is the solution to the Dirichlet or Neumann problems in a random domain $D^\epsilon (\omega )$, the corresponding shape derivative $u'(\omega )$ is the solution to the following homogeneous Dirichlet or Neumann problems in the reference domain $D^0$:

For a.a. $\omega \in \Omega$, find $u'(\omega )\in H^1_w(D^0)$ satisfying

$$\begin{aligned} \triangle u'(\omega ) = 0 \quad \text {in } D^0, \end{aligned}$$

(4.4)

with either the Dirichlet boundary condition

$$\begin{aligned} u'(\omega ) = G_D(\omega ) \quad \text {on } \Gamma ^0, \end{aligned}$$

(4.5)

or the Neumann boundary condition

$$\begin{aligned} \dfrac{\partial u'(\omega )}{\partial {\varvec{n}}^0} = G_N(\omega ) \quad \text {on } \Gamma ^0, \end{aligned}$$

(4.6)

and the vanishing condition

$$\begin{aligned} \left| u'({\varvec{x}},\omega )\right| = {\mathcal {O}}\left( \left\| {{\varvec{x}}}\right\| _{}^{-1}\right) \quad \text {as } \left| {{\varvec{x}}}\right| _{}\rightarrow \infty . \end{aligned}$$

(4.7)

Note here that the boundary conditions $G_D(\omega )$ and $G_N(\omega )$ are given by

$$\begin{aligned} G_D(\omega )&= \left( - \frac{\partial u^0}{\partial {\varvec{n}}^0} + \frac{\partial g_D}{\partial {\varvec{n}}^0} \right) \kappa (\omega ), \end{aligned}$$

(4.8)

$$\begin{aligned} G_N (\omega )&= \nabla _{\Gamma ^0}\cdot \Big ( \kappa (\omega ) \nabla _{\Gamma ^0}u^0 \Big ) + \frac{\partial g_N}{\partial {\varvec{n}}^0} \kappa (\omega ) + {{\,\mathrm{{div }}\,}}_{\Gamma ^0}({\varvec{n}}^0) g_N\kappa (\omega ) + f\kappa (\omega ); \end{aligned}$$

(4.9)

see (3.47) and (3.28).

Recall the definition of the single- and double-layer potentials:

$$\begin{aligned} {{\mathscr {V}}w({\varvec{x}})}= & {} {\frac{1}{4\pi }} \int _{\Gamma ^0} \frac{1}{\left| {{\varvec{x}}-{\varvec{y}}}\right| _{}}\, w({\varvec{y}}) \,d\sigma _{{\varvec{y}}} \quad \text {and}\quad \\ {{\mathscr {W}}v({\varvec{x}})}= & {} {\frac{1}{4\pi }} \int _{\Gamma ^0} \frac{\partial }{\partial \varvec{n}_{{\varvec{y}}}^0} \frac{1}{\left| {{\varvec{x}}-{\varvec{y}}}\right| _{}}\, v({\varvec{y}}) \,d\sigma _{{\varvec{y}}}, \quad {\varvec{x}}\in D^0, \end{aligned}$$

for $w \in H^{-1/2}(\Gamma ^0)$ and $v \in H^{1/2}(\Gamma ^0)$. It is well known that (see, e.g., [16]) the solution $u'_D$ of (4.4), (4.5) and (4.7) and the solution $u_N'$ of (4.4), (4.6) and (4.7) can be represented, respectively, as

$$\begin{aligned} u_D'(\cdot ,\omega ) = {\mathscr {V}}\phi _D(\cdot ,\omega ) \quad \text {and}\quad u_N'(\cdot ,\omega ) = {\mathscr {W}}\phi _N(\cdot ,\omega ) \quad \text {in } D^0, \end{aligned}$$

(4.10)

where $\phi _D \in H^{-1/2}(\Gamma ^0)$ and $\phi _N \in H^{1/2}(\Gamma ^0)/{{\mathbb {R}}}$ are, respectively, unique solutions of

$$\begin{aligned} {\mathcal {V}}\phi _D(\cdot ,\omega ) = G_D(\cdot ,\omega ) \quad \text {and}\quad \mathcal{D}\phi _N(\cdot ,\omega ) = G_N(\cdot ,\omega ) \quad \text {on } \Gamma ^0. \end{aligned}$$

