The Poisson equation involving surface measures

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