The Poisson equation involving surface measures

Abstract

We prove the (optimal) $W^{1,\infty }$-regularity of weak solutions to the equation $-\Delta u=Q{\mathcal{H}}^{n-1}└\Gamma$ 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.

40 pages, comments welcome!