Interface and mixed boundary value problems on $n$-dimensional polyhedral domains

Abstract

Let $\mu \in {\mathbb{Z}}_{+}$ be arbitrary. We prove a well-posedness result for mixed boundary value/interface problems of second-order, positive, strongly elliptic operators in weighted Sobolev spaces ${\mathcal{K}}_a^m(\Omega )$ on a bounded, curvilinear polyhedral domain $\Omega$ in a manifold $M$ of dimension $n$. The typical weight $\eta$ that we consider is the (smoothed) distance to the set of singular boundary points of $\partial \Omega$. Our model problem is $Pu:=-\mathrm{div}(A\nabla u)=f$, in $\Omega ,u=0$ on ${\partial }_D\Omega$, and $D_{\nu }^{P}u=0$ on ${\partial }_{\nu }\Omega$, where the function $A\ge ϵ>0$ is piece-wise smooth on the polyhedral decomposition $\bar{\Omega }={\cup }_j{\bar{\Omega }}_j$, and $\partial \Omega ={\partial }_D\Omega \cup {\partial }_N\Omega$ is a decomposition of the boundary into polyhedral subsets corresponding, respectively, to Dirichlet and Neumann boundary conditions. If there are no interfaces and no adjacent faces with Neumann boundary conditions, our main result gives an isomorphism $P:{\mathcal{K}}_{a+1}^{\mu +1}(\Omega )\cap \{u=0$ on ${\partial }_D\Omega ,D_{\nu }^{P}u=0$ on ${\partial }_N\Omega \}\to {\mathcal{K}}_{a-1}^{\mu -1}(\Omega )$ for $\mu \ge 0$ and $|a|<\eta$, for some $\eta >0$ that depends on $\Omega$ and $P$ but not on $\mu$. If interfaces are present, then we only obtain regularity on each subdomain ${\Omega }_j$. Unlike in the case of the usual Sobolev spaces, $\mu$ can be arbitrarily large, which is useful in certain applications. An important step in our proof is a regularity result, which holds for general strongly elliptic operators that are not necessarily positive. The regularity result is based, in turn, on a study of the geometry of our polyhedral domain when endowed with the metric $(dx/\eta {)}^2$, where $\eta$ is the weight (the smoothed distance to the singular set). The well-posedness result applies to positive operators, provided the interfaces are smooth and there are no adjacent faces with Neumann boundary conditions.