Polyharmonic equations involving surface measures

This article studies (optimal) $W^{2m-1,\mathrm{\infty }}$-regularity for the polyharmonic equation $(-\mathrm{\Delta }{)}^{m}u=Q{\mathcal{H}}^{n-1}\mathrm{\Gamma }$, where $\mathrm{\Gamma }$ is a (suitably regular) $(n-1)$-dimensional submanifold of ${\mathbb{R}}^n$, ${\mathcal{H}}^{n-1}$ is the Hausdorff measure, and $Q$ is some suitably regular density. We extend findings in [9], where the second-order equation $-\mathrm{d}\mathrm{i}\mathrm{v}(A(x)\mathrm{\nabla }u)=Q{\mathcal{H}}^{n-1}\mathrm{\Gamma }$ is studied. As an application we derive (optimal) $W^{3,\mathrm{\infty }}$-regularity for solutions of the biharmonic Alt-Caffarelli problem in two dimensions.