A second-order shape optimization algorithm for solving the exterior Bernoulli free boundary problem using a new boundary cost functional

Abstract

The exterior Bernoulli problem is rephrased into a shape optimization problem using a new type of objective function called the Dirichlet-data-gap cost function which measures the $L^2$-distance between the Dirichlet data of two state functions. The first-order shape derivative of the cost function is explicitly determined via the chain rule approach. Using the same technique, the second-order shape derivative of the cost function at the solution of the free boundary problem is also computed. The gradient and Hessian informations are then used to formulate an efficient second-order gradient-based descent algorithm to numerically solve the minimization problem. The feasibility of the proposed method is illustrated through various numerical examples.

Appendix 1: Shape derivative of the adjoint state \(p_{{\mathrm{N}}}\)

Let us first introduce some notations and present some properties of the operator $T_t$ (see Sect. 3) that will be useful to our analysis. For $t \in (0, \varepsilon )$ ($\varepsilon > 0$ sufficiently small), the transformation $T_t$ is invertible and $T_t, T^{-1}_t \in {\mathscr {D}}^1({\mathbb {R}}^2, {\mathbb {R}}^2)$ (see, e.g., [10, Lemma 11]). In addition, the Jacobian matrix of the transformation $T_t = T_t({\mathbf {V}})$ associated with the velocity field ${\mathbf {V}}$ denoted by $\det \,DT_t(X)$ is strictly positive. Here, we shall use the notations $(DT_t)^{-1}$ and $(DT_t)^{-\top }$ to denote the inverse and inverse transpose of the Jacobian matrix $DT_t$, respectively. Also, for convenience, we write ${A_t} = \det \,DT_t(X) (DT_t^{-1})(DT_t)^{-\top }$ and ${w_t} = \det \,DT_t(X) |(DT_t)^{-\top } {\mathbf {n}}|$ which represent the Jacobian matrix of $T_t$ with respect to the boundary $\partial \Omega$.

The following lemma, whose proof can be found in [22, 71], will also be essential to our analysis.

Lemma A.1

Let${\mathbf {V}}$be a fixed vector field in$\Theta$ (see (9)) and${I} = (-t_0, t_0)$, with$t_0 > 0$sufficiently small. Then, the following regularity properties of$T_t$hold

• (i) $t \mapsto \det \,DT_t(X) \in C^1({I}, C({\bar{\Omega }}))$

• (ii) $t \mapsto {A_t} \in C^1({I}, C^1({\bar{\Omega }}))$

• (iii) $t \mapsto {w_t} \in C^1({I}, C(\Sigma ))$

• (iv) $\lim _{t\searrow 0} {w_t} = 1$

• (v) $\dfrac{{\mathrm{d}}}{{\mathrm{d}}t}{w_t}|_{t=0}=w'(0)={\mathrm{div}}_\Sigma {\mathbf {V}}$

• (vi) $\dfrac{{\mathrm{d}}}{{\mathrm{d}}t}{A_t}|_{t=0}=A'(0)$,

where$A'(0)=({\mathrm{div}}\, {\mathbf {V}}){\mathbf {I}}_2 - (D {\mathbf {V}}+(D {\mathbf {V}})^{\top })$and the limits defining the derivatives at$t = 0$exist uniformly in$x \in {\bar{\Omega }}$.

Before we derive the shape derivative of $p_{{\mathrm{N}}}$, and for completeness, let us first prove the unique solvability of the adjoint problem on the perturbed domain $\Omega _t$.

Lemma A.2

For any$t>0$sufficiently small, the variational problem: find$p_{{\mathrm{N}}}^t \in H^1(\Omega )$such that$p_{{\mathrm{N}}}^t = 0$on$\Gamma$and

$$\begin{aligned} \int _{\Omega }{{A_t} \nabla p_{{\mathrm{N}}}^t \cdot \nabla \varphi }{\, \mathrm{d} x} - \int _{\Sigma }{{w_t}u_{\mathrm{N}}^t \varphi }{\, \mathrm{d} \sigma } = 0, \quad \forall \varphi \in H^1_{\Gamma ,0}(\Omega ). \end{aligned}$$

(66)

admits a unique solution$p_{{\mathrm{N}}}^t$in$H^1(\Omega )$.

