Polyharmonic equations involving surface measures
Marius Müller · Interfaces and Free Boundaries Mathematical Analysis Computation and Applications · 2023
This article studies (optimal) W^{2m-1,\infty} -regularity for the polyharmonic equation (-\Delta)^m u = Q \; \mathcal{H}^{n-1} {\huge \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. As an application, we derive (optimal) W^{3,\infty} -regularity for solutions of the biharmonic Alt–Caffarelli problem in two dimensions.