Optimal Sobolev regularity for second order divergence elliptic operators on domains with buried boundary parts

Abstract

We study the regularity of solutions of elliptic second order boundary value problems on a bounded domain $Omega$ in $mathbb R^3$. The coefficients are not necessarily continuous and the boundary conditions may be mixed, i.e. Dirichlet on one part $D$ of the boundary and Neumann on the complementing part. The peculiarity is that $D$ is partly `buried' in $Omega$ in the sense that the topological interior of $Omega cup D$ properly contains $Omega$. The main result is that the singularity of the solution along the border of the buried contact behaves exactly as the singularity for the solution of a mixed boundary value problem along the border between the Dirichlet and the Neumann boundary part.