Proof

We first note that the variational problem being examined is obtained from the problem: find $p_{\mathrm{N}t}\in H^1(\Omega _t)$ such that $p_{\mathrm{N}t}= 0$ on $\Gamma$ and

$$\begin{aligned} \int _{\Omega _t}{\nabla p_{\mathrm{N}t}\cdot \nabla \varphi }{\, \mathrm{d} x_t} - \int _{\Sigma _t}{u_{\mathrm{N}t}\varphi }{\, \mathrm{d} \sigma _t} = 0, \quad \forall \varphi \in H^1_{\Gamma ,0}(\Omega _t), \end{aligned}$$

(67)

via the application of domain and boundary transformation formulas (see, e.g., [71, Proposition 2.46–2.47]). In fact, the functions $\phi _t:\Omega _t \rightarrow {\mathbb {R}}$ and $\phi ^t: \Omega \rightarrow {\mathbb {R}}$ are related through the equation $\phi ^t = \phi _t \circ T_t$. Hence, if $p_{\mathrm{N}t}$ solves the variational equation (67), then $p_{{\mathrm{N}}}^t = p_{\mathrm{N}t}\circ T_t$ satisfies (66). In addition, the boundary condition $p_{{\mathrm{N}}}^t = p_{\mathrm{N}t}\circ T_t = 0$ on $\Gamma$ implies that $p_{{\mathrm{N}}}^t$ is actually in $H^1_{\Gamma ,0}(\Omega )$.

Now, consider the bilinear form $b_t(\cdot ,\cdot ): {\mathbf {H}}^1_{\Gamma ,0}(\Omega ) \rightarrow {\mathbb {R}}$ defined by

$$\begin{aligned} b_t(\phi ^t, \varphi ) = \int _{\Omega }{{A_t} \nabla \phi ^t\cdot \nabla \varphi }{\, \mathrm{d} x}, \quad \forall \phi ^t, \varphi \in H^1_{\Gamma ,0}(\Omega ). \end{aligned}$$

(68)

Note that, as a consequence of Lemma A.1, ${A_t}$ is bounded. Hence, it is clear that $b_t(\cdot ,\cdot )$ is continuous because $|b_t(\phi ^t, \varphi )| = \left| \int _{\Omega }{{A_t} \nabla \phi ^t\cdot \nabla \varphi }{\, \mathrm{d} x} \right| \lesssim \Vert {A_t}\Vert _{L^{\infty }(\Omega )} \Vert \phi ^t\Vert _{H^1(\Omega )} |\varphi |_{H^1(\Omega )}.$ Moreover, $b_t(\cdot ,\cdot )$ is coercive. Indeed, from the fact that ${A_t} \rightarrow {\mathbf {I}}$ uniformly on ${\bar{\Omega }}$ as $t \rightarrow 0$, we know that there exist sufficiently small values for $t > 0$ such that $\Vert {A_t} - {\mathbf {I}}\Vert _{L^{\infty }(\Omega )} < 1$. So, we have

$$\begin{aligned} b_t(\phi ^t, \phi ^t)&= \int _{\Omega }{{A_t} \nabla \phi ^t\cdot \nabla \phi ^t}{\, \mathrm{d} x} = \left| \int _{\Omega }{ ( {A_t} - {\mathbf {I}}) \nabla \phi ^t\cdot \nabla \phi ^t + | \nabla \phi ^t |^2}{\, \mathrm{d} x} \right| \\&\geqslant \Vert \nabla \phi ^t \Vert ^2_{L^2(\Omega )} - \Vert {A_t} - {\mathbf {I}}\Vert _{L^{\infty }(\Omega )} \Vert \nabla \phi ^t \Vert ^2_{L^2(\Omega )} \\&\gtrsim |\nabla \phi ^t|^2_{H^1(\Omega )}. \end{aligned}$$

Next, we consider the functional $\omega :H^1_{\Gamma ,0}(\Omega ) \rightarrow {\mathbb {R}}$ defined by $\langle \omega , \varphi \rangle = \int _{\Sigma }{{w_t}u_{\mathrm{N}}^t\varphi }{\, \mathrm{d} \sigma }$. Evidently, this functional is continuous because of the boundedness of $|{w_t}|_{\infty }$ and due to the sequence of inequalities

$$\begin{aligned} \left| \int _{\Sigma }{{w_t}u_{\mathrm{N}}^t\varphi }{\, \mathrm{d} \sigma } \right| \lesssim |{w_t}|_{\infty } \Vert u_{\mathrm{N}}^t\Vert _{L^2(\Sigma )} \Vert \varphi \Vert _{L^2(\Sigma )} \lesssim |{w_t}|_{\infty } \Vert u_{\mathrm{N}}^t\Vert _{H^1(\Omega )} |\varphi |_{H^1(\Omega )}. \end{aligned}$$

Thus, by Lax–Milgram theorem, the function $p_{{\mathrm{N}}}^t$, vanishing on $\Gamma$, is the unique solution to the variational equation (66) in $H^1(\Omega )$. This proves the lemma. $\square$