(4.11)

Here (see, e.g., [16, Sections 1.2, 1.4])

$$\begin{aligned} {\mathcal {V}}w({\varvec{x}})&:= \frac{1}{4\pi } \int _{\Gamma ^0{\setminus } \{{\varvec{x}}\}} \frac{1}{|{\varvec{x}}- {\varvec{y}}|} w({\varvec{y}}) ds_{{\varvec{y}}}&\quad \text {for } {\varvec{x}}\in \Gamma ^0, \text { for all } w\in H^{-1/2}(\Gamma ^0), \\ \mathcal{D}v({\varvec{x}})&:= -(\varvec{n}_{{\varvec{x}}}^0 \times \nabla _{{\varvec{x}}}) {\mathcal {V}}( \varvec{n}_{{\varvec{y}}}^0 \times \nabla _{{\varvec{y}}} v)({\varvec{x}})&\quad \text {for } {\varvec{x}}\in \Gamma ^0, \text { for all } v\in H^{1/2}(\Gamma ^0), \end{aligned}$$

It follows from (4.10) and (4.11) that

$$\begin{aligned} u_D'(\cdot ,\omega ) = {\mathscr {V}}{\mathcal {V}}^{-1} G_D(\cdot ,\omega ) \quad \text {and}\quad u_N'(\cdot ,\omega ) = {\mathscr {W}}\mathcal{D}^{-1} G_N(\cdot ,\omega ) \quad \text {on } \Gamma ^0. \end{aligned}$$

Tensorizing and integrating the above equations over $\omega \in \Omega$, we obtain

$$\begin{aligned} {\mathcal {M}}^k[u_D']({\varvec{x}}^1,{\varvec{x}}^2,\ldots ,{\varvec{x}}^k)&= {\mathscr {V}}^{(k)} \left[ {\mathcal {V}}^{-1}\right] ^{(k)} {\mathcal {M}}^k [G_D] ({\varvec{x}}^1,{\varvec{x}}^2,\ldots ,{\varvec{x}}^k) \\ {\mathcal {M}}^k[u_N']({\varvec{x}}^1,{\varvec{x}}^2,\ldots ,{\varvec{x}}^k)&= {\mathscr {W}}^{(k)} \left[ \mathcal{D}^{-1}\right] ^{(k)} {\mathcal {M}}^k [G_N] ({\varvec{x}}^1,{\varvec{x}}^2,\ldots ,{\varvec{x}}^k), \end{aligned}$$

for any $({\varvec{x}}^1,{\varvec{x}}^2,\ldots , {\varvec{x}}^k)\in (D^0)^k$. This together with Theorem 4.1 yields the following result:

Corollary 4.2

Under the assumptions of Theorem 4.1, for any compact subset $K\subset \subset D^0$ there holds

$$\begin{aligned} {\mathcal {M}}^k[u^\epsilon - {\mathbb {E}}[u^\epsilon ]] = {\left\{ \begin{array}{ll} \epsilon ^k {\mathscr {V}}^{(k)} \left[ {\mathcal {V}}^{-1}\right] ^{(k)} {\mathcal {M}}^k[G_D] + o(\epsilon ^k) \quad &{} \text {in} \quad H^1_{\mathrm{mix}}(K^k) \quad \text {(Dirichlet case)} \\ \epsilon ^k {\mathscr {W}}^{(k)} \left[ \mathcal{D}^{-1}\right] ^{(k)} {\mathcal {M}}^k[G_N] + o(\epsilon ^k) \quad &{} \text {in} \quad H^1_{\mathrm{mix}}(K^k) \quad \text {(Neumann case)} \end{array}\right. } \end{aligned}$$

