The Poisson equation involving surface measures

TL;DR

This paper establishes optimal regularity results for solutions to a Poisson equation with surface measure data, with implications for free boundary problems like the biharmonic Alt-Caffarelli problem.

Contribution

It proves the optimal $W^{1, abla}^ ext{infty}$-regularity of solutions to a Poisson equation involving surface measures and discusses the optimality of assumptions.

Findings

Proves optimal regularity of solutions with surface measure data.

Analyzes the necessity of assumptions on surface and coefficient functions.

Applies results to free boundary problems such as the biharmonic Alt-Caffarelli problem.

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.