Proposition A.1

Let$\Omega$be a bounded$C^{2,1}$domain. The shape derivative of the adjoint state variable$p_{{\mathrm{N}}}\in H^3(\Omega )$at$\Omega = \Omega ^*$satisfying the mixed Dirichlet–Neumann problem (16) is a solution to the following mixed boundary value problem:

$$\begin{aligned} -\Delta p_{ {\mathrm{N}} W}' = 0 \ {\mathrm{in}} \ \Omega ^*,\quad p_{ \mathrm{N} W}' = 0 \ {\mathrm{on}} \ \Gamma ,\quad \partial _{{\mathbf {n}}}{p_{ {\mathrm{N}} W}'} = u_{ {\mathrm{N}} W}' + \lambda {\mathbf {W}}\cdot {\mathbf {n}}\quad {\mathrm{on}} \ \Sigma ^*. \end{aligned}$$

Proof

The proof mainly contains two parts; we first prove the existence of the material derivative of $p_{{\mathrm{N}}}$, then we formally proceed on the derivation of its shape derivative.

Step 1. Existence of the material derivative of $p_{{\mathrm{N}}}$. The variational formulation of (16) on the reference domain $\Omega$ is given as follows: find $p_{{\mathrm{N}}}\in H^1_{\Gamma ,0}(\Omega )$ such that

$$\begin{aligned} \int _{\Omega }{\nabla p_{{\mathrm{N}}}\cdot \nabla \varphi }{\, \mathrm{d} x} - \int _{\Sigma }{u_{\mathrm{N}}\varphi }{\, \mathrm{d} \sigma } = 0, \quad \forall \varphi \in H^1_{\Gamma ,0}(\Omega ). \end{aligned}$$

(69)

Subtracting (66) with $t = 0$ from the case where $t > 0$, for all $\varphi \in H^1_{\Gamma ,0}(\Omega )$, we obtain

$$\begin{aligned} \int _{\Omega }{\{ {A_t} \nabla p_{{\mathrm{N}}}^t - \nabla p_{{\mathrm{N}}}^t + \nabla p_{{\mathrm{N}}}^t - \nabla p_{{\mathrm{N}}}\} \cdot \nabla \varphi }{\, \mathrm{d} x} - \int _{\Sigma }{ \{{w_t}u_{\mathrm{N}}^t - u_{\mathrm{N}}^t + u_{\mathrm{N}}^t - u_{\mathrm{N}}\} \varphi }{\, \mathrm{d} \sigma } = 0. \end{aligned}$$

Hence, we have a unique solution $p_{{\mathrm{N}}}^t - p_{{\mathrm{N}}}\in H^1_{\Gamma ,0}(\Omega )$ to the variational equation

$$\begin{aligned}&\int _{\Omega }{ \nabla (p_{{\mathrm{N}}}^t - p_{{\mathrm{N}}}) \cdot \nabla \varphi }{\, \mathrm{d} x} = - \int _{\Omega }{ ( {A_t} - {\mathbf {I}}) \nabla p_{{\mathrm{N}}}^t \cdot \nabla \varphi }{\, \mathrm{d} x} \nonumber \\&\quad + \int _{\Sigma }{({w_t} - 1)u_{\mathrm{N}}^t\varphi }{\, \mathrm{d} \sigma } + \int _{\Sigma }{(u_{\mathrm{N}}^t - u_{\mathrm{N}}) \varphi }{\, \mathrm{d} \sigma }, \end{aligned}$$

(70)

for all $\varphi \in H^1_{\Gamma ,0}(\Omega )$. We note that $\nabla p_{{\mathrm{N}}}^t$ is uniformly bounded in $L^2(\Omega ;{\mathbb {R}}^2)$ and we have the convergence $\nabla p_{{\mathrm{N}}}^t \rightarrow \nabla p_{{\mathrm{N}}}$ also in that space. Indeed, using the boundedness of $\Vert {A_t}\Vert _{L^{\infty }(\Omega )}$ from below, we get the estimate

$$\begin{aligned} \Vert \nabla p_{{\mathrm{N}}}^t\Vert _{L^2(\Omega )}^2 \lesssim \int _{\Omega }{{A_t} \nabla p_{{\mathrm{N}}}^t\cdot \nabla p_{{\mathrm{N}}}^t}{\, \mathrm{d} x} = \int _{\Sigma }{{w_t}u_{\mathrm{N}}^t p_{{\mathrm{N}}}^t}{\, \mathrm{d} \sigma } \lesssim |{w_t}|_{\infty } \Vert u_{\mathrm{N}}^t\Vert _{H^1(\Omega )} \Vert p_{{\mathrm{N}}}^t\Vert _{L^2(\Omega )}. \end{aligned}$$

