Optimal regularity for elliptic transmission problems including $C^1$ interfaces
Johannes Elschner, Joachim Rehberg, G. Schmidt · Interfaces and Free Boundaries Mathematical Analysis Computation and Applications · 2007
We prove an optimal regularity result for elliptic operators $- abla \cdot \mu abla:W^{1,q}\_0 \rightarrow W^{-1,q}$ 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.