and

$$\begin{aligned} {\mathcal {M}}^k[u^\epsilon - u^0] = {\left\{ \begin{array}{ll} \epsilon ^k {\mathscr {V}}^{(k)} \left[ {\mathcal {V}}^{-1}\right] ^{(k)} {\mathcal {M}}^k[G_D] + o(\epsilon ^k) \quad &{} \text {in} \quad H^1_{\mathrm{mix}}(K^k) \quad \text {(Dirichlet case)} \\ \epsilon ^k {\mathscr {W}}^{(k)} \left[ \mathcal{D}^{-1}\right] ^{(k)} {\mathcal {M}}^k[G_N] + o(\epsilon ^k) \quad &{} \text {in} \quad H^1_{\mathrm{mix}}(K^k) \quad \text {(Neumann case)}. \end{array}\right. } \end{aligned}$$

5 Examples

For illustrative purposes, we consider in this section the boundary value problems (1.1) and (1.4) with the Dirichlet and Neumann boundary conditions (1.2) and (1.3). The reference surface $\Gamma ^0$ is the unit sphere ${\mathbb {S}}$ and the random surfaces under consideration are the spheres $\Gamma ^\epsilon (\omega )$ which are given by $\Gamma ^\epsilon (\omega ) = \{{\varvec{x}}+\epsilon \alpha (\omega ){\varvec{n}}^0({\varvec{x}}) : {\varvec{x}}\in {\mathbb {S}}\}$. This means $\kappa ({\varvec{x}},\omega ) = \alpha (\omega )$ for all ${\varvec{x}}\in \Gamma ^0 = {\mathbb {S}}$, where the random parameter $\alpha (\omega )$ is the truncated Laplace distribution whose probability distribution function is given by

$$\begin{aligned} \rho (t) = \frac{e}{2(e-1)} e^{-\left| t\right| }, \quad t\in [-1,1]. \end{aligned}$$

(5.1)

Note here that $\alpha (\omega )$ is centered, i.e.,

$$\begin{aligned} {\mathbb {E}}[\kappa ]= {\mathbb {E}}[\alpha ] = 0. \end{aligned}$$

(5.2)

5.1 Dirichlet Problem

We first consider the Dirichlet problem

$$\begin{aligned} \begin{aligned} \Delta u^\epsilon ({\varvec{x}},\omega )&= 0, \quad {\varvec{x}}\in D^\epsilon (\omega ), \\ u^\epsilon ({\varvec{x}},\omega )&= \frac{x_1+x_2+x_3}{\sqrt{x_1^2+x_2^2+x_3^2}} \quad \text {on} \ \Gamma ^\epsilon (\omega ), \\ \left| u^\epsilon ({\varvec{x}},\omega )\right|&= {\mathcal {O}}\left( \left| {\varvec{x}}\right| ^{-1}\right) \quad \text {as} \ \left| {\varvec{x}}\right| \rightarrow +\infty . \end{aligned} \end{aligned}$$

(5.3)

The solution of the Dirichlet problem is given by

$$\begin{aligned} u^\epsilon ({\varvec{x}},\omega ) = \frac{(x_1+x_2+x_3)(1+\epsilon \alpha (\omega ))^2}{(x_1^2+x_2^2+x_3^2)^{\frac{3}{2}}}, \quad {\varvec{x}}\in D^\epsilon (\omega ). \end{aligned}$$

(5.4)

In particular, the solution of the Dirichlet problem on the reference surface $\Gamma ^0$ is given by

$$\begin{aligned} u^0({\varvec{x}}) = \frac{x_1+x_2+x_3}{(x_1^2+x_2^2+x_3^2)^{\frac{3}{2}}}, \quad {\varvec{x}}\in D^0. \end{aligned}$$

(5.5)

Integrating both sides of (5.4) with respect to $\omega \in \Omega$, noting (5.5) and (5.1), we derive

$$\begin{aligned} {\mathbb {E}}[u^\epsilon ({\varvec{x}},\cdot )]&= \int _{\omega \in \Omega } \frac{(x_1+x_2+x_3)(1+\epsilon \alpha (\omega ))^2}{(x_1^2+x_2^2+x_3^2)^{\frac{3}{2}}} \,d{\mathbb {P}}(\omega ) \nonumber \\&= \frac{e}{2(e-1)} \int _{-1}^1 \frac{(x_1+x_2+x_3)(1+\epsilon t)^2}{(x_1^2+x_2^2+x_3^2)^{\frac{3}{2}}} e^{-\left| t\right| }\,dt \nonumber \\&= u^0({\varvec{x}}) + \frac{(2e -5)u^0({\varvec{x}})}{e-1}\epsilon ^2, \quad {\varvec{x}}\in D^0. \end{aligned}$$