Because $u_{\mathrm{N}}^t$ is uniformly bounded in $H^1(\Omega )$ (cf. [10, Theorem 23], see also [50, Proposition 3.1]), the uniform boundedness of $\nabla p_{{\mathrm{N}}}^t$ in $L^2(\Omega ;{\mathbb {R}}^2)$ immediately follows, and so the convergence $\nabla p_{{\mathrm{N}}}^t \rightarrow \nabla p_{{\mathrm{N}}}$ in $L^2(\Omega ;{\mathbb {R}}^2)$. Next, we divide both sides of (70) by t and denote $\phi ^t := \frac{1}{t} (p_{{\mathrm{N}}}^t - p_{{\mathrm{N}}})$ to obtain

$$\begin{aligned}&\int _{\Omega }{ \nabla \phi ^t \cdot \nabla \varphi }{\, \mathrm{d} x} = - \int _{\Omega }{ \left( \frac{{A_t} - {\mathbf {I}}}{t} \right) \nabla p_{{\mathrm{N}}}^t \cdot \nabla \varphi }{\, \mathrm{d} x} + \int _{\Sigma }{\left( \frac{ {w_t} - 1 }{t} \right) u_{\mathrm{N}}^t\varphi }{\, \mathrm{d} \sigma } \\&\quad + \int _{\Sigma }{\left( \frac{u_{\mathrm{N}}^t - u_{\mathrm{N}}}{t}\right) \varphi }{\, \mathrm{d} \sigma }, \end{aligned}$$

for all $\varphi \in H^1_{\Gamma ,0}(\Omega )$. We choose a sequence $\{t_n\}$ such that $t_n \rightarrow 0$ as $n \rightarrow \infty$. Our goal is to show that the limit $\lim _{n \rightarrow \infty } \phi ^t$ exists. Using the boundedness of $\frac{1}{t_n} \left( {A_t} - {\mathbf {I}} \right)$ and $\frac{1}{t_n} \left( {w_t} - 1\right)$ in $L^{\infty }$, we deduce that $\nabla p_{{\mathrm{N}}}^{t_n}$ is bounded in $L^2(\Omega ;{\mathbb {R}}^2)$, and thus the boundedness of $\phi ^{t_n}$ in $H^1_{\Gamma ,0}(\Omega )$. Hence, we can extract a subsequence, which we still denote by $\{t_n\}$, such that $\lim _{n \rightarrow \infty } t_n = 0$. Moreover, there exists an element $\phi$ of $H^1_{\Gamma ,0}(\Omega )$ such that $\phi ^{t_n} \rightharpoonup \phi$ weakly in $H^1_{\Gamma ,0}(\Omega )$. From the convergences $\nabla p_{{\mathrm{N}}}^{t_n} \rightarrow \nabla p_{{\mathrm{N}}}$ in $L^2(\Omega ;{\mathbb {R}}^2)$ and $u_{\mathrm{N}}^{t_n} \rightarrow u_{\mathrm{N}}$ in $L^2(\Sigma )$, together with Lemma A.1(v)–(vi), we get

$$\begin{aligned} \int _{\Omega }{ \nabla \phi \cdot \nabla \varphi }{\, \mathrm{d} x} = - \int _{\Omega }{ A \nabla p_{{\mathrm{N}}}\cdot \nabla \varphi }{\, \mathrm{d} x} + \int _{\Sigma }{ u_{\mathrm{N}}\varphi {\mathrm{div}}_{\Sigma } {\mathbf {W}}}{\, \mathrm{d} \sigma } + \int _{\Sigma }{ {\dot{u}}_{\mathrm{N}}\varphi }{\, \mathrm{d} \sigma }, \end{aligned}$$

for all $\varphi \in H^1_{\Gamma ,0}(\Omega )$, where ${\dot{u}}_{\mathrm{N}}= \lim _{t\searrow 0} \frac{1}{t} (u_{\mathrm{N}}^t - u_{\mathrm{N}})$ which is exactly the material derivative of $u_{\mathrm{N}}$ at $t = 0$ in the direction ${\mathbf {W}}$. This function exists and is actually an element of $H^1_{\Gamma ,0}(\Omega )$ as shown, for example, in [9]. Hence, the above equation admits a unique solution in $H^1(\Omega )$ and we deduce that $\phi ^{t_n} \rightharpoonup \phi$ for any sequence $\{t_n\}$. This implies the strong convergence of $\phi ^{t_n}$ to $\phi$ in $L^2(\Sigma )$. Now, taking $\varphi = \phi ^{t_n} \in H^1_{\Gamma ,0}(\Omega )$, we obtain

