Qualitative behaviour of solutions for the two-phase Navier–Stokes equations with surface tension

Abstract

The two-phase free boundary value problem for the isothermal Navier–Stokes system is studied for general bounded geometries in absence of phase transitions, external forces and boundary contacts. It is shown that the problem is well-posed in an $L_p$-setting, and that it generates a local semiflow on the induced state manifold. If the phases are connected, the set of equilibria of the system forms a $(n+1)$-dimensional manifold, each equilibrium is stable, and it is shown that global solutions which do not develop singularities converge to an equilibrium as time goes to infinity. The latter is proved by means of the energy functional combined with the generalized principle of linearized stability.

Appendix A: Transmission problems

In this section we provide some results, concerning the existence and uniqueness of solutions to the transmission problem

$$\begin{aligned} \begin{array}{rcll} \lambda q-\varDelta q&= f,&\quad x\in \varOmega \backslash \varGamma \\ {[\![\rho q]\!]}&= g,&\quad x\in \varGamma ,\\ {[\![\partial _{\nu _\varGamma } q]\!]}&= h_1,&\quad x\in \varGamma ,\\ \delta \partial _{\nu _\varOmega }q+(1-\delta )q&= h_{2,\delta },&\quad x\in \partial \varOmega ,\ \delta \in \{0,1\}, \end{array} \end{aligned}$$

(8.1)

where $\lambda \ge 0$,

$$\begin{aligned} \rho (x):=\rho _1\chi _{\varOmega _1}(x)+\rho _2\chi _{\varOmega _2}(x),\quad x\in \varOmega \backslash \varGamma , \end{aligned}$$

and $\rho _j>0$. To be precise, we will study (8.1) in different functional analytic settings. We begin by stating the result for the ’classical’ case, i.e. if the basic space is given by $L_p(\varOmega )$.

Theorem 6

Let $\varOmega \subset \mathbb{R }^n$ open, $1<p<\infty , f\in L_p(\varOmega ), g\in W_p^{2-1/p}(\varGamma ), h_1\in W_p^{1-1/p}(\varGamma )$ and $h_{2,\delta }\in W_p^{2-\delta -1/p}(\partial \varOmega ), \delta \in \{0,1\}$ be given. Then, for each $\lambda >0$, there exists a unique solution $q\in H_p^2(\varOmega \backslash \varGamma )$ of (8.1) and a constant $C_1>0$ such that

$$\begin{aligned} |q|_{H_p^2(\varOmega \backslash \varGamma )}\le C_1\left(|f|_{L_p(\varOmega )}+|g|_{W_p^{2-1/p}(\varGamma )}+|h_1|_{W_p^{1-1/p}(\varGamma )}+|h_{2,\delta }|_{W_p^{2-\delta -1/p}(\partial \varOmega )}\right). \end{aligned}$$

If in addition $J=[0,a], f=f(t,x), f\in H_p^1(J;L_p(\varOmega ))$ and $g=h_1=h_{2,\delta }=0$, then for each $\lambda >0$, there exists a unique solution $q\in H_p^1(J;H_p^2(\varOmega \backslash \varGamma ))$, and the estimate

$$\begin{aligned} ||q||_{H_p^1(J;H_p^2(\varOmega \backslash \varGamma ))}\le C_2||f||_{H_p^1(J;L_p(\varOmega ))} \end{aligned}$$

holds with some constant $C_2>0$.

Proof

The first assertion basically follows from [12], since the Lopatinskii–Shapiro condition is satisfied at $\varGamma$ and $\partial \varOmega$. The second assertion follows from the first one by differentiating (8.1) w.r.t. $t$ and by employing the uniqueness of the solution of (8.1).

We will also need a result for the case $\lambda =0$. To this end, let $\varOmega \subset \mathbb{R }^n$ be a bounded domain, $g=h_1=h_{2,\delta }=0$ and $f\in L_p(\varOmega )$. Define $A_\delta$ by $A_\delta q=-\varDelta q$, with domain

$$\begin{aligned}&D(A_\delta )=\{q\in H_p^2(\varOmega \backslash \varGamma ):[\![\rho q]\!]=[\![\partial _{\nu _{\varGamma }}q]\!]=0 \ \text{ on}\ \varGamma ,\\&\qquad \qquad \qquad \qquad \delta \partial _{\nu _\varOmega }q+(1-\delta )q=0,\ \text{ on} \ \partial \varOmega \},\ \delta \in \{0,1\}. \end{aligned}$$

Since

the resolvent of $A_\delta$ is compact and therefore the spectral set $\sigma (A_\delta )$ consists solely of a countably infinite sequence of isolated eigenvalues. In case $\delta =1$ it can be readily checked that $0$ is a simple eigenvalue of $A_1$, hence $L_p(\varOmega )=N(A_1)\oplus R(A_1)$. The kernel $N(A_1)$ of $A_1$ is given by $N(A_1)=\mathbb K {\small 1}\!\!1_\rho$, where

$$\begin{aligned} {\small 1}\!\!1_\rho (x):=\chi _{\varOmega _1}(x)+\frac{\rho _1}{\rho _2}\chi _{\varOmega _2}(x),\quad x\in \varOmega \backslash \varGamma . \end{aligned}$$

and $R(A_1)=\{f\in L_p(\varOmega ):(f|{\small 1}\!\!1_\rho )=0\}$. Therefore (8.1) has a unique solution $q\in H_p^2(\varOmega \backslash \varGamma )\ominus \mathbb K {\small 1}\!\!1_\rho$, provided $(f|{\small 1}\!\!1_\rho )=0$. In case of Dirichlet boundary conditions, i.e. $\delta =0$, it holds that $N(A_0)=\{0\}$, hence or each $f\in L_p(\varOmega )$, the system (8.1) admits a unique solution $q\in H_p^2(\varOmega \backslash \varGamma )$.

Theorem 7

Let $\varOmega \subset \mathbb{R }^n$ a bounded domain, $1<p<\infty , f\in L_p(\varOmega ), g=h_1=h_{2,\delta }=0$ and $\lambda =0$. Then the following assertions hold

• 1. If $\delta =0$, then there exists a unique solution $q\in H_p^2(\varOmega \backslash \varGamma )$ of (8.1).

• 2. If $\delta =1$ and $(f|{\small 1}\!\!1_\rho )=0$, then there exists a unique solution $q\in H_p^2(\varOmega \backslash \varGamma )\ominus \mathbb K {\small 1}\!\!1_\rho$.

If in addition $J=[0,a], f=f(t,x)$ and $f\in H_p^1(J;L_p(\varOmega ))$ s.t. $f(t,\cdot )\in R(A_\delta )$ for a.e. $t\in J$, then $q\in H_p^1(J;H_p^2(\varOmega \backslash \varGamma )\ominus N(A_\delta ))$.

1.1 A.1 A weak transmission problem

Here we study the (weak) transmission problem

$$\begin{aligned} (\nabla q|\nabla \phi )_{L_2(\varOmega )}&= (f|\nabla \phi )_{L_2(\varOmega )},\quad \phi \in H_{p^{\prime }}^1(\varOmega ),\\ {[\![\rho q]\!]}&= g,\quad x\in \varGamma , \end{aligned}$$

where $\varOmega \subset \mathbb{R }^n$ is open and bounded with $\partial \varOmega \in C^2$. We want to show that this problem admits a unique solution $q\in \dot{H}_p^1(\varOmega \backslash \varGamma )$, that satisfies the estimate

$$\begin{aligned} |\nabla q|_{L_p(\varOmega )}\le C\left(|f|_{L_p(\varOmega ;\mathbb{R }^n)}+|g|_{W_p^{1-1/p}(\varGamma )}\right)\!, \end{aligned}$$

provided $f\in L_p(\varOmega ;\mathbb{R }^n)$ and $g\in W_p^{1-1/p}(\varGamma )$. We will first treat the case $f=0, g\in W_p^{2-1/p}(\varGamma )$ and consider the problem

$$\begin{aligned} \begin{aligned} \lambda (q|\phi )+(\nabla q|\nabla \phi )_{L_2(\varOmega )}&=0,\quad \phi \in H_{p^{\prime }}^1(\varOmega ),\\ {[\![\rho q]\!]}&=g,\quad x\in \varGamma . \end{aligned} \end{aligned}$$

(8.2)

with $\lambda >0$. Theorem 6 then yields a strong unique solution $q\in H_p^2(\varOmega \backslash \varGamma )$ of (8.1) with $f=h_1=h_2=0$ which is also the unique solution of (8.2). This follows from integration by parts. Our aim is to derive an estimate which is of the form

$$\begin{aligned} |q|_{H_p^1(\varOmega \backslash \varGamma )}\le C|g|_{W_p^{1-1/p}(\varGamma )}, \end{aligned}$$

which will be done by a localization argument. For this purpose we consider first the following auxiliary transmission problem

$$\begin{aligned} \begin{aligned} \lambda q-\varDelta q&=f,\quad x\in \dot{\mathbb{R }}^{n},\\ {[\![\rho q]\!]}&=g,\quad x\in \mathbb{R }^{n-1},\\ {[\![\partial _\nu q]\!]}&=h,\quad x\in \mathbb{R }^{n-1}, \end{aligned} \end{aligned}$$