(5.6)

This agrees with our result (4.1) in Theorem 4.1 where the error is of order ${\mathcal {O}}(\epsilon ^2)$.

From (5.4) and (5.6), we deduce

$$\begin{aligned}&\mathrm{{Covar}}_{u^\epsilon }({\varvec{x}},{\varvec{y}}) \nonumber \\&\quad = \int _{\omega \in \Omega } \left( u^\epsilon ({\varvec{x}},\omega ) - {\mathbb {E}}[u^\epsilon ({\varvec{x}},\cdot )]\right) \left( u^\epsilon ({\varvec{y}},\omega ) - {\mathbb {E}}[u^\epsilon ({\varvec{y}},\cdot )]\right) \,d{\mathbb {P}}(\omega ) \nonumber \\&\quad = \frac{e}{2(e-1)} \int _{-1}^1 \frac{(x_1+x_2+x_3)(y_1+y_2+y_3)}{\sqrt{x_1^2+x_2^2+x_3^2}^3 \sqrt{y_1^2+y_2^2+y_3^2}^3} \left( (1+\epsilon t)^2 - 1 + \frac{(2e-5)\epsilon ^2}{e-1} \right) e^{-\left| t\right| }\,dt \nonumber \\&\quad = \epsilon ^2 \frac{(x_1+x_2+x_3)(y_1+y_2+y_3)}{\sqrt{ x_1^2+x_2^2+x_3^2}^3 \sqrt{y_1^2+y_2^2+y_3^2}^3} \left( \frac{8e - 20}{e-1} + \epsilon ^2 \frac{40-69e + 20e^2}{(e-1)^2} \right) . \end{aligned}$$

(5.7)

On the other hand, from (4.8), (5.3) and (5.5), we have $G_D({\varvec{x}},\omega ) = 2(x_1 + x_2 + x_3)\,\alpha (\omega )$ for ${\varvec{x}}\in \Gamma ^0$. By (4.5), the shape derivative $u'(\omega )$ of the solution to the random problem (5.3) is the solution to the following problem

$$\begin{aligned} \begin{aligned} \Delta u'(\cdot ,\omega )&= 0 \quad \text {in} \ D^0 \\ u'(\cdot ,\omega )&= 2(x_1 + x_2 + x_3)\,a(\omega ) \quad \text {on} \ \Gamma ^0 \\ \left| u'({\varvec{x}},\omega )\right|&= {\mathcal {O}}\left( \left| {\varvec{x}}\right| ^{-1}\right) \quad \text {as} \ \left| {\varvec{x}}\right| \rightarrow \infty . \end{aligned} \end{aligned}$$

It can be shown that

$$\begin{aligned} u'({\varvec{x}},\omega ) = \frac{2(x_1 + x_2 + x_3)\,\alpha (\omega )}{\sqrt{x_1^2+x_2^2+x_3^2}^3}, \quad {\varvec{x}}\in D^0. \end{aligned}$$

(5.8)

The correlation of $u'(\cdot ,\omega )$ is then

$$\begin{aligned} \mathrm{{Cor}}_{u'}({\varvec{x}},{\varvec{y}}) = \frac{(x_1+x_2+x_3)(y_1+y_2+y_3)}{\sqrt{ x_1^2+x_2^2+x_3^2}^3 \sqrt{y_1^2+y_2^2+y_3^2}^3} \frac{8e - 20}{e-1} \end{aligned}$$