$$\begin{aligned} \lim _{t_n \rightarrow 0} |\phi ^{t_n}|_{H^1(\Omega )}^2&= - \lim _{t_n \rightarrow 0} \left\{ \int _{\Omega }{ \left( \frac{A({t_n}) - {\mathbf {I}}}{{t_n}} \right) \nabla p_{{\mathrm{N}}}^{t_n} \cdot \nabla \phi ^{t_n}}{\, \mathrm{d} x} \right\} \\&\quad + \lim _{t_n \rightarrow 0} \left\{ \int _{\Sigma }{\left( \frac{ w({t_n}) - 1 }{{t_n}} \right) u_{\mathrm{N}}^{t_n} \phi ^{t_n}}{\, \mathrm{d} \sigma } \right\} \\&\quad + \lim _{t_n \rightarrow 0} \left\{ \int _{\Sigma }{\left( \frac{u_{\mathrm{N}}^{t_n} - u_{\mathrm{N}}}{{t_n}}\right) \phi ^{t_n}}{\, \mathrm{d} \sigma } \right\} \\&= - \int _{\Omega }{ A \nabla p_{{\mathrm{N}}}\cdot \nabla \phi }{\, \mathrm{d} x} + \int _{\Sigma }{ u_{\mathrm{N}}\phi \mathrm{div}_{\Sigma } {\mathbf {W}}}{\, \mathrm{d} \sigma } + \int _{\Sigma }{ {\dot{u}}_{\mathrm{N}}\phi }{\, \mathrm{d} \sigma } = |\phi |_{H^1(\Omega )}. \end{aligned}$$

The norm convergence and the weak convergence of $\phi ^{t_n}$ in $H^1_{\Gamma ,0}(\Omega )$ implies the strong convergence of $\phi ^{t_n}$ to $\phi \in H^1_{\Gamma ,0}(\Omega )$. This guarantees the existence of the material derivative of $p_{{\mathrm{N}}}$.

Step 2. Computing the shape derivative of $p_{{\mathrm{N}}}$ at $\Omega = \Omega ^*$along the deformation field ${\mathbf {W}}$. From the previous step, we showed the existence of the material derivative of $p_{{\mathrm{N}}}$ in $H^1_{\Gamma ,0}(\Omega )$. Denoting this derivative by ${\dot{p}}_{\mathrm{N}}$, we know that it satisfies the variational equation

$$\begin{aligned}&\int _{\Omega }{ \nabla {\dot{p}}_{\mathrm{N}}\cdot \nabla \varphi }{\, \mathrm{d} x} = - \int _{\Omega }{ A \nabla p_{{\mathrm{N}}}\cdot \nabla \varphi }{\, \mathrm{d} x} + \int _{\Sigma }{ u_{\mathrm{N}}\varphi {\mathrm{div}}_{\Sigma } {\mathbf {W}}}{\, \mathrm{d} \sigma } \nonumber \\&\quad + \int _{\Sigma }{ {\dot{u}}_{\mathrm{N}}\varphi }{\, \mathrm{d} \sigma }, \quad \forall \varphi \in H^1_{\Gamma ,0}(\Omega ). \end{aligned}$$

(71)

In addition, it is clear that ${\dot{p}}_{\mathrm{N}}= 0$ on $\Gamma$. Applying Green’s formula to the above variational form, we get

$$\begin{aligned}&- \int _{\Omega }{ \varphi \Delta {\dot{p}}_{\mathrm{N}}}{\, \mathrm{d} x} + \int _{\Sigma }{\varphi \partial _{{\mathbf {n}}}{{\dot{p}}_{\mathrm{N}}}}{\, \mathrm{d} \sigma } = \int _{\Omega }{ \varphi {\mathrm{div}}(A \nabla p_{{\mathrm{N}}})}{\, \mathrm{d} x} - \int _{\Sigma }{ \varphi A \partial _{{\mathbf {n}}}{p_{{\mathrm{N}}}}}{\, \mathrm{d} \sigma }\\&\quad + \int _{\Sigma }{ u_{\mathrm{N}}\varphi {\mathrm{div}}_{\Sigma } {\mathbf {W}}}{\, \mathrm{d} \sigma } + \int _{\Sigma }{ {\dot{u}}_{\mathrm{N}}\varphi }{\, \mathrm{d} \sigma }, \quad \forall \varphi \in H^1_{\Gamma ,0}(\Omega ). \end{aligned}$$

