Regularity of Lipschitz free boundaries in the alt-Caffarelli problem

Abstract

In this work we study the regularity of Lipschitz free boundaries in the Alt-Caffarelli problem. We prove that Lipschitz free boundaries are $C^{1, \alpha}$ by exploiting the rescaling invariance of the problem and the initial Lipschitz regularity of the boundary. Moreover, we also show that $C^{1, \alpha}$ boundaries are smooth, which combined with the previous result implies that Lipschitz free boundaries are smooth.