This together with (5.7) gives

$$\begin{aligned} \mathrm{{Covar}}_{u^\epsilon }({\varvec{x}},{\varvec{y}}) = \epsilon ^2\mathrm{{Cor}}_{u'}({\varvec{x}},{\varvec{y}}) + {\mathcal {O}}(\epsilon ^4), \end{aligned}$$

which agrees with Theorem 4.1. Furthermore, for any integer $k\ge 2$, we deduce from (5.4), (5.6) and (5.8) that for any $({\varvec{x}}^1,{\varvec{x}}^2,\ldots ,{\varvec{x}}^k)\in (D^0)^k$,

$$\begin{aligned} {\mathcal {M}}^k[u^\epsilon - {\mathbb {E}}[u^\epsilon ]]&({\varvec{x}}^1,{\varvec{x}}^2,\ldots ,{\varvec{x}}^k) = \int _{\omega \in \Omega } \prod _{i=1}^k \left( u^\epsilon ({\varvec{x}}^i,\omega ) - {\mathbb {E}}[u^\epsilon ({\varvec{x}}^i,\cdot )] \right) \,d{\mathbb {P}}(\omega ) \nonumber \\&= \frac{e}{2(e-1)} \int _{-1}^1 \prod _{i=1}^k \left( 2\epsilon u^0({\varvec{x}}^i) t + \epsilon ^2 u^0({\varvec{x}}^i) \Big [t^2-\frac{2e-5}{e-1}\Big ] \right) e^{-\left| t\right| }\,dt \nonumber \\&= \frac{e}{2(e-1)} \epsilon ^k \prod _{i=1}^k u^0({\varvec{x}}^i) \int _{-1}^1 \left( 2t + \epsilon \Big [t^2-\frac{2e-5}{e-1}\Big ] \right) ^k e^{-\left| t\right| }\,dt \nonumber \\&= \epsilon ^k \prod _{i=1}^k u^0({\varvec{x}}^i) \frac{e}{2(e-1)} \int _{-1}^1 \sum _{i=0}^k \left( {k\atopwithdelims ()i} 2^i t^i + \epsilon ^{k-i} \Big [t^2-\frac{2e-5}{e-1}\Big ]^{k-i} \right) e^{-\left| t\right| }\,dt. \end{aligned}$$

(5.9)

Tensorizing and integrating both sides of (5.8) give

$$\begin{aligned} {\mathcal {M}}^k [u']({\varvec{x}}^1,{\varvec{x}}^2,\ldots ,{\varvec{x}}^k)&= 2^k \prod _{i=1}^k u^0({\varvec{x}}^i) \frac{e}{2(e-1)} \int _{-1}^1 t^k e^{-\left| t\right| }\,dt. \end{aligned}$$

(5.10)

It follows from (5.9) and (5.10) that

$$\begin{aligned} {\mathcal {M}}^k[u^\epsilon - {\mathbb {E}}[u^\epsilon ]]({\varvec{x}}_1,{\varvec{x}}_2,\ldots ,{\varvec{x}}_k) = {\left\{ \begin{array}{ll} \epsilon ^k {\mathcal {M}}^k[u'] + {\mathcal {O}}(\epsilon ^{k+1}) &{} \text {if }k \text { is odd} \\ \epsilon ^k {\mathcal {M}}^k[u'] + {\mathcal {O}}(\epsilon ^{k+2}) &{} \text {if }k \text { is even}. \end{array}\right. } \end{aligned}$$

This agrees with Theorem 4.1.

5.2 Neumann Problem

We then consider the Neumann problem

$$\begin{aligned} \begin{aligned} \Delta u^\epsilon (\cdot ,\omega )&= 0 \quad \text {in} \ D^\epsilon (\omega ), \\ \frac{\partial u^\epsilon (\cdot ,\omega )}{\partial {\varvec{n}}^\epsilon }&= 1 \quad \text {on} \ \Gamma ^\epsilon (\omega ), \\ \left| u^\epsilon ({\varvec{x}},\omega )\right|&= {\mathcal {O}}\left( \left| {\varvec{x}}\right| ^{-1}\right) \quad \text {as} \ \left| {\varvec{x}}\right| \rightarrow +\infty . \end{aligned} \end{aligned}$$

