On elliptic equations involving surface measures

TL;DR

This paper establishes optimal Lipschitz regularity for solutions to elliptic PDEs with measure-valued surface measures, relevant for free boundary problems and involving regularity assumptions on coefficients and hypersurfaces.

Contribution

It proves optimal Lipschitz regularity for weak solutions of elliptic equations with surface measure data, under specific regularity conditions on the hypersurface and coefficients.

Findings

Solutions are Lipschitz continuous under optimal conditions.

The regularity assumptions on data are shown to be sharp.

The results apply to free boundary problems involving surface measures.

Abstract

We show optimal Lipschitz regularity for very weak solutions of the (measure-valued) elliptic PDE $-\mathrm{div}(A(x) \nabla u) = Q \; \mathcal{H}^{n-1} \llcorner \Gamma$ 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.