First, let us choose $\varphi \in H^1_0(\Omega )$. Then, we have $- \int _{\Omega }{ \varphi \Delta {\dot{p}}_{\mathrm{N}}}{\, \mathrm{d} x} = \int _{\Omega }{ \varphi {\mathrm{div}}(A \nabla p_{{\mathrm{N}}})}{\, \mathrm{d} x}$. Since, $H^1_0(\Omega )$ is dense in $L^2(\Omega )$, we obtain $- \Delta {\dot{p}}_{\mathrm{N}}= {\mathrm{div}}(A \nabla p_{{\mathrm{N}}})$ in $\Omega$. Next, we choose $\varphi \in H^1_{\Gamma ,0}(\Omega )$ such that $\varphi$ is arbitrary in $\Sigma$. This gives us

$$\begin{aligned} \int _{\Sigma }{\varphi \partial _{{\mathbf {n}}}{{\dot{p}}_{\mathrm{N}}}}{\, \mathrm{d} \sigma } = - \int _{\Sigma }{ \varphi A \partial _{{\mathbf {n}}}{p_{{\mathrm{N}}}}}{\, \mathrm{d} \sigma } + \int _{\Sigma }{ u_{\mathrm{N}}\varphi {\mathrm{div}}_{\Sigma } {\mathbf {W}}}{\, \mathrm{d} \sigma } + \int _{\Sigma }{ {\dot{u}}_{\mathrm{N}}\varphi }{\, \mathrm{d} \sigma }. \end{aligned}$$

Because the traces of functions in $H^1_{\Gamma ,0}(\Omega )$ are dense in $L^2(\Sigma )$, we arrive at $\partial _{{\mathbf {n}}}{{\dot{p}}_{\mathrm{N}}} = -A \partial _{{\mathbf {n}}}{p_{{\mathrm{N}}}} + u_{\mathrm{N}}{\mathrm{div}}_{\Sigma } {\mathbf {W}}+ {\dot{u}}_{\mathrm{N}}$ on $\Sigma$. Summarizing these results, we see that ${\dot{p}}_{\mathrm{N}}$ satisfies the following boundary value problem:

$$\begin{aligned} -\Delta {\dot{p}}_{\mathrm{N}}= {\mathrm{div}}(A \nabla p_{{\mathrm{N}}}) \ {\mathrm{in}} \ \Omega ,\quad {\dot{p}}_{\mathrm{N}}= 0 \ {\mathrm{on}} \ \Gamma ,\quad \partial _{{\mathbf {n}}}{{\dot{p}}_{\mathrm{N}}} = -A \partial _{{\mathbf {n}}}{p_{{\mathrm{N}}}} + u_{\mathrm{N}}{\mathrm{div}}_{\Sigma } {\mathbf {W}}+ {\dot{u}}_{\mathrm{N}}\quad {\mathrm{on}} \ \Sigma . \end{aligned}$$

From above equations, and due to the fact that ${\mathbf {W}}$ vanishes on $\Gamma$, we immediately obtain [in view of the identity (12)] $p_{{\mathrm{N}}}' = {\dot{p}}_{\mathrm{N}}- {\mathbf {W}}\cdot \nabla p_{{\mathrm{N}}}= 0$ on $\Gamma$. Now, we consider $\varphi \in H^2(\Omega )$. Note that for $C^{1,1}$ domain, we have that $u_{\mathrm{N}}\in H^2(\Omega )$ (see [10, Theorem 29] and also [50]). Hence, $u_{\mathrm{N}}\in H^{3/2}(\Sigma )$ which, in turn, means that $p_{{\mathrm{N}}}\in H^2(\Omega )$ by standard elliptic regularity theory. Given this regularity of $p_{{\mathrm{N}}}$ and since $-\Delta p_{{\mathrm{N}}}= 0$ in $\Omega$, we can therefore write $- \int _{\Omega }{ A \nabla p_{{\mathrm{N}}}\cdot \nabla \varphi }{\, \mathrm{d} x}$ as follows (see [50, Lemma 4.1])

$$\begin{aligned}&- \int _{\Omega }{ A \nabla p_{{\mathrm{N}}}\cdot \nabla \varphi }{\, \mathrm{d} x} = \int _{\Omega }{ \nabla ({\mathbf {W}}\cdot \nabla p_{{\mathrm{N}}}) \cdot \nabla \varphi }{\, \mathrm{d} x} + \int _{\Sigma }{ \partial _{{\mathbf {n}}}{p_{{\mathrm{N}}}} ({\mathbf {W}}\cdot \nabla \varphi ) }{\, \mathrm{d} \sigma } \nonumber \\&\quad - \int _{\Sigma }{ (\nabla p_{{\mathrm{N}}}\cdot \nabla \varphi ) {\mathbf {W}}\cdot {\mathbf {n}}}{\, \mathrm{d} \sigma }, \end{aligned}$$

