On elliptic equations involving surface measures

Abstract

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