Optimal Sobolev Regularity for Second Order Divergence Elliptic Operators on Domains with Buried Boundary Parts

We study the regularity of solutions of elliptic second order boundary value problems on a bounded domain $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$ in the sense that the topological interior of $\mathrm{\Omega }\cup D$ properly contains $\mathrm{\Omega }$. 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.