(8.3)

with data $f\in L_p(\mathbb R ^{n}), g\in W_p^{2-1/p}(\mathbb{R }^{n-1})$ and $h\in W_{p}^{1-1/p}(\mathbb{R }^n)$, which will play an important role in the forthcoming localization procedure. Solve the full space problem

$$\begin{aligned} \lambda q-\varDelta q=f,\quad x\in \mathbb R ^{n}, \end{aligned}$$

to obtain a unique solution $q_1=(\lambda -\varDelta )^{-1}f\in H_p^2(\mathbb{R }^{n})$, provided $\mathrm{Re}\lambda >0$. In the sequel we will always assume that $\lambda$ is real and $\lambda \ge 1$. In particular, it follows that

$$\begin{aligned} \lambda ^{1/2}|q_1|_{L_p(\mathbb{R }^{n})}+|\nabla q_1|_{L_p(\mathbb{R }^{n})}\le C|f|_{H_p^{-1}(\mathbb{R }^{n})}, \end{aligned}$$

(8.4)

with some constant $C>0$ being independent of $\lambda \ge 1$, since

$$\begin{aligned} \lambda ^{1/2}|(\lambda -\varDelta )^{-1}f|_{L_p(\mathbb{R }^{n})}&\le C\lambda ^{1/2}||(I-\varDelta )^{1/2}(\lambda -\varDelta )^{-1} ||_{\mathcal{B }(L_p;L_p)}|f|_{H_p^{-1}(\mathbb{R }^n)}\\&\le C||(I-\varDelta )^{1/2}(\lambda -\varDelta )^{-1/2} ||_{\mathcal{B }(L_p;L_p)}|f|_{H_p^{-1}(\mathbb{R }^n)}\\&\le C|f|_{H_p^{-1}(\mathbb{R }^n)}, \end{aligned}$$

and

$$\begin{aligned} |\nabla (\lambda -\varDelta )^{-1}f|_{L_p(\mathbb{R }^{n})}&\le C ||(I-\varDelta ) (\lambda -\varDelta )^{-1}||_{\mathcal{B }(L_p;L_p)}|f|_{H_p^{-1}(\mathbb{R }^n)}\\&\le C|f|_{H_p^{-1}(\mathbb{R }^n)}, \end{aligned}$$

since the norm

$$\begin{aligned} ||(I-\varDelta )^{\alpha }(\lambda -\varDelta )^{-\alpha }||_{\mathcal{B }(L_p;L_p)},\quad \alpha \in \{1/2,1\}, \end{aligned}$$

is independent of $\lambda \ge 1$, which follows e.g. from functional calculus. The shifted function $q_2=q-q_1$ should now solve the reduced problem

$$\begin{aligned} \begin{aligned} \lambda q_2-\varDelta q_2&=0,\quad x\in \dot{\mathbb{R }}^{n},\\ {[\![\rho q_2]\!]}&=\tilde{g},\quad x\in \mathbb{R }^{n-1},\\ {[\![\partial _\nu q_2]\!]}&=h,\quad x\in \mathbb{R }^{n-1}, \end{aligned} \end{aligned}$$

(8.5)

with a modified function $\tilde{g}\in W_p^{2-1/p}(\mathbb{R }^{n-1})$. Let $x=(x^{\prime },y)\in \mathbb{R }^n\times \mathbb{R }$ and define $L:=(\lambda -\varDelta _n)^{1/2}$, where $\varDelta _n$ denotes the Laplacian with respect to the first $n-1$ variables $x^{\prime }$ and with domain $D(L)=H_p^1(\mathbb{R }^{n-1})$. Let furthermore

$$\begin{aligned} \rho (x^{\prime },y)=\rho _2\chi _{\{y>0\}}(x^{\prime },y)+\rho _1\chi _{\{y<0\}}(x^{\prime },y),\ (x^{\prime },y)\in \mathbb{R }^{n-1}\times \mathbb{R }. \end{aligned}$$

We make the following ansatz to find a solution of (8.5)

$$\begin{aligned} q_2(y):=\left\{ \begin{array}{ll} e^{-Ly}a_+,&y>0,\\ e^{Ly}a_-,&y<0,\\ \end{array}\right. \end{aligned}$$

(8.6)

where $a_-,a_+$ have to be determined. The first transmission condition in (8.5) yields $\rho _2a_+-\rho _1a_-=\tilde{g}$, whereas the second condition implies $-L(a_+ + a_-)=h$, hence $a_+ +a_-=-L^{-1}h$. Observe that $\tilde{g},L^{-1}h\in W_p^{2-1/p}(\mathbb{R }^{n-1})$. Therefore we may solve this linear system of equations to the result

$$\begin{aligned} a_-=-\frac{1}{\rho _1+\rho _2}\left(\tilde{g}+\rho _2 L^{-1}h\right),\ a_+=\frac{1}{\rho _1+\rho _2}\left(\tilde{g}+\rho _2 L^{-1}h\right)-L^{-1}h.\qquad \end{aligned}$$

(8.7)

In other words, the solution of (8.5) (hence of (8.3)) is uniquely determined and $a_-,a_+\in W_p^{2-1/p}(\mathbb{R }^{n-1})$. Since $|Le^{\pm L\cdot }\cdot |_{L_p(\mathbb{R }^{n-1}\times \mathbb{R }_\mp )}$ is an equivalent norm in $W_p^{1-1/p}(\mathbb{R }^{n-1})$ and the corresponding constants are independent of $\lambda \ge 1$, we obtain first

$$\begin{aligned} \lambda ^{1/2}|q_2|_{L_p(\mathbb{R }^{n})}= \lambda ^{1/2}|L^{-1}Lq_2|_{L_p(\mathbb{R }^{n})} \le C\left(|a_+|_{W_p^{1-1/p} (\mathbb{R }^{n-1})}+|a_-|_{W_p^{1-1/p}(\mathbb{R }^{n-1})}\right). \end{aligned}$$

Concerning $\nabla q_2$ in $L_p(\mathbb{R }^{n})$, we estimate as follows

$$\begin{aligned} |\nabla _{x^{\prime }} q_2|_{L_p(\mathbb{R }^n)}&\le C|L_0 q_2|_{L_p(\mathbb{R }^n)}=C|L_0L^{-1}L q_2|_{L_p(\mathbb{R }^n)}\\&\le C||L_0L^{-1}||_{\mathcal{B }(L_p,L_p)}|Lq_2|_{L_p(\mathbb{R }^n)}\\&\le C\left(|a_+|_{W_p^{1-1/p}(\mathbb{R }^{n-1})}+|a_-|_ {W_p^{1-1/p}(\mathbb{R }^{n-1})}\right), \end{aligned}$$

with $L_0:=(I-\varDelta _{x^{\prime }})^{1/2}$. Here the norm $||L_0L^{-1}||_{\mathcal{B }(L_p,L_p)}$ does not depend on $\lambda \ge 1$, which is a consequence of the functional calculus. The estimate for $\partial _yq_2$ in $L_p(\mathbb{R }^{n})$ is even simpler, since

$$\begin{aligned} |\partial _y q_2|_{L_p(\mathbb{R }^n)}=|Lq_2|_{L_p(\mathbb{R }^n)}\le C\left(|a_+|_{W_p^{1-1/p}(\mathbb{R }^{n-1})}+|a_-|_{W_p^{1-1/p}(\mathbb{R }^{n-1})}\right). \end{aligned}$$

This yields the estimate

$$\begin{aligned} \lambda ^{1/2}|q_2|_{L_p(\mathbb{R }^{n})}+|\nabla q_2|_{L_p(\mathbb{R }^n)}\le C\left(|\tilde{g}|_{W_p^{1-1/p}(\mathbb{R }^{n-1})}+|L^{-1}h|_{W_p^{1-1/p}(\mathbb{R }^{n-1})}\right). \end{aligned}$$

