Regularity for $C^{1,\alpha }$ Interface Transmission Problems

Abstract

We study the existence, uniqueness, and optimal regularity of solutions to transmission problems for harmonic functions with $C^{1,\alpha }$ interfaces. For this, we develop a novel geometric stability argument based on the mean value property.

Appendix

A special Lipschitz domain $\Omega$ in $\mathbb {R}^n$ is a set of the form

$$\begin{aligned} \Omega =\{(x',x_n)\in \mathbb {R}^n:x'\in \mathbb {R}^{n-1},\,x_n>\psi (x')\}, \end{aligned}$$

where $\psi \in {\text {Lip}}(\mathbb {R}^{n-1})$, that is, there exists $M>0$ such that

$$\begin{aligned} |\psi (x')-\psi (y')|\le M|x'-y'|\qquad \hbox {for all}~x',y'\in \mathbb {R}^{n-1}. \end{aligned}$$

In other words, $\Omega$ is the set of points lying above the graph of a Lipschitz function $\psi$. Then, by Rademacher’s Theorem, $\psi$ is Fréchet differentiable almost everywhere with $\Vert \nabla \psi \Vert _{L^\infty (\mathbb {R}^{n-1})}\le M$. On $\partial \Omega$ we thus have

$$\begin{aligned} \mathrm{d}H^{n-1}\big |_{\partial \Omega }=\sqrt{1+|\nabla \psi (x')|^2}\,\mathrm{d}x'\quad \hbox {and}\quad \nu (x',\psi (x'))=\frac{(\nabla \psi (x'),-1)}{\sqrt{1+|\nabla \psi (x')|^2}}, \end{aligned}$$

where $x=(x',\psi (x'))\in \partial \Omega$. For a measurable function f on $\partial \Omega$, we have

$$\begin{aligned} \int _{\partial \Omega }f(x)\,\mathrm{d}H^{n-1}= \int _{\mathbb {R}^{n-1}}f(x',\psi (x'))\sqrt{1+|\nabla \psi (x')|^2}\,\mathrm{d}x'. \end{aligned}$$

For more details see [6, 11].

A bounded Lipschitz domain in $\mathbb {R}^n$ is a bounded domain $\Omega$ such that the boundary $\partial \Omega$ can be covered by finitely many open balls $B_j$ in $\mathbb {R}^n$, $j=1,\ldots ,J$, centered at $\partial \Omega$, such that

$$\begin{aligned} B_j\cap \Omega =B_j\cap \Omega _j,\quad j=1,\ldots ,J, \end{aligned}$$

where $\Omega _j$ are rotations of suitable special Lipschitz domains given by Lipschitz functions $\psi _j$. One may then assume that $\partial \Omega \cap B_j$ can be represented in local coordinates by $x_n=\psi _j(x')$, where $\psi _j$ is a Lipschitz function on $\mathbb {R}^{n-1}$ with $\psi _j(0')=0$. Recall also that if $\psi$ is a Lipschitz function defined on an set $A\subset \mathbb {R}^{n-1}$, with Lipschitz constant M, then there exists an extension $\overline{\psi }:\mathbb {R}^{n-1}\rightarrow \mathbb {R}$ of $\psi$ such that $\overline{\psi }=\psi$ on A and the Lipschitz constant of $\overline{\psi }$ does not exceed M, see [6].

Let $\Omega _0=\Omega \cap \big (\bigcup _{j=1}^JB_j\big )^c$. A partition of unity $\{\xi _j\}_{j=0}^J$ subordinated to $\{\Omega _0,B_1,\ldots ,B_J\}$ is a family of nonnegative smooth functions $\xi _j$ on $\mathbb {R}^n$ such that

$$\begin{aligned} \xi _0\in C^\infty _c(\Omega _0)\quad \xi _j\in C^\infty _c(B_j),~j=1,\ldots ,J\quad \hbox {and}\quad \sum \nolimits _{j=0}^J\xi _j(x)=1\quad \hbox {for all}~x\in \overline{\Omega }. \end{aligned}$$

It follows that $0\le \xi _j\le 1$, $j=0,1,\ldots ,J$. Obviously the family $\{\xi _j\}_{j=1}^J$ is a partition of unity subordinated to the open cover $\{B_1,\ldots ,B_J\}$ of $\partial \Omega$ and $\sum _{j=1}^J\xi _j(x)=1$ for every $x\in \partial \Omega$.

Let $f:\Gamma \rightarrow \mathbb {R}$ be a measurable function, where $\Gamma =\partial \Omega$ is the boundary of a bounded Lipschitz domain $\Omega$. Consider the balls $B_j$, $j=1, \ldots ,J$, that cover $\Gamma$ as above, and the corresponding Lipschitz functions $\psi _j:\mathbb {R}^{n-1}\rightarrow \mathbb {R}$. Let $\{\xi _j\}_{j=1}^J$ be a smooth partition of unity subordinated to the open cover $\{B_j\}_{j=1}^J$ of $\Gamma$. Then

$$\begin{aligned} \int _\Gamma f\,\mathrm{d}H^{n-1}=\sum _{j=1}^J\int _\Gamma \xi _jf\,\mathrm{d}H^{n-1}=\sum _{j=1}^J\int _{B_j\cap \Gamma }\xi _jf\,\mathrm{d}H^{n-1}. \end{aligned}$$

Let us consider each one of the terms in the sum above separately. We study the following situation: let B be a ball and let $\bar{f}:B\cap \Gamma \rightarrow \mathbb {R}$ of compact support in $B\cap \Gamma$. Let $\psi :\mathbb {R}^{n-1}\rightarrow \mathbb {R}$ be a Lipschitz function such that $\psi (B_1')=B\cap \Gamma$. Then, by extending trivially $\bar{f}$ to the rest of the graph of $\psi$ and using the coarea formula [6, 11],

$$\begin{aligned} \int _{B\cap \Gamma }\bar{f}\,\mathrm{d}H^{n-1}&=\int _{\psi (B_1')}\bar{f}\,\mathrm{d}H^{n-1}=\int _{\psi (\mathbb {R}^{n-1})}\bar{f}\,\mathrm{d}H^{n-1}\\&=\int _{\mathbb {R}^{n-1}}\bar{f}(y',\psi (y'))\sqrt{1+|\nabla \psi (y')|^2}\,\mathrm{d}y' \\&=\int _{B_1'}\bar{f}(y',\psi (y'))\sqrt{1+|\nabla \psi (y')|^2}\,\mathrm{d}y'. \end{aligned}$$

Remark. The contents of this work are part of the second author’s PhD dissertation. She presented these results at the AMS Fall Central Sectional Meeting at University of Michigan, Ann Arbor (Oct. 2018), the Barcelona Analysis Conference at Universitat de Barcelona (Jun. 2019), and the Midwest Geometry Conference at Iowa State University (Sep. 2019).