Polyharmonic equations involving surface measures

Abstract

This article studies (optimal) $W^{2m-1,\infty }$-regularity for the polyharmonic equation

where $\Gamma$ 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. As an application, we derive (optimal) $W^{3,\infty }$-regularity for solutions of the biharmonic Alt–Caffarelli problem in two dimensions.