On elliptic equations involving surface measures

We show optimal Lipschitz regularity for very weak solutions of the (measure-valued) elliptic PDE $-\mathrm{div}(A(x) \nabla u) = Q \; \mathcal{H}^{n-1} \llcorner \Gamma$ in a smooth domain $\Omega \subset \mathbb{R}^n$. Here $\Gamma$ is a $C^{1,\alpha}$-regular hypersurface, $Q\in C^{0,\alpha}$ is a density on $\Gamma$, and the coefficient matrix $A$ is symmetric, uniformly elliptic and $W^{1,q}$-regular $(q>n)$. We also discuss optimality of these assumptions on the data. The equation can be understood as a special coupling of two $A$-harmonic functions with an interface $\Gamma$. As such it plays an important role in several free boundary problems, as we shall discuss.