For each fixed $\lambda \ge 1$ the operator $L^{-1}$ is bounded and linear from $W_p^{-1/p}(\mathbb{R }^{n-1})$ to $W_p^{1-1/p}(\mathbb{R }^{n-1})$, where $W_p^{-1/p}(\mathbb{R }^{n-1})$ is the topological dual space of $W_{p^{\prime }}^{1/p}(\mathbb{R }^{n-1})$, and $1/p+1/p^{\prime }=1$. We want to show that the bound of $L^{-1}$ is independent of $\lambda \ge 1$. This can be seen as follows. We have

$$\begin{aligned} |L^{-1}h|_{W_p^{1}(\mathbb{R }^{n-1})}\le C|L_0L^{-1}h|_{L_p(\mathbb{R }^{n-1})}\le C||L_0L^{-1}||_{\mathcal{B }(L_p,L_p)}|h|_{L_p(\mathbb{R }^{n-1})} \end{aligned}$$

which holds for all $h\in L_p(\mathbb{R }^{n-1})$, since $|L_0\cdot |_{L_p(\mathbb{R }^{n-1})}$ is an equivalent norm in $W_p^1(\mathbb{R }^{n-1})$. On the other hand we have

$$\begin{aligned} |L^{-1}h|_{L_p(\mathbb{R }^{n-1})}&= |L_0L_0^{-1}L^{-1}h|_{L_p(\mathbb{R }^{n-1})}= |L_0L^{-1}L_0^{-1}h|_{L_p(\mathbb{R }^{n-1})}\\&\le ||L_0L^{-1}||_{\mathcal{B }(L_p,L_p)}|L_0^{-1}h|_{L_p(\mathbb{R }^{n-1})}\\&\le C||L_0L^{-1}||_{\mathcal{B }(L_p,L_p)}|h|_{W_p^{-1}(\mathbb{R }^{n-1})} \end{aligned}$$

for all $h\in W_p^{-1}(\mathbb{R }^{n-1})$, since $|L_0^{-1}\cdot |_{L_p(\mathbb{R }^{n-1})}$ is an equivalent norm in $W_p^{-1}(\mathbb{R }^{n-1})$ and since $L^{-1}$ and $L_0^{-1}$ are commuting operators. Finally we apply the real interpolation method to obtain

$$\begin{aligned} |L^{-1}h|_{W_p^{1-1/p}(\mathbb{R }^{n-1})}\le C|h|_{W_p^{-1/p}(\mathbb{R }^{n-1})}, \end{aligned}$$

for all $h\in W_p^{-1/p}(\mathbb{R }^{n-1})$, where the constant $C>0$ is independent of $\lambda \ge 1$. In summary we derived the a priori estimate

$$\begin{aligned} \lambda ^{1/2}|q_2|_{L_p(\mathbb{R }^n)}+|\nabla q_2|_{L_p(\mathbb R ^{n})}\le C\left(|\tilde{g}|_{W_p^{1-1/p}(\mathbb{R }^{n-1})} +|h|_{W_p^{-1/p}(\mathbb{R }^{n-1})}\right), \end{aligned}$$

for the solution of (8.5), hence

$$\begin{aligned}&\lambda ^{1/2}|q|_{L_p(\mathbb{R }^n)}+|\nabla q|_{L_p(\mathbb R ^{n})}\nonumber \\&\qquad \le C\left(|f|_{H_p^{-1}(\mathbb{R }^{n})}+ |g|_{W_p^{1-1/p}(\mathbb{R }^{n-1})}+|h|_{W_p^{-1/p}(\mathbb{R }^{n-1})}\right) \end{aligned}$$

(8.8)

for the solution of (8.3), since

$$\begin{aligned} |\tilde{g}|_{W_p^{1-1/p}(\mathbb{R }^{n-1})}&\le |g|_{W_p^{1-1/p}(\mathbb{R }^{n-1})}+|[\![\rho q_1]\!]|_{W_p^{1-1/p}(\mathbb{R }^{n-1})}\\&\le |g|_{W_p^{1-1/p}(\mathbb{R }^{n-1})}+C|f|_{H_p^{-1}(\mathbb{R }^n)}, \end{aligned}$$

by (8.4). Consider now a bounded domain $\varOmega \subset \mathbb{R }^{n}$ with $\partial \varOmega \in C^2$ and let $\varGamma \subset \varOmega$ be a hypersurface such that $\varGamma \in C^2, \varGamma \cap \partial \varOmega =\emptyset$ and such that $\varGamma$ divides the set $\varOmega$ into two disjoint regions $\varOmega _1,\varOmega _2$, where $\partial \varOmega _1=\varGamma$ and $\partial \varOmega _2=\partial \varOmega \cup \varGamma$. Since $\bar{\varOmega }$ is compact, we may cover it by a union of finitely many open sets $U_k,\ k=0,\ldots ,N$ which are subject to the following conditions

• $\partial \varOmega \subset U_0$ and $U_0\cap \varGamma =\emptyset$;

• $U_1\subset \varOmega _1$ and $U_1\cap \varGamma =\emptyset$;

• $U_k\cap \varGamma \ne \emptyset , U_k\cap \partial \varOmega =\emptyset \ k=2,\ldots ,N$ and

  $$\begin{aligned} \bigcup _{k=2}^N U_k\supset \varGamma . \end{aligned}$$

For $k\ge 2$, the sets $U_k$ may be balls with a fixed but arbitrarily small radius $r>0$. Let $\{\varphi _k\}_{k=0}^N$ be a partition of unity, such that $\mathrm{supp}\,\varphi _k\subset U_k$ and $0\le \varphi _k(x)\le 1$ for all $x\in \bar{\varOmega }$. Consider the transmission problem

$$\begin{aligned} \begin{aligned} \lambda q-\varDelta q&=0,&x\in \varOmega \backslash \varGamma \\ {[\![\rho q]\!]}&=g,&x\in \varGamma ,\\ {[\![\partial _{\nu _\varGamma } q]\!]}&=0,&x\in \varGamma ,\\ \partial _\nu q&=0,&x\in \partial \varOmega , \end{aligned} \end{aligned}$$

(8.9)

where $g\in W_p^{2-1/p}(\varGamma )$. Set $q_k=q\varphi _k$ and $g_k=g\varphi _k$. By Theorem 6 there exists a unique solution $q\in H_p^2(\varOmega \backslash \varGamma )$ of (8.9), if e.g. $\lambda \ge 1$. Multiplying (8.9) by $\varphi _0$ yields

$$\begin{aligned} \begin{aligned} \lambda q_0-\varDelta q_0&=-2(\nabla q|\nabla \varphi _0)-q\varDelta \varphi _0,\quad x\in \varOmega ,\\ \partial _\nu q_0&=q\partial _\nu \varphi _0,\quad x\in \partial \varOmega , \end{aligned} \end{aligned}$$

(8.10)

which is an elliptic boundary value problem in $\varOmega$. Denote by $(F_0,G_0)$ the right-hand side of (8.10). By [30, Theorem 3.3.4], there exists a common bounded extension operator $E$ from $L_p(\varOmega )$ resp. $H_p^{-1}(\varOmega )$ to $L_p(\mathbb{R }^{n})$ resp. $H_p^{-1}(\mathbb{R }^{n})$. Solve the equation

$$\begin{aligned} \lambda q_0^1-\varDelta q_0^1=E F_0,\quad x\in \mathbb{R }^{n}. \end{aligned}$$

The solution is given by $q_0^1=(\lambda -\varDelta )^{-1}EF_0$ and we have the estimate

$$\begin{aligned} \lambda ^{1/2}|q_0^1|_{L_p(\mathbb{R }^n)}+|\nabla q_0^1|_{L_p(\mathbb{R }^{n})}\le C|EF_0|_{H_p^{-1}(\mathbb{R }^{n})}\le C|F_0|_{H_p^{-1}(\varOmega )}\le C|q|_{L_p(\varOmega )}, \end{aligned}$$

as we have already shown. Note that since $F_0\in L_p(\varOmega )$, it holds that

$$\begin{aligned} |q_0^1|_{H_p^2(\mathbb{R }^n)}=|(\lambda -\varDelta )^{-1}EF_0|_{H_p^2(\mathbb{R }^n)}\le C|F_0|_{L_p(\varOmega )}\le C|q|_{H_p^1(\varOmega )}, \end{aligned}$$

and $C>0$ does not depend on $\lambda \ge 1$. In particular, the real interpolation method yields

$$\begin{aligned} |q_0^1|_{W_p^{1+s}(\mathbb{R }^n)}\le C|q|_{W_p^s(\varOmega )},\ s\in [0,1]. \end{aligned}$$

The shifted function $q_0^2=q_0-q_0^1$ solves the problem

$$\begin{aligned} \begin{aligned} \lambda q_0^2-\varDelta q_0^2&=0,\qquad x\in \varOmega ,\\ \partial _\nu q_0^2&=G_0^2,\quad x\in \partial \varOmega , \end{aligned} \end{aligned}$$

(8.11)

with some modified function $G_0^2\in W_p^{1-1/p}(\partial \varOmega )$. By [1, Theorem 9.2], there exists a bounded solution operator $S_0^2:W_p^{-1/p}(\partial \varOmega )\rightarrow H_p^1(\varOmega )$ such that $q_0^2=S_0^2G_0^2$ and there exists a constant $C>0$ being independent of $\lambda \ge 1$ such that

$$\begin{aligned} \lambda ^{1/2}|q_0^2|_{L_p(\varOmega )}+|\nabla q_0^2|_{L_p(\varOmega )}\le C|G_0^2|_{W_p^{-1/p}(\partial \varOmega )}. \end{aligned}$$

This yields

$$\begin{aligned} \lambda ^{1/2}|q_0|_{L_p(\varOmega )}+|\nabla q_0|_{L_p(\varOmega )}\le C(|(\nabla q|\nabla \varphi _0)|_{H_p^{-1}(\varOmega )}+|q\varDelta \varphi _0|_{H_p^{-1}(\varOmega )}\\ +|q\partial _\nu \varphi _0|_{W_p^{-1/p}(\partial \varOmega )}+|\partial _\nu q_0^1|_{W_p^{-1/p}(\partial \varOmega )}). \end{aligned}$$

Since $\varphi _0$ is smooth and compactly supported and since $\nu \in C^1(\partial \varOmega )$, we have

$$\begin{aligned} \lambda ^{1/2}|q_0|_{L_p(\varOmega )}+|\nabla q_0|_{L_p(\varOmega )}\le C|q|_{W_p^s(\varOmega )}, \end{aligned}$$

