Polyharmonic equations involving surface measures

TL;DR

This paper investigates optimal regularity results for polyharmonic equations with surface measure sources, extending previous work and applying findings to the biharmonic Alt-Caffarelli problem in two dimensions.

Contribution

It extends regularity results for polyharmonic equations with surface measure sources and applies these to the biharmonic Alt-Caffarelli problem.

Findings

Established $W^{2m-1, ext{infinity}}$-regularity for polyharmonic equations with surface measure sources.

Derived $W^{3, ext{infinity}}$-regularity for solutions to the biharmonic Alt-Caffarelli problem in 2D.

Abstract

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