(72)

for all $\varphi \in H^2(\Omega )$. Hence, using the identity (12), we have the equation

$$\begin{aligned} \int _{\Omega }{ \nabla {\dot{p}}_{\mathrm{N}}\cdot \nabla \varphi }{\, \mathrm{d} x} = \int _{\Omega }{ \nabla p_{{\mathrm{N}}}' \cdot \nabla \varphi }{\, \mathrm{d} x} + \int _{\Omega }{ \nabla ({\mathbf {W}}\cdot \nabla p_{{\mathrm{N}}}) \cdot \nabla \varphi }{\, \mathrm{d} x}, \quad \forall \varphi \in H^1_{\Gamma ,0}(\Omega ). \end{aligned}$$

Combining this equation with (71) and (72) yields

$$\begin{aligned}&\int _{\Omega }{ \nabla ({\mathbf {W}}\cdot \nabla p_{{\mathrm{N}}}) \cdot \nabla \varphi }{\, \mathrm{d} x} + \int _{\Sigma }{ \partial _{{\mathbf {n}}}{p_{{\mathrm{N}}}} ({\mathbf {W}}\cdot \nabla \varphi ) }{\, \mathrm{d} \sigma } - \int _{\Sigma }{ (\nabla p_{{\mathrm{N}}}\cdot \nabla \varphi ) {\mathbf {W}}\cdot {\mathbf {n}}}{\, \mathrm{d} \sigma }\\&\qquad + \int _{\Sigma }{ u_{\mathrm{N}}\varphi {\mathrm{div}}_{\Sigma } {\mathbf {W}}}{\, \mathrm{d} \sigma } + \int _{\Sigma }{ {\dot{u}}_{\mathrm{N}}\varphi }{\, \mathrm{d} \sigma }\\&\quad = \int _{\Omega }{ \nabla p_{{\mathrm{N}}}' \cdot \nabla \varphi }{\, \mathrm{d} x} + \int _{\Omega }{ \nabla ({\mathbf {W}}\cdot \nabla p_{{\mathrm{N}}}) \cdot \nabla \varphi }{\, \mathrm{d} x}, \quad \forall \varphi \in H^2 \cap H^1_{\Gamma ,0}(\Omega ). \end{aligned}$$

Applying Green’s formula on the right side of the above equation we arrive at

$$\begin{aligned}&- \int _{\Omega }{ \varphi \Delta p_{{\mathrm{N}}}' }{\, \mathrm{d} x} + \int _{\Sigma }{ \varphi \partial _{{\mathbf {n}}}{p_{{\mathrm{N}}}'} }{\, \mathrm{d} \sigma } = \int _{\Sigma }{ \partial _{{\mathbf {n}}}{p_{{\mathrm{N}}}} ({\mathbf {W}}\cdot \nabla \varphi ) }{\, \mathrm{d} \sigma } - \int _{\Sigma }{ (\nabla p_{{\mathrm{N}}}\cdot \nabla \varphi ) {\mathbf {W}}\cdot {\mathbf {n}}}{\, \mathrm{d} \sigma }\\&\quad + \int _{\Sigma }{ u_{\mathrm{N}}\varphi {\mathrm{div}}_{\Sigma } {\mathbf {W}}}{\, \mathrm{d} \sigma } + \int _{\Sigma }{ {\dot{u}}_{\mathrm{N}}\varphi }{\, \mathrm{d} \sigma }, \quad \forall \varphi \in H^2 \cap H^1_{\Gamma ,0}(\Omega ). \end{aligned}$$

Now, we choose $\varphi \in C^{\infty }_0(\Omega )$. This leads us to $- \Delta p_{{\mathrm{N}}}' = 0$ in $\Omega$. Moreover, we get

$$\begin{aligned} \int _{\Sigma }{ \varphi \partial _{{\mathbf {n}}}{p_{{\mathrm{N}}}'} }{\, \mathrm{d} \sigma } = \int _{\Sigma }{ (u_{\mathrm{N}}{\mathbf {W}}- \nabla p_{{\mathrm{N}}}{\mathbf {W}}\cdot {\mathbf {n}})\cdot \nabla \varphi }{\, \mathrm{d} \sigma } + \int _{\Sigma }{ u_{\mathrm{N}}\varphi {\mathrm{div}}_{\Sigma } {\mathbf {W}}}{\, \mathrm{d} \sigma } + \int _{\Sigma }{ {\dot{u}}_{\mathrm{N}}\varphi }{\, \mathrm{d} \sigma }. \end{aligned}$$

