Optimal regularity for elliptic transmission problems including $C^1$ interfaces

Abstract

We prove an optimal regularity result for elliptic operators $-\nabla \cdot \mu \nabla :W_0^{1,q}\to W^{-1,q}$ for a $q>3$ in the case when the coefficient function $\mu$ has a jump across a $C^1$ interface and is continuous elsewhere. A counterexample shows that the $C^1$ condition cannot be relaxed in general. Finally, we draw some conclusions for corresponding parabolic operators.