(8.12)

for some $s\in (1/p,1)$, since

$$\begin{aligned} |q|_{W_p^{-1/p}(\partial \varOmega )}\le C|q|_{L_p(\partial \varOmega )}\le C|q|_{W_p^s(\varOmega )},\quad s\in (1/p,1), \end{aligned}$$

and

$$\begin{aligned} |\partial _\nu q_0^1|_{W_p^{-1/p}(\partial \varOmega )}\le C|\partial _\nu q_0^1|_{L_p(\partial \varOmega )}\le C|q_0^1|_{W_p^{1+s}(\varOmega )}\le C|q|_{W_p^s(\varOmega )}. \end{aligned}$$

In a next step we multiply (8.9) by $\varphi _1$ to obtain the full space problem

$$\begin{aligned} \lambda q_1-\varDelta q_1=-2(\nabla q|\nabla \varphi _1)-q\varDelta \varphi _1,\quad x\in \mathbb{R }^{n}. \end{aligned}$$

(8.13)

This problem admits a unique solution $q_1=(\lambda -\varDelta )^{-1}F_1$, provided $\lambda \ge 1$, where $S_1=(\lambda -\varDelta )^{-1}:H_p^{-1}(\mathbb{R }^{n})\rightarrow H_p^1(\mathbb{R }^{n})$ is bounded and $F_1$ denotes the right-hand side of (8.13). As before we obtain the estimate

$$\begin{aligned} \lambda ^{1/2}|q_1|_{L_p(\mathbb{R }^n)}+|\nabla q_1|_{L_p(\mathbb{R }^n)}\le C|q|_{L_p(\varOmega )}, \end{aligned}$$

(8.14)

with $C>0$ being independent of $\lambda \ge 1$.

We turn now to the charts $U_k, k=2,\ldots ,N$. Multiplying (8.9) by $\varphi _k, k=2,\ldots ,N$, we obtain the pure transmission problem

$$\begin{aligned} \begin{aligned} \lambda q_k-\varDelta q_k&=-2(\nabla q|\nabla \varphi _k)-q\varDelta \varphi _k,\quad x\in \mathbb{R }^n\backslash \varGamma ,\\ {[\![\rho q_k]\!]}&=g_k,\quad x\in \varGamma ,\\ {[\![\partial _\nu q_k]\!]}&=[\![q]\!]\partial _\nu \varphi _k,\quad x\in \varGamma . \end{aligned} \end{aligned}$$

(8.15)

Let $x_0\in \varGamma$. Then there exists $k\in \{2,\ldots ,N\}$ such that $x_0\in U_{k}$. After a translation and a rotation of coordinates we may assume that $x_0=0$ and that the normal $\nu (x_0)$ at $x_0$ which points from $\varOmega _1$ to $\varOmega _2$ is given by $\nu (x_0)=[0,\ldots ,0,-1]^\mathsf{T}$. Consider a graph $\eta \in C^2(\mathbb{R }^{n-1})$ with compact support such that

$$\begin{aligned} \{(x^{\prime },x_{n})\in U_{k}\subset \mathbb{R }^{n-1}\times \mathbb{R }:x_{n}=\eta (x^{\prime })\}=\varGamma \cap U_{k}. \end{aligned}$$

Note that, since $\nabla _{x^{\prime }}\eta (0)=0$, we may choose $|\nabla _{x^{\prime }}\eta |_\infty$ as small as we wish, by decreasing the size of the chart $U_{k}$. Let $q(x^{\prime },x_{n})=v(x^{\prime },x_{n}-\eta (x^{\prime }))$, where $(x^{\prime },x_{n})\in U_{k}$. We define a new coordinate by $y=x_{n}-\eta (x^{\prime }), (x^{\prime },x_{n})\in U_{k}$. Then we obtain

$$\begin{aligned} \varDelta q(x^{\prime },x_{n})&= \varDelta _y v(x^{\prime },y)-2\partial _y\left(\nabla _{x^{\prime }}v(x^{\prime },y)|\nabla _{x^{\prime }}\eta (x^{\prime })\right)\\&\quad +\partial _y^2v(x^{\prime },y)|\nabla _{x^{\prime }}\eta |^2-\partial _y v(x^{\prime },y)\varDelta _{x^{\prime }}\eta (x^{\prime }) \end{aligned}$$

and

$$\begin{aligned}{}[\![\partial _\nu q]\!]=-\sqrt{1+|\nabla _{x^{\prime }}\eta |^2}[\![\partial _y v]\!]+\frac{1}{\sqrt{1+|\nabla _{x^{\prime }}\eta |^2}}\left([\![\nabla _{x^{\prime }}v]\!]|\nabla _{x^{\prime }}\eta \right), \end{aligned}$$

since the normal at $x\in U_{k}\cap \varGamma$ is given by

$$\begin{aligned} \nu (x^{\prime },\eta (x^{\prime }))=\frac{1}{\sqrt{1+|\nabla _{x^{\prime }}\eta |^2}} [(\nabla _{x^{\prime }}\eta )^\mathsf{T},-1]^\mathsf{T}. \end{aligned}$$

Let $(\varTheta u)(x^{\prime },y):=q(x^{\prime },y+\eta (x^{\prime }))=v(x^{\prime },y)$ with inverse $(\varTheta ^{-1} v)(x^{\prime },x_{n+1})=v(x^{\prime },x_{n+1}-\eta (x^{\prime }))=q(x^{\prime },x_{n+1})$. Applying the $C^2$-diffeomorphism $\varTheta$ to (8.15) and considering the terms on the right-hand side of (8.15) which depend on $u$ as given functions $(f_k,g_k,h_k)$ yields the problem

$$\begin{aligned} \begin{aligned} \lambda v_k-\varDelta _y v_k&=F(f_k,v_k,\varphi _k,\eta ),\quad (x^{\prime },y)\in \dot{\mathbb{R }}^{n},\\ {[\![\rho v_k]\!]}&=G(g_k),\quad x^{\prime }\in \mathbb R ^{n-1},\ y=0,\\ {[\![\partial _y v_k]\!]}&=H(v_k,\varphi _k,\eta ),\quad x^{\prime }\in \mathbb R ^{n-1},\ y=0. \end{aligned} \end{aligned}$$

(8.16)

which is of the form (8.3). Here

$$\begin{aligned} F(f_k,v_k,\varphi _k,\eta ):=-2\partial _y\left(\nabla _{x^{\prime }}v_k|\nabla _{x^{\prime }}\eta \right) +\partial _y^2v_k|\nabla _{x^{\prime }}\eta |^2-\partial _y v_k\varDelta _{x^{\prime }}\eta , \end{aligned}$$

$G(g_k):=\varTheta g_k$ and

$$\begin{aligned} H(h_k,v_k,\varphi _k,\eta ):=\frac{1}{1+|\nabla _{x^{\prime }}\eta |^2} \left([\![\nabla _{x^{\prime }}v_k]\!]|\nabla _{x^{\prime }}\eta \right) \end{aligned}$$

We want to apply (8.8) to (8.16) and estimate as follows.

$$\begin{aligned} |\partial _y\left(\nabla _{x^{\prime }}v_k|\nabla _{x^{\prime }}\eta \right) |_{W_p^{-1}(\mathbb{R }^{n})}&\le C|(I-\varDelta _y)^{-1/2}\partial _y\left(\nabla _{x^{\prime }}v_k| \nabla _{x^{\prime }}\eta \right)|_{L_p(\mathbb{R }^{n})}\\&\le C|\left(\nabla _{x^{\prime }}v_k|\nabla _{x^{\prime }}\eta \right)|_{L_p(\mathbb{R }^{n})}\\&\le C|\nabla _{x^{\prime }}\eta |_\infty |v_k|_{W_p^1(\dot{\mathbb{R }}^{n})}. \end{aligned}$$