Observe that $(u_{\mathrm{N}}{\mathbf {W}}- \nabla p_{{\mathrm{N}}}{\mathbf {W}}\cdot {\mathbf {n}})\cdot {\mathbf {n}}= 0$. Hence, we can replace $\nabla \varphi |_{\Sigma }$ by the tangential gradient $\nabla _{\Sigma } \varphi$. Using the tangential Green’s formula (see equation 21) thrice, noting that ${\mathbf {W}}\cdot {\mathbf {n}}\nabla _{\Sigma }p_{{\mathrm{N}}}\cdot {\mathbf {n}}= 0$, and then using the relation ${\dot{u}}_{\mathrm{N}}= u_{\mathrm{N}}' + {\mathbf {W}}\cdot \nabla u_{\mathrm{N}}$, we obtain

$$\begin{aligned}&\int _{\Sigma }{ \varphi \partial _{{\mathbf {n}}}{p_{{\mathrm{N}}}'} }{\, \mathrm{d} \sigma } = \int _{\Sigma }{ \varphi \mathrm{div}_{\Sigma }(\nabla p_{{\mathrm{N}}}{\mathbf {W}}\cdot {\mathbf {n}}) }{\, \mathrm{d} \sigma } + \int _{\Sigma }{ {\dot{u}}_{\mathrm{N}}\varphi }{\, \mathrm{d} \sigma }\\&\quad = \int _{\Sigma }{ \varphi \kappa (\nabla p_{{\mathrm{N}}}{\mathbf {W}}\cdot {\mathbf {n}})\cdot {\mathbf {n}}}{\, \mathrm{d} \sigma } - \int _{\Sigma }{ (\nabla _{\Sigma } \varphi \cdot \nabla p_{{\mathrm{N}}}) {\mathbf {W}}\cdot {\mathbf {n}}}{\, \mathrm{d} \sigma } + \int _{\Sigma }{ {\dot{u}}_{\mathrm{N}}\varphi }{\, \mathrm{d} \sigma }\\&\quad = \int _{\Sigma }{ \varphi \kappa u_{\mathrm{N}}{\mathbf {W}}\cdot {\mathbf {n}}}{\, \mathrm{d} \sigma } - \int _{\Sigma }{ (\nabla _{\Sigma } \varphi \cdot \nabla _{\Sigma } p_{{\mathrm{N}}}) {\mathbf {W}}\cdot {\mathbf {n}}}{\, \mathrm{d} \sigma } + \int _{\Sigma }{ {\dot{u}}_{\mathrm{N}}\varphi }{\, \mathrm{d} \sigma }\\&\quad = \int _{\Sigma }{ \varphi \kappa u_{\mathrm{N}}{\mathbf {W}}\cdot {\mathbf {n}}}{\, \mathrm{d} \sigma } + \int _{\Sigma }{ \varphi \mathrm{div}_{\Sigma }(\nabla _{\Sigma }p_{{\mathrm{N}}}{\mathbf {W}}\cdot {\mathbf {n}}) }{\, \mathrm{d} \sigma } + \int _{\Sigma }{ (u_{\mathrm{N}}' + {\mathbf {W}}\cdot \nabla u_{\mathrm{N}}) \varphi }{\, \mathrm{d} \sigma }, \end{aligned}$$

for all $\varphi \in H^2 \cap H^1_{\Gamma ,0}(\Omega )$. Since the trace of functions from $H^2(\Omega )$ is dense in $L^2(\Sigma )$, we deduce the boundary condition on for $p_{{\mathrm{N}}}'$ given by $\partial _{{\mathbf {n}}}{p_{{\mathrm{N}}}'} = {\mathrm{div}}_{\Sigma }(\nabla _{\Sigma }p_{{\mathrm{N}}}{\mathbf {W}}\cdot {\mathbf {n}}) + \kappa u_{\mathrm{N}}{\mathbf {W}}\cdot {\mathbf {n}}+ u_{\mathrm{N}}' + {\mathbf {W}}\cdot \nabla u_{\mathrm{N}}.$ Summarizing these results, and letting $\Omega = \Omega ^*$, we get

$$\begin{aligned} -\Delta p_{{\mathrm{N}}}' = 0 \ {\mathrm{in}} \ \Omega ^*,\quad p_{{\mathrm{N}}}' = 0 \ {\mathrm{on}} \ \Gamma ,\quad \partial _{{\mathbf {n}}}{p_{{\mathrm{N}}}'} = u_{\mathrm{N}}' + \lambda {\mathbf {W}}\cdot {\mathbf {n}}\quad {\mathrm{on}} \ \Sigma ^*, \end{aligned}$$

as desired. $\square$

It is worth remarking that the existence of the shape derivative $p_{{\mathrm{N}}}'$ of $p_{{\mathrm{N}}}$ can only be justified if $u_{\mathrm{N}}$ is $H^3$-regular. Hence, we require that $\Omega$ be at least of class $C^{2,1}$ so that $u_{\mathrm{N}}$ (as well as $u_{\mathrm{R}}$) is in $H^3(\Omega )$ (see, e.g., [10, Theorem 29]).