The Poisson equation involving surface measures

Abstract

We prove the (optimal) $W^{1,\infty}$-regularity of weak solutions to the equation $-\Delta u = Q \; \mathcal{H}^{n-1} \llcorner \Gamma$ in a domain $\Omega \subset \mathbb{R}^n$ with Dirichlet boundary conditions, where $\Gamma \subset \subset \Omega$ is a compact (Lipschitz) manifold and $Q \in L^\infty(\Gamma)$. We also discuss optimality and necessity of the assumptions on $Q$ and $\Gamma$. Our findings can be applied to study the regularity of solutions for several free boundary problems, in particular the biharmonic Alt-Caffarelli Problem.