In the same way we obtain

$$\begin{aligned} |\partial _y^2v_k|\nabla _{x^{\prime }}\eta |^2|_{W_p^{-1}(\mathbb{R }^{n})}\le C|\nabla _{x^{\prime }}\eta |_\infty ^2|v_k|_{W_p^1(\dot{\mathbb{R }}^{n})}, \end{aligned}$$

whereas

$$\begin{aligned} |\partial _y v_k\varDelta _{x^{\prime }}\eta |_{W_p^{-1}(\mathbb{R }^{n})}\le C |v_k|_{L_p(\mathbb{R }^{n})}, \end{aligned}$$

since $\eta$ is smooth. Concerning the terms in the Neumann transmission condition, we obtain by trace theory

$$\begin{aligned} |\left([\![\nabla _{x^{\prime }}v_k]\!]| \nabla _{x^{\prime }}\eta \right)|_{W_p^{-1/p}(\mathbb{R }^{n-1})}&\le C|\nabla _{x^{\prime }}\eta |_{C^\alpha (\mathbb{R }^{n-1})}|\nabla _{x^{\prime }}v_k|_{W_p^{-1/p}(\mathbb{R }^{n-1})}\\&\le C|\nabla _{x^{\prime }}\eta |_{C^\alpha (\mathbb{R }^{n-1})} |v_k|_{W_p^{1-1/p}(\mathbb{R }^{n-1})}\\&\le C|\nabla _{x^{\prime }}\eta |_{C^\alpha (\mathbb{R }^{n-1})}|v_k|_{W_p^{1} (\dot{\mathbb{R }}^{n})}, \end{aligned}$$

where $\alpha \in (1/p,1)$. These estimates show that the right-hand side of (8.16) may be estimated by terms that are either of lower order or of highest order, but the higher order terms carry a factor of the form $|\nabla _{x^{\prime }}\eta |_\infty ^\theta , \theta >0$, which becomes small, by decreasing the size of the chart $U_k$. Applying perturbation theory it follows that there exists $\lambda _0\ge 1$ such that for each chart $U_k, k=2,\ldots ,N$, the linear problem (8.16) has a bounded solution operator

$$\begin{aligned} S_k:W_p^{-1}(\mathbb{R }^{n})\times W_p^{1-1/p}(\mathbb{R }^{n-1})\times W_p^{-1/p}(\mathbb{R }^{n-1})\rightarrow W_p^1(\dot{\mathbb{R }}^{n}), \end{aligned}$$

provided $\lambda \ge \lambda _0$. This in turn yields that $\varTheta ^{-1}S_k\varTheta$ is the corresponding solution operator for problem (8.15), i.e. we have

$$\begin{aligned} q_k=(\varTheta ^{-1}S_k\varTheta )(F_k,G_k,H_k), \end{aligned}$$

for each $k=2,\ldots ,N$, where $(F_k,G_k,H_k)$ denotes the right-hand side of (8.15). Since $\varTheta$ is a $C^2$-diffeomorphism, we obtain the estimate

$$\begin{aligned} \lambda ^{1/2}|q_k|_{L_p(\varOmega )}+|\nabla q_k|_{L_p(\varOmega )}\le C\left(|g|_{W_p^{1-1/p}(\varGamma )}+|q|_{W_p^s(\varOmega \backslash \varGamma )}\right), \end{aligned}$$

(8.17)

for some $s\in (1/p,1)$ and for each $k=2,\ldots ,N$. Here the constant $C>0$ does not depend on $\lambda \ge \lambda _0$, as we have already shown in the investigation of (8.3). Let us introduce

$$\begin{aligned} |v|_{\lambda ,W_p^1(\varOmega )}:=| \lambda |^{1/2}|v|_{L_p(\varOmega )}+|\nabla v|_{L_p(\varOmega )},\quad \lambda \ge 1,\ v\in W_p^1(\varOmega \backslash \varGamma ), \end{aligned}$$

which is an equivalent norm in $W_p^1(\varOmega \backslash \varGamma )$. This yields

$$\begin{aligned} |q|_{\lambda ,W_p^1(\varOmega )}\le \sum _{k=0}^N |q_k|_{\lambda ,W_p^1(\varOmega )}\le C\Big (|g|_{W_p^{1-1/p}(\varGamma )}+|q|_{W_p^s(\varOmega )}\Big ). \end{aligned}$$

with constants $C,M>0$, being independent of $\lambda$. Since $s\in (1/p,1)$ we may apply interpolation theory to the result

$$\begin{aligned} |q|_{W_p^s(\varOmega )}&\le \varepsilon |q|_{W_p^1(\varOmega )}+C(\varepsilon )|q|_{L_p(\varOmega )}\\&\le \varepsilon |q|_{\lambda ,W_p^1(\varOmega )}+C(\varepsilon )|q|_{L_p(\varOmega )}\\&\le \left(\varepsilon +C(\varepsilon )/|\lambda |^{1/2}\right) |q|_{\lambda ,W_p^1(\varOmega )}, \end{aligned}$$

since by assumption $\lambda \ge 1$. Choosing first $\varepsilon >0$ small enough and then $\lambda \ge 1$ sufficiently large, we finally obtain the estimate

$$\begin{aligned} |q|_{W_p^1(\varOmega )}\le C|g|_{W_p^{1-1/p}(\varGamma )} \end{aligned}$$

(8.18)

for the strong solution $q\in W_p^2(\varOmega \backslash \varGamma )$ of (8.9). Now we want to reduce the regularity of $g$. Fix $g\in W_p^{1-1/p}(\varGamma )$. Then there exists a sequence $(g_m)\subset W_p^{2-1/p}(\varGamma )$, such that $g_m\rightarrow g$ as $m\rightarrow \infty$ in $W_p^{1-1/p}(\varGamma )$. We denote by $q_m\in W_p^2(\varOmega \backslash \varGamma )$ the corresponding solutions of (8.9). Then it follows from (8.18) that $(q_m)$ is a Cauchy sequence in $W_p^{1}(\varOmega \backslash \varGamma )$. Therefore the limit $\lim _{m\rightarrow \infty } q_m=:q_\infty$ exists and $q_\infty \in W_p^{1}(\varOmega \backslash \varGamma )$ is the unique weak solution of (8.9) for sufficiently large $\lambda \ge 1$.

Lemma 1

Let $1<p<\infty , 1/p+1/p^{\prime }=1$ and let $g\in W_p^{1-1/p}(\varGamma )$ be given. Then there exists $\lambda _0\ge 1$ such that the problem

$$\begin{aligned} \lambda (q|\phi )_{L_2(\varOmega )}+(\nabla q|\nabla \phi )_{L_2(\varOmega )}&= 0,\quad \phi \in H_{p^{\prime }}^1(\varOmega ),\\ {[\![\rho q]\!]}&= g,\quad x\in \varGamma , \end{aligned}$$

has a unique solution $q\in H_p^1(\varOmega \backslash \varGamma )$, provided $\lambda \ge \lambda _0$. Moreover, the solution $q\in H_p^1(\varOmega \backslash \varGamma )$ satisfies the estimate

$$\begin{aligned} |q|_{H_p^{1}(\varOmega )}\le C|g|_{W_p^{1-1/p}(\varGamma )}. \end{aligned}$$

(8.19)

In a next step we consider the problem

$$\begin{aligned} \begin{aligned} \lambda (q|\phi )_{L_2(\varOmega )}+(\nabla q|\nabla \phi )_{L_2(\varOmega )}&=(f|\nabla \phi )_{L_2(\varOmega )},\quad \phi \in H_{p^{\prime }}^1(\varOmega ),\\ {[\![\rho q]\!]}&=0,\quad x\in \varGamma , \end{aligned} \end{aligned}$$

(8.20)

where $f\in L_p(\varOmega ;\mathbb{R }^n)$ is given. Observe that the mapping $\psi _f:H_{p^{\prime }}^1(\varOmega \backslash \varGamma )\rightarrow \mathbb{R }$ defined by

$$\begin{aligned} \psi _f(\phi ):=\langle \psi _f,\phi \rangle :=\int _{\varOmega }(f|\nabla \phi ) dx, \end{aligned}$$

is linear and continuous, since

$$\begin{aligned} |\psi _f(\phi )|\le |f|_{L_p(\varOmega ;\mathbb{R }^n)}|\phi |_{H_{p^{\prime }}^1(\varOmega )}, \end{aligned}$$

hence $\psi _f\in \left(H_{p^{\prime }}^1(\varOmega \backslash \varGamma )\right)^*$. With the help of the Dirichlet form

$$\begin{aligned} a:H_p^1(\varOmega \backslash \varGamma )\times H_{p^{\prime }}^1(\varOmega \backslash \varGamma )\rightarrow \mathbb{R },\quad a(q,v):=\int _{\varOmega }\nabla q\cdot \nabla v dx, \end{aligned}$$

we define an operator $A:H_p^1(\varOmega \backslash \varGamma )\rightarrow \left(H_{p^{\prime }}^1(\varOmega \backslash \varGamma )\right)^*$ by means of

$$\begin{aligned} \langle Aq,v\rangle :=a(q,v), \end{aligned}$$

with domain $D(A)=\{q\in H_p^1(\varOmega \backslash \varGamma ):[\![\rho q]\!]=0\ \text{ on}\,\,\varGamma \}$. Making use of these definitions, we may rewrite (8.20) in the abstract form

$$\begin{aligned} \lambda q+Aq=\psi _f,\quad \text{ in} \left(H_{p^{\prime }}^1(\varOmega \backslash \varGamma )\right)^*. \end{aligned}$$

(8.21)

Since

the resolvent of $A$ is compact and therefore the spectral set $\sigma (A)$ consists solely of a countably infinite sequence of isolated eigenvalues. By a bootstrap argument it is easily seen that the corresponding eigenfunctions are smooth. Hence, defining $A_2$ to be the part of $A$ in $L_2(\varOmega \backslash \varGamma )$ with domain $D(A_2)=\{q\in D(A):Au\in L_2(\varOmega \backslash \varGamma )\}$, it follows that $\sigma (A)=\sigma (A_2)$. Integrating by parts, we obtain

$$\begin{aligned} D(A_2)=\{q\in H_2^2(\varOmega \backslash \varGamma ):[\![\rho q]\!]=0, [\![\partial _{\nu _\varGamma } q]\!]=0\ \text{ on}\,\,\varGamma ,\ \partial _\nu q=0\ \text{ on}\,\,\partial \varOmega \} \end{aligned}$$

and $A_2 q=-\varDelta q$ in $\varOmega \backslash \varGamma$. Let $\lambda \in \sigma (-A)=\sigma (-A_2)$ and let $q\in D(A_2)$ be a corresponding eigenfunction. Then $q$ satisfies the problem

$$\begin{aligned} \begin{aligned} \lambda q-\varDelta q&=0,\quad x\in \varOmega \backslash \varGamma ,\\ {[\![\rho q]\!]}&=0,\quad x\in \varGamma ,\\ {[\![\partial _{\nu _\varGamma } q]\!]}&=0,\quad x\in \varGamma ,\\ \partial _\nu q&=0,\quad x\in \partial \varOmega . \end{aligned} \end{aligned}$$