(5.11)

The solution to the problem (5.11) is given by

$$\begin{aligned} u^\epsilon ({\varvec{x}},\omega ) = -\frac{(1 + \epsilon \alpha (\omega ))^2}{\sqrt{x_1^2+x_2^2+x_3^2}}, \quad {\varvec{x}}\in D^\epsilon (\omega ). \end{aligned}$$

(5.12)

The solution to the problem with the reference surface is given by

$$\begin{aligned} u^0({\varvec{x}}) = -\frac{1}{\sqrt{x_1^2+x_2^2+x_3^2}}, \quad {\varvec{x}}\in D^0. \end{aligned}$$

(5.13)

Integrating both sides of (5.12), we obtain

$$\begin{aligned} {\mathbb {E}}[u^\epsilon ({\varvec{x}},\cdot )]&= - \int _{\omega \in \Omega } \frac{(1 + \epsilon \alpha (\omega ))^2}{\sqrt{x_1^2+x_2^2+x_3^2}} \,dP(\omega ) = -\frac{e}{2(e-1)} \int _{-1}^1 \frac{(1 + \epsilon t)^2}{\sqrt{x_1^2+x_2^2+x_3^2}} e^{-\left| t\right| }\,dt \nonumber \\&= -\frac{1}{\sqrt{x_1^2+x_2^2+x_3^2}} - \frac{\epsilon ^2}{3\sqrt{x_1^2+x_2^2+x_3^2}} \nonumber \\&= u^0({\varvec{x}}) - \frac{\epsilon ^2u^0({\varvec{x}})}{3}, \quad {\varvec{x}}\in D^0. \end{aligned}$$

(5.14)

This agrees with Theorem 4.1, and the error is of order ${\mathcal {O}}(\epsilon ^2)$. From (5.12) to (5.14), we derive

$$\begin{aligned} {\mathcal {M}}^k[&u^\epsilon - {\mathbb {E}}[u^\epsilon ]] ({\varvec{x}}_1,{\varvec{x}}_2,\ldots ,{\varvec{x}}_k) = \int _{\omega \in \Omega } \prod _{i=1}^k u^0({\varvec{x}}^i) \left( (1+ \epsilon a(\omega ))^2 - 1 + \frac{\epsilon ^2}{3} \right) \nonumber \\&= \epsilon ^k \prod _{i=1}^k u^0({\varvec{x}}^i) \frac{e}{2(e-1)} \int _{-1}^1 \left( 2t + \epsilon (\frac{1}{3}+t^2) \right) ^k e^{-\left| t\right| }\,dt \nonumber \\&= \epsilon ^k \prod _{i=1}^k u^0({\varvec{x}}^i) \frac{e}{2(e-1)} \int _{-1}^1 \left( \sum _{i=1}^k {k\atopwithdelims ()i} 2^i t^i\, \epsilon ^{k-i} \big ( \frac{1}{3} + t^2 \big )^{k-i} \right) e^{-\left| t\right| }\,dt \end{aligned}$$

(5.15)

for any $({\varvec{x}}^1,{\varvec{x}}^2,\ldots ,{\varvec{x}}^k)\in (D^0)^k$. Following the result in Theorem 3.7, the shape derivative $u'$ of the solution to the random Neumann problem (5.11) is the solution of the problem

$$\begin{aligned} \begin{aligned} -\Delta u'(\cdot ,\omega )&= 0 \quad \text {in} \ D^0 \\ \frac{\partial u'(\cdot ,\omega )}{\partial {\varvec{n}}^0}&= 2\alpha (\omega ) \quad \text {on} \ \Gamma ^0 \\ \left| u'({\varvec{x}},\omega )\right|&= {\mathcal {O}}\left( \left| {\varvec{x}}\right| ^{-1}\right) \quad \text {as} \ \left| {\varvec{x}}\right| \rightarrow +\infty . \end{aligned} \end{aligned}$$

(5.16)

The solution to (5.16) is given by

$$\begin{aligned} u'({\varvec{x}},\omega ) = 2\alpha (\omega ) u^0({\varvec{x}}), \quad {\varvec{x}}\in D^0. \end{aligned}$$

