On elliptic equations involving surface measures

We show optimal Lipschitz regularity for very weak solutions of the (measure-valued) elliptic PDE $-\mathrm{d}\mathrm{i}\mathrm{v}(A(x)\mathrm{\nabla }u)=Q{\mathcal{H}}^{n-1}\mathrm{\Gamma }$ in a smooth domain $\mathrm{\Omega }\subset {\mathbb{R}}^n$. Here $\mathrm{\Gamma }$ is a $C^{1,\alpha }$-regular hypersurface, $Q\in C^{0,\alpha }$ is a density on $\mathrm{\Gamma }$, 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 $\mathrm{\Gamma }$. As such it plays an important role in several free boundary problems, as we shall discuss.