(8.22)

Multiplying (8.22)$_{1}$ by $\rho q$ and integrating by parts, we obtain by (8.22)$_{2,3,4}$

$$\begin{aligned} -\lambda \int _{\varOmega \backslash \varGamma }\rho |q|^2dx&= -\int _{\varOmega \backslash \varGamma }\rho q \varDelta q dx=-\rho _1\int _{\varOmega _1} q_1 \varDelta q_1 dx-\rho _2\int _{\varOmega _2} q_2 \varDelta q_2 dx\\&= \rho _1|\nabla q_1|_2^2+\rho _2|\nabla q_2|_2^2+\int _\varGamma \left(\partial _{\nu _\varGamma } q_2 \rho _2q_2-\partial _{\nu _\varGamma } q_1 \rho _1q_1\right) d\varGamma \\&= \rho _1|\nabla q_1|_2^2+\rho _2|\nabla q_2|_2^2+\int _\varGamma \partial _{\nu _\varGamma } q_2 [\![\rho q]\!] d\varGamma \\&= \rho _1|\nabla q_1|_2^2+\rho _2|\nabla q_2|_2^2\ge 0, \end{aligned}$$

where $q_j$ denotes the part of $q$ in $\varOmega _j$. In particular it follows that $\lambda$ is real and $\lambda \le 0$ for all $\lambda \in \sigma (-A)$ and if $\lambda =0$ then $q_1$ and $q_2$ are both equal to a constant in $\varOmega _1$ and $\varOmega _2$, respectively, satisfying the identity $\rho _1 q_1=\rho _2 q_2$. In other words the eigenvalue $\lambda =0$ is simple and the kernel $N(A)=N(A_2)$ is given by

$$\begin{aligned} N(A)=\mathbb{K }{\small 1}\!\!1_\rho ,\quad {\small 1}\!\!1_\rho (x):=\chi _{\varOmega _1}(x) +\frac{\rho _1}{\rho _2}\chi _{\varOmega _2}(x),\ x\in \varOmega . \end{aligned}$$

Therefore, spectral theory implies $\left(H_{p^{\prime }}^1(\varOmega \backslash \varGamma )\right)^*=N(A)\oplus R(A)$ and $H_p^1(\varOmega \backslash \varGamma )=N(A)\oplus Y$, where $Y$ is a closed subspace of $H_p^1(\varOmega \backslash \varGamma )$. Note that these decompositions reduce the linear operator $A$. It follows that the equation $Aq=F$ has a unique solution $q\in Y\subset H_p^1(\varOmega \backslash \varGamma )$ if and only if $F\in R(A)$, or equivalently $\langle F,{\small 1}\!\!1_\rho \rangle =0$. If $c\in \mathbb{K }$, then any other solution $\tilde{q}\in H_p^1(\varOmega \backslash \varGamma )$ of $A\tilde{q}=F$ is given by $\tilde{q}=q+c{\small 1}\!\!1_\rho$ and we have the estimate

$$\begin{aligned} |\nabla \tilde{q}|_{L_p(\varOmega )}\le C|F|_{\left(H_{p^{\prime }}^1(\varOmega \backslash \varGamma )\right)^*}. \end{aligned}$$

Lemma 2

Let $1<p<\infty , 1/p+1/p^{\prime }=1$ and let $f\in L_p(\varOmega ;\mathbb{R }^n)$ be given. Then the problem

$$\begin{aligned} (\nabla q|\nabla \phi )_{L_2(\varOmega )}&= (f|\nabla \phi )_{L_2(\varOmega )},\quad \phi \in H_{p^{\prime }}^1(\varOmega ),\\ {[\![\rho q]\!]}&= 0,\quad x\in \varGamma , \end{aligned}$$

has a unique solution $q\in \dot{H}_p^1(\varOmega \backslash \varGamma )$, satisfying the estimate

$$\begin{aligned} |\nabla q|_{L_p(\varOmega )}\le C|f|_{L_p(\varOmega ;\mathbb{R }^n)}. \end{aligned}$$

For the final step, let $v\in H_p^1(\varOmega \backslash \varGamma )$ be the unique solution of

$$\begin{aligned} \lambda _0 (v|\phi )_{L_2(\varOmega )}+(\nabla v|\nabla \phi )_{L_2(\varOmega )}&= 0,\quad \phi \in H_{p^{\prime }}^1(\varOmega ),\\ {[\![\rho v]\!]}&= g,\quad x\in \varGamma , \end{aligned}$$

which is well-defined, thanks to Lemma 1. With the help of this solution $v$, we define a functional $\psi _v\in \left(H_{p^{\prime }}^1(\varOmega \backslash \varGamma )\right)^*$ by

$$\begin{aligned} \psi _v(\phi ):=\int _{\varOmega }\nabla v\cdot \nabla \phi dx. \end{aligned}$$

By definition it holds that $\psi _v({\small 1}\!\!1_\rho )=0$. Since also $\psi _f({\small 1}\!\!1_\rho )=0$ for all $f\in L_p(\varOmega ;\mathbb{R }^n)$, Lemma 2 yields a unique solution $w\in \dot{H}_p^1(\varOmega \backslash \varGamma )$ of

$$\begin{aligned} (\nabla w|\nabla \phi )_{L_2(\varOmega )}&= \psi _f(\phi )-\psi _v(\phi ),\quad \phi \in H_{p^{\prime }}^1(\varOmega ),\\ {[\![\rho w]\!]}&= 0,\quad x\in \varGamma . \end{aligned}$$

Finally, the sum $q:=v+w\in \dot{H}_p^1(\varOmega \backslash \varGamma )$ is the unique solution of

$$\begin{aligned} (\nabla q|\nabla \phi )_{L_2(\varOmega )}&= \psi _f(\phi )=(f|\nabla \phi )_ {L_2(\varOmega )},\quad \phi \in H_{p^{\prime }}^1(\varOmega ),\\ {[\![\rho q]\!]}&= g,\quad x\in \varGamma \end{aligned}$$

and we have the estimate

$$\begin{aligned} |\nabla q|_{L_p(\varOmega )}\le C\left(|f|_{L_p(\varOmega ;\mathbb{R }^n)} +|g|_{W_p^{1-1/p}(\varGamma )}\right). \end{aligned}$$

Theorem 8

Let $1<p<\infty , 1/p+1/p^{\prime }=1, f\in L_p(\varOmega ;\mathbb{R }^n)$ and $g\in W_p^{1-1/p}(\varGamma )$ be given. Then the problem

$$\begin{aligned} (\nabla q|\nabla \phi )_{L_2(\varOmega )}&= (f|\nabla \phi ) _{L_2(\varOmega )}, \quad \phi \in H_{p^{\prime }}^1(\varOmega ),\\ {[\![\rho q]\!]}&= g,\quad x\in \varGamma \end{aligned}$$

has a unique solution $u\in \dot{H}_p^1(\varOmega \backslash \varGamma )$ satisfying the estimate

$$\begin{aligned} |\nabla q|_{L_p(\varOmega )}\le C_1\left(|f|_{L_p(\varOmega ;\mathbb{R }^n)}+|g|_{W_p^{1-1/p}(\varGamma )}\right). \end{aligned}$$

If $J=[0,a], f=f(t,x), f\in H_p^1(J;L_p(\varOmega ;\mathbb{R }^n)), g=0$, then $q\in H_p^1(J;\dot{H}_p^1(\varOmega \backslash \varGamma ))$ and

$$\begin{aligned} ||\nabla q||_{H_p^1(J;L_p(\varOmega ))}\le C_2||f||_{H_p^1(J;L_p(\varOmega ;\mathbb{R }^n))}. \end{aligned}$$

1.2 A.2 Higher regularity in the bulk phases

The next problem we consider, is about higher regularity in the bulk phases $\varOmega \backslash \varGamma$. To be precise, we study the elliptic transmission problem

$$\begin{aligned} \begin{aligned} \lambda q-\varDelta q&=f,\quad x\in \varOmega \backslash \varGamma ,\\ {[\![\rho q]\!]}&=0,\quad x\in \varGamma ,\\ {[\![\partial _{\nu _\varGamma } q]\!]}&=0,\quad x\in \varGamma ,\\ \delta \partial _{\nu _{\varOmega }} q+(1-\delta ) q&=0,\quad x\in \partial \varOmega ,\ \delta \in \{0,1\}, \end{aligned} \end{aligned}$$