This implies

$$\begin{aligned} {\mathcal {M}}^k[u']({\varvec{x}}^1,{\varvec{x}}^2,\ldots ,{\varvec{x}}^k)&= 2^k \prod _{i=1}^k u^0({\varvec{x}}^i) \frac{e}{2(e-1)} \int _{-1}^1 t^k e^{-\left| t\right| }\,dt. \end{aligned}$$

(5.17)

Comparing (5.15) and (5.17), we conclude

$$\begin{aligned} {\mathcal {M}}^k[u^\epsilon -{\mathbb {E}}[u^\epsilon ]] = \epsilon ^k {\mathcal {M}}^k[u'] + {\mathcal {O}}(\epsilon ^{k+1}), \end{aligned}$$

which agrees with Theorem 4.1.

6 Numerical Experiments

In this section, we will find approximate solutions to the covariance of random solutions to the following Dirichlet problem

$$\begin{aligned} \begin{aligned} \Delta u^\epsilon ({\varvec{x}},\omega )&= 0, \quad {\varvec{x}}\in D^\epsilon (\omega ) \\ u^\epsilon ({\varvec{x}},\omega )&= -1, \quad {\varvec{x}}\in \Gamma ^\epsilon (\omega ) \\ \left| u^\epsilon ({\varvec{x}},\omega )\right|&= {\mathcal {O}}\left( \left| {\varvec{x}}\right| ^{-1}\right) \quad \text {as} \ \left| {\varvec{x}}\right| \rightarrow \infty , \end{aligned} \end{aligned}$$

(6.1)

where $\Gamma ^\epsilon (\omega ) = \{ {\varvec{x}}+ \epsilon \,\alpha (\omega )\,{\varvec{n}}^0({\varvec{x}}) : {\varvec{x}}\in {\mathbb {S}}\}$ and the uniform perturbation of the interface $\alpha (\omega )\sim \rho _1(t)=~1/2$. Approximation spaces which will be used are spaces of spherical splines, see, e.g., [1,2,3, 17]. The solution to (6.1) is

$$\begin{aligned} u^\epsilon ({\varvec{x}},\omega ) = - \frac{1+\epsilon \alpha (\omega )}{\sqrt{x_1^2 + x_2^2 + x_3^2}}, \quad {\varvec{x}}\in D^\epsilon (\omega ). \end{aligned}$$

In particular, the solution with respect to the reference domain $D^0$ is

$$\begin{aligned} u^0({\varvec{x}}) = - \frac{1}{\sqrt{x_1^2+x_2^2+x_3^2}}, \quad {\varvec{x}}\in D^0. \end{aligned}$$

The mean value of the solutions on the random perturbed domains is

$$\begin{aligned} {\mathbb {E}}[u^\epsilon ({\varvec{x}},\cdot )]&= - \frac{1}{2} \int _{-1}^1 \frac{1+\epsilon t}{ \sqrt{x_1^2+x_2^2+x_3^2} }\,dt = - \frac{1}{\sqrt{x_1^2+x_2^2+x_3^2}} = u^0({\varvec{x}}), \quad {\varvec{x}}\in D^0. \end{aligned}$$

The shape derivative of the solutions is the solution to the following problem

$$\begin{aligned} \begin{aligned} \Delta u'({\varvec{x}})&= 0, \quad {\varvec{x}}\in D^0, \\ u'({\varvec{x}})&= -\alpha (\omega ), \quad {\varvec{x}}\in \Gamma ^0, \\ \left| u'({\varvec{x}})\right|&= {\mathcal {O}}(\left| {\varvec{x}}\right| ^{-1}) \ \text {as} \ \left| {\varvec{x}}\right| \rightarrow \infty . \end{aligned} \end{aligned}$$

