Polyharmonic equations involving surface measures

Abstract

This article studies (optimal) $W^{2m-1,\infty }$-regularity for the polyharmonic equation $(-\Delta {)}^{m}u=Q{\mathcal{H}}^{n-1}└\Gamma$, 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. We extend findings in [9], where the second-order equation $-\mathrm{div}(A(x)\nabla u)=Q{\mathcal{H}}^{n-1}└\Gamma$ is studied. As an application we derive (optimal) $W^{3,\infty }$-regularity for solutions of the biharmonic Alt-Caffarelli problem in two dimensions.

15 pages, 1 figure, comments welcome!