(8.23)

where $f\in L_p(\varOmega )\cap W_p^s(\varOmega \backslash \varGamma ), s>0$, is given and $\lambda \ge 1$. It is our aim to find a unique solution $q\in W_p^{2+s}(\varOmega \backslash \varGamma )$ of (8.23). Note that by Theorem 6 there exists a unique solution $q\in H_p^2(\varOmega \backslash \varGamma )$ of (8.23). Moreover, there exists a constant $C>0$ being independent of $\lambda \ge 1$ such that the estimate

$$\begin{aligned} | q|_{H_p^2(\varOmega \backslash \varGamma )}\le C|f|_{L_p(\varOmega )} \end{aligned}$$

(8.24)

is valid. Thus, it remains to show that in addition $q\in W_p^{2+s}(\varOmega \backslash \varGamma )$, provided $f\in L_p(\varOmega )\cap W_p^s(\varOmega \backslash \varGamma )$. For this purpose let $\partial \varOmega \in C^3$ and cover the compact set $\bar{\varOmega }$ by a union of finitely many open sets $U_k,\ k=0,\ldots ,N$ which are subject to the following conditions

• $\partial \varOmega \subset U_0$ and $U_0\cap \varGamma =\emptyset$;

• $U_1\subset \varOmega _1$ and $U_1\cap \varGamma =\emptyset$;

• $U_k\cap \varGamma \ne \emptyset , U_k\cap \partial \varOmega =\emptyset \ k=2,\ldots ,N$ and

  $$\begin{aligned} \bigcup _{k=2}^N U_k\supset \varGamma . \end{aligned}$$

For $k\ge 2$, the sets $U_k$ may be balls with a fixed but arbitrarily small radius $r>0$. As before, let $\{\varphi _k\}_{k=0}^N$ be a partition of unity, such that $\mathrm{supp}\,\varphi _k\subset U_k$ and $0\le \varphi _k(x)\le 1$ for all $x\in \bar{\varOmega }$. Let $q_k:= q\varphi _k$ and $f_k:=f\varphi _{k}$.

Multiplying (8.23) by $\varphi _0$ yields the problem

$$\begin{aligned} \begin{aligned} \lambda q_0-\varDelta q_0&=f_0-2(\nabla q|\nabla \varphi _0)- q\varDelta \varphi _0,\quad x\in \varOmega ,\\ \delta \partial _{\nu _{\varOmega }} q_0+(1-\delta ) q_0&=\delta q\partial _{\nu _{\varOmega }}\varphi _0,\quad x\in \partial \varOmega ,\ \delta \in \{0,1\}. \end{aligned} \end{aligned}$$

(8.25)

Since $\varphi _0$ is smooth and $q\in H_p^2(\varOmega )$, the right-hand side $(F_0,G_0)$ in (8.25) is in $W_p^s(\varOmega )\times W_p^{1+s-1/p}(\partial \varOmega )$, at least for $s\in (0,1]$. It follows from [31, Theorem 5.5.1 & Remark 5.5.2/2] that $q_0\in W_p^{2+s}(\varOmega ), s\in [0,1]$ and

$$\begin{aligned} | q_0|_{W_p^{2+s}(\varOmega )}\le C(|F_0|_{W_p^s(\varOmega )}+|G_0|_{W_p^{1+s-1/p}(\partial \varOmega )})\le C\left(|f|_{W_p^{s}(\varOmega \backslash \varGamma )}+|f|_{L_p(\varOmega )}\right), \end{aligned}$$

by (8.24), where the constant $C>0$ does not depend on $\lambda \ge 1$. Multiplying (8.23) by $\varphi _1$ we obtain the full space problem

$$\begin{aligned} \lambda q_1-\varDelta q_1=f_1-2(\nabla q|\nabla \varphi _1)- q\varDelta \varphi _1,\quad x\in \mathbb{R }^n, \end{aligned}$$

(8.26)

with a right-hand side in $W_p^s(\mathbb{R }^n), s\in (0,1]$, which we denote by $F_1$. Then the solution of (8.26) is given by

$$\begin{aligned} q_1=(\lambda -\varDelta )^{-1}F_1. \end{aligned}$$

If $\alpha \in \{0,1\}$ and $F_1\in H_p^\alpha (\mathbb{R }^n)$ then $q_1\in H_p^{2+\alpha }(\mathbb{R }^n)$ and

$$\begin{aligned} | q_1|_{H_p^{2+\alpha }(\mathbb{R }^n)}&\le C|(I-\varDelta )^{1+\alpha /2} q_1|_{L_p(\mathbb{R }^n)}=C|(I-\varDelta )^{1+\alpha /2}(\lambda -\varDelta ) ^{-1}F_1|_{L_p(\mathbb{R }^n)}\\&= C|(I-\varDelta )(\lambda -\varDelta )^{-1}(I-\varDelta ) ^{\alpha /2}F_1|_{L_p(\mathbb{R }^n)}\\&\le C||(I-\varDelta )(\lambda -\varDelta )^{-1}|| _{\mathcal{B }(L_p,L_p)}|(I-\varDelta )^{\alpha /2}F_1|_{L_p(\mathbb{R }^n)}\\&\le C||(I-\varDelta )(\lambda -\varDelta )^{-1}||_{\mathcal{B }(L_p,L_p)} |F_1|_{H_p^\alpha (\mathbb{R }^n)}, \end{aligned}$$

since $|(I-\varDelta )^{1+\alpha /2}\cdot |_{L_p(\mathbb{R }^n)}$ is an equivalent norm in $H_p^{2+\alpha }(\mathbb{R }^n), \alpha \in \{0,1\}$. Note that the term

$$\begin{aligned} ||(I-\varDelta )(\lambda -\varDelta )^{-1}||_{\mathcal{B }(L_p,L_p)} \end{aligned}$$

is independent of $\lambda \ge 1$, which follows e.g. from functional calculus. The real interpolation method and (8.24) then yield the estimate

$$\begin{aligned} | q_1|_{W_p^{2+s}(\mathbb{R }^n)}\le C|F_1|_{W_p^s(\mathbb{R }^n)}\le C\left(|f|_{W_p^{s}(\varOmega \backslash \varGamma )}+|f|_{L_p(\varOmega )}\right)\!, \end{aligned}$$

for $s\in (0,1]$, where $C>0$ does not depend on $\lambda \ge 1$. Next, we multiply (8.23) by $\varphi _k, k\in \{2,\ldots ,N\}$, to obtain the pure transmission problems

$$\begin{aligned} \begin{aligned} \lambda q_k-\varDelta q_k&=f_k-2(\nabla q|\nabla \varphi _k)- q\varDelta \varphi _k,\quad x\in \mathbb{R }^n\backslash \varGamma ,\\ {[\![\rho q_k]\!]}&=0,\quad x\in \varGamma ,\\ {[\![\partial _\nu q_k]\!]}&=[\![ q]\!]\partial _\nu \varphi _k,\quad x\in \varGamma , \end{aligned} \end{aligned}$$

(8.27)

with some function $f_k\in W_p^s(\mathbb{R }^n\backslash \varGamma )\cap L_p(\mathbb{R }^n)$. For each fixed $k\in \{2,\ldots ,N\}$ we may use the transformation described above, to reduce (8.27) to the problem

$$\begin{aligned} \begin{aligned} \lambda \psi -\varDelta \psi&=F,\quad x^{\prime }\in \mathbb{R }^{n-1},\ y\in \dot{\mathbb{R }},\\ {[\![\rho \psi ]\!]}&=0,\quad x^{\prime }\in \mathbb{R }^{n-1},\ y=0,\\ {[\![\partial _y \psi ]\!]}&=G,\quad x^{\prime }\in \mathbb{R }^{n-1},\ y=0, \end{aligned} \end{aligned}$$

(8.28)

with given functions $F\in W_p^{s}(\dot{\mathbb{R }}^n)$ and $G\in W_p^{1+s-1/p}(\mathbb{R }^{n-1}), s\in (0,1]$. First we remove the inhomogeneity $F$. To this end we solve the Dirichlet problems

$$\begin{aligned} \lambda \psi -\varDelta _{x^{\prime }}\psi ^+-\partial _y^2\psi ^+=F^+,\ x^{\prime }\in \mathbb{R }^{n-1},\ y>0,\quad \psi ^+(x^{\prime },0)=0, \end{aligned}$$

and

$$\begin{aligned} \lambda \psi -\varDelta _{x^{\prime }}\psi ^--\partial _y^2\psi ^-=F^-,\ x^{\prime }\in \mathbb{R }^{n-1},\ y<0,\quad \psi ^-(x^{\prime },0)=0, \end{aligned}$$

where $F^+:=F|_{y>0}$ and $F^-:=F|_{y<0}$. Let $\psi ^{\pm }\in W_p^{2+s}(\dot{\mathbb{R }}^n)$ be defined as

$$\begin{aligned} \psi ^{\pm }(x^{\prime },y):= {\left\{ \begin{array}{ll} \psi ^+(x^{\prime },y),&\ y>0,\\ \psi ^-(x^{\prime },y),&\ y<0. \end{array}\right.} \end{aligned}$$