We shall find an approximation for the correlation $\mathrm{{Cor}}_{u'}$ of the shape derivative, which is given by

$$\begin{aligned} \mathrm{{Cor}}_{u_D'}({\varvec{x}},{\varvec{y}})&= \left( {\mathscr {V}}^{(2)} \mathrm{{Cor}}_{\phi _D} \right) ({\varvec{x}},{\varvec{y}}), \quad {\varvec{x}},{\varvec{y}}\in D^0, \end{aligned}$$

(6.2)

where $\mathrm{{Cor}}_{\phi _D}$ is the solution of the following equation

$$\begin{aligned} {\mathcal {V}}^{(2)} \mathrm{{Cor}}_{\phi _D} = \mathrm{{Cor}}_{G_D} \ \text {on} \ {\mathbb {S}}^2. \end{aligned}$$

(6.3)

Approximate solution to (6.3) can be found as follows: Find $\Phi _D^X\in \left( S_1^0(\Delta _X)\right) ^{(2)}\subset \left( H^{-1/2}({\mathbb {S}})\right) ^{(2)}$ satisfying

$$\begin{aligned} \left\langle \left\langle { {\mathcal {V}}^{(2)} \Phi _D^X},{\phi }\right\rangle \right\rangle&= \left\langle \left\langle { \mathrm{{Cor}}_{G_D} },{\phi }\right\rangle \right\rangle \quad \forall \phi \in \left( S_1^0(\Delta _X)\right) ^{(2)}. \end{aligned}$$

Here, $\left( S_1^0(\Delta _X)\right) ^{(2)}$ is the tensor product $S_1^0(\Delta _X)\otimes S_1^0(\Delta _X)$, where $S_1^0(\Delta _X)$ is the space of continuous spherical splines of degree 1 defined on a spherical triangulation $\Delta _X$, see [1,2,3, 17]. The inner product $\left\langle \left\langle {\cdot },{\cdot }\right\rangle \right\rangle$ is the $\left( H^{1/2}({\mathbb {S}})\right) ^{(2)}-\left( H^{-1/2}({\mathbb {S}})\right) ^{ (2)}$-duality pairing. Once $\Phi _D^X$ is found, an approximate value of $\mathrm{{Cor}}_{u_D'}$, denoted by $\mathrm{{Cor}}_{u_D'}^X$, is computed by using (6.2) as follows

$$\begin{aligned} \mathrm{{Cor}}_{u_D'}^X({\varvec{x}},{\varvec{y}})&= \left( {\mathscr {V}}^{(2)} \Phi _D^X \right) ({\varvec{x}},{\varvec{y}}). \end{aligned}$$

We compute $\mathrm{{Cor}}_{u_D'}^X({\varvec{x}},{\varvec{y}})$ and then evaluate the relative error

$$\begin{aligned} E({\varvec{x}},{\varvec{y}}) := \left| \left( \mathrm{{Covar}}_{u^\epsilon }({\varvec{x}},{\varvec{y}}) - \epsilon ^2\mathrm{{Cor}}_{u_D'}^X({\varvec{x}},{\varvec{y}})\right) \Big /\mathrm{{Covar}}_{u^\epsilon } ({\varvec{x}},{\varvec{y}})\right| , \end{aligned}$$

(6.4)

where the sets X are of different sizes and ${\varvec{x}}$, ${\varvec{y}}$ are points outside the reference surface ${\mathbb {S}}$. In Tables 1 and 2, the relative errors $E({\varvec{x}},{\varvec{y}})$ (6.4) where ${\varvec{x}}= {\varvec{y}}= (1.5,0,0)$ and ${\varvec{x}}= (1.1,-0.2,0.2)$, ${\varvec{y}}= (1.1,-0.2,-0.2)$ are computed by using spherical splines of order 1. The results show that when the numbers of vertices which create the triangulation $\Delta _X$ increase, more accurate approximate solutions are obtained.

Table 1 Relative error $E({\varvec{x}},{\varvec{y}})$ of the covariance where ${\varvec{x}}={\varvec{y}}= (1.5,0,0)$

Table 2 Relative error $E({\varvec{x}},{\varvec{y}})$ of the covariance where ${\varvec{x}}= (1.1,-0.2,0.2)$, ${\varvec{y}}= (1.1,-0.2,-0.2)$