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We prove the (optimal) $W^{1,\infty}$-regularity of weak solutions to theequation $-\Delta u = Q \; \mathcal{H}^{n-1} \llcorner \Gamma$ in a domain$\Omega \subset \mathbb{R}^n$ with Dirichlet boundary conditions, where $\Gamma\subset \subset \Omega$ is a compact (Lipschitz) manifold and $Q \inL^\infty(\Gamma)$.  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 severalfree boundary problems, in particular the biharmonic Alt-Caffarelli Problem.

The Poisson equation involving surface measures