Since $\psi ^+(x^{\prime },0)=\psi ^-(x^{\prime },0)=0$ and $[\![\rho \tilde{\psi }]\!]=[\![\rho ]\!]\psi ^{\pm }=0$, the shifted function $\tilde{\psi }:=\psi -\psi ^{\pm }$ solves the problem

$$\begin{aligned} \begin{aligned} \lambda \tilde{\psi }-\varDelta \tilde{\psi }&=0,\quad x^{\prime }\in \mathbb{R }^{n-1},\ y\in \dot{\mathbb{R }},\\ {[\![\rho \tilde{\psi }]\!]}&=0,\quad x^{\prime }\in \mathbb{R }^{n-1},\ y=0,\\ {[\![\partial _y \tilde{\psi }]\!]}&=\tilde{G},\quad x^{\prime }\in \mathbb{R }^{n-1},\ y=0, \end{aligned} \end{aligned}$$

(8.29)

where $\tilde{G}:=G-[\![\partial _y \psi ^{\pm }]\!]\in W_p^{1+s-1/p}(\mathbb{R }^{n-1})$. According to (8.6) and (8.7) the unique solution of (8.29) is given by

$$\begin{aligned} \tilde{\psi }(y)=-\frac{1}{\rho _1+\rho _2}L^{-1} {\left\{ \begin{array}{ll} \rho _1e^{-Ly}\tilde{G},&\ y>0,\\ \rho _2e^{Ly}\tilde{G},&\ y<0, \end{array}\right.} \end{aligned}$$

(8.30)

where $L:=(\lambda -\varDelta _{x^{\prime }})^{1/2}$ with domain $D(L)=H_p^1(\mathbb{R }^{n-1})$. Assume for a moment that $\tilde{G}\in W_p^{2-1/p}(\mathbb{R }^{n-1})$. Then it follows from semigroup theory and (8.30) that the solution of (8.29) satisfies the estimates

$$\begin{aligned} |\tilde{\psi }|_{H_p^3(\dot{\mathbb{R }}^n)}\le C|\tilde{G}|_{W_p^{2-1/p}(\mathbb{R }^{n-1})} \end{aligned}$$

as well as

$$\begin{aligned} |\tilde{\psi }|_{H_p^2(\dot{\mathbb{R }}^n)}\le C|\tilde{G}|_{W_p^{1-1/p}(\mathbb{R }^{n-1})}, \end{aligned}$$

where the constant $C>0$ does not depend on $\lambda \ge 1$. This can be seen as in the proof of Lemma 1. Applying the real interpolation method yields

$$\begin{aligned} |\tilde{\psi }|_{W_p^{2+s}(\dot{\mathbb{R }}^n)}\le C|\tilde{G}|_{W_p^{1+s-1/p}(\mathbb{R }^{n-1})}, \end{aligned}$$

for some $s\in (0,1]$ and if $\tilde{G}\in W_p^{1+s-1/p}(\mathbb{R }^{n-1})$. We have thus shown that the transmission problem (8.28) has a unique solution $\psi \in W_p^{2+s}(\dot{\mathbb{R }}^{n})$ if and only if $F\in W_p^s(\dot{\mathbb{R }}^n)$ and $G\in W_p^{1+s-1/p}(\mathbb{R }^{n-1})$. By perturbation theory, there exists $\lambda _0\ge 1$ such that (8.27) has a unique solution $q_k\in W_p^{2+s}(\mathbb{R }^n\backslash \varGamma ), s\in (0,1]$, satisfying the estimate

$$\begin{aligned} | q_k|_{W_p^{2+s}(\mathbb{R }^n\backslash \varGamma )}&\le C\Big (|f_k|_{W_p^s(\mathbb{R }^n\backslash \varGamma )}+|(\nabla q|\nabla \varphi _k)|_{W_p^s(\mathbb{R }^n\backslash \varGamma )} +| q\varDelta \varphi _k|_{W_p^s(\mathbb{R }^n\backslash \varGamma )}\\&\qquad +|[\![ q]\!]\partial _\nu \varphi _k|_{W_p^{s+1-1/p}(\varGamma )}\Big ), \end{aligned}$$

provided $\lambda \ge \lambda _0$. By the smoothness of $\varphi _k$ and by (8.24) we obtain the estimate

$$\begin{aligned} | q_k|_{W_p^{2+s}(\mathbb{R }^n\backslash \varGamma )}\le C\left(|f|_{W_p^s(\varOmega \backslash \varGamma )}+| q|_{W_p^{1+s}(\varOmega )}\right) \le C\left(|f|_{W_p^s(\varOmega \backslash \varGamma )}+|f|_{L_p(\varOmega )}\right)\!, \end{aligned}$$

valid for all $k\in \{2,\ldots ,N\}$ and $s\in (0,1]$. Since $\{\varphi _k\}_{k=0}^N$ is a partition of unity, we obtain

$$\begin{aligned} | q|_{W_p^{2+s}(\varOmega \backslash \varGamma )}\le \sum _{k=0}^N| q_k|_{W_p^{2+s}(\varOmega \backslash \varGamma )}\le C\left(|f|_{W_p^s(\varOmega \backslash \varGamma )}+|f|_{L_p(\varOmega )}\right)\!, \end{aligned}$$

showing that $q\in W_p^{2+s}(\varOmega \backslash \varGamma ), s\in (0,1]$. It is easy to extend this result to the case $\lambda \in [0,\lambda _0)$. To this end, let $f\in L_p(\varOmega )\cap W_p^s(\varOmega \backslash \varGamma )\cap R(A_\delta ), s>0$, where $A_\delta :H_p^2(\varOmega \backslash \varGamma )\rightarrow L_p(\varOmega )$ was defined at the beginning of Sect. 3. Note that $R(A_\delta )=\{f\in L_p(\varOmega ):(f|{\small 1}\!\!1_\rho )_2=0\}$ if $\delta =1$ and $\lambda =0$ and $R(A_\delta )=L_p(\varOmega )$ if either $\delta =0$ and $\lambda \ge 0$ or $\delta =1$ and $\lambda >0$. Consider the solution $q\in H_p^2(\varOmega \backslash \varGamma )$ of (8.23) with $\lambda \in [0,\lambda _0)$, which is well-defined thanks to Theorem 6 and which satisfies the estimate (8.24). Rewriting (8.23)$_1$ as

$$\begin{aligned} \lambda _0 q-\varDelta q=f+(\lambda _0-\lambda ) q, \end{aligned}$$

we may regard the new right-hand side $f+(\lambda _0-\lambda ) q$ as a given function, say $\tilde{f}\in W_p^s(\varOmega \backslash \varGamma ), s\in (0,1]$. The above result for (8.23) then yields the estimate

$$\begin{aligned} | q|_{W_p^{2+s}(\varOmega \backslash \varGamma )}&\le C\left(|\tilde{f}|_{W_p^s(\varOmega \backslash \varGamma )}+|\tilde{f}|_{L_p(\varOmega )}\right)\\&\le C\left(|f|_{W_p^s(\varOmega \backslash \varGamma )}+|f|_{L_p(\varOmega )}\right)\!, \end{aligned}$$

since

$$\begin{aligned} | q|_{W_p^s(\varOmega \backslash \varGamma )}\le C| q|_{H_p^2(\varOmega \backslash \varGamma )}\le C|f|_{L_p(\varOmega )}, \end{aligned}$$

by the smoothness of $q$ and by (8.24). If $s>1$ and $f\in L_p(\varOmega )\cap W_p^s(\varOmega \backslash \varGamma )$, then $q\in H_p^3(\varOmega \backslash \varGamma )$, since $f\in L_p(\varOmega )\cap H_p^1(\varOmega \backslash \varGamma )$. This additional regularity for $q$ and the preceding steps allow us to conclude that $q\in W_p^{2+s}(\varOmega \backslash \varGamma )$, at least for $s\in [1,2]$. By an obvious argument it follows that $q\in W_p^{2+s}(\varOmega \backslash \varGamma )$ for each fixed $s>0$, provided $f\in L_p(\varOmega )\cap W_p^{s}(\varOmega \backslash \varGamma )$. This yields the following result.

Theorem 9

Let $\varOmega \subset \mathbb{R }^n$ be a bounded domain with boundary $\partial \varOmega \in C^{2+s}$, let $1<p<\infty , s>0$ and $f\in L_p(\varOmega )\cap W_p^s(\varOmega \backslash \varGamma )$. Then the following assertions hold.

• 1. If $\delta =1$ and $\lambda =0$, then there exists a unique solution $q\in W_p^{2+s}(\varOmega \backslash \varGamma )\ominus \mathbb K {\small 1}\!\!1_\rho$ of (8.23), provided that $(f|{\small 1}\!\!1_\rho )=0$.

• 2. If either $\delta =1$ and $\lambda >0$ or $\delta =0$ and $\lambda \ge 0$, then there exists a unique solution $q\in W_p^{2+s}(\varOmega \backslash \varGamma )$ of (8.23).

If in addition $J=[0,a], f=f(t,x)$ and $f\in H_p^1(J;L_p(\varOmega )\cap W_p^{2+s}(\varOmega \backslash \varGamma ))$ s.t. $f(t,\cdot )\in R(A_\delta )$ for a.e. $t\in J$, then $q\in H_p^1(J;W_p^{2+s}(\varOmega \backslash \varGamma )\ominus N(A_\delta ))$.