Regularity of Lipschitz free boundaries for the thin one-phase problem

TL;DR

This paper proves that the free boundary in the thin one-phase problem is mostly smooth, with only small singular sets, and establishes regularity results for minimizers and viscosity solutions.

Contribution

It demonstrates the regularity and structure of free boundaries in the thin one-phase problem, including smoothness and measure properties, and extends results to viscosity solutions.

Findings

Free boundary has locally finite ^{n-1} measure.

Free boundary is a C^{2,lpha} surface except on a small singular set.

Lipschitz free boundaries of viscosity solutions are also C^{2,lpha}.

Abstract

We study regularity properties of the free boundary for the thin one-phase problem which consists of minimizing the energy functional $$\label{E} E(u,\Omega) = \int_\Omega |\nabla u|^2 dX + \mathcal{H}^n(\{u>0\} \cap \{x_{n+1} = 0\}), \quad \Omega \subset \R^{n+1},$$ among all functions $u\ge 0$ which are fixed on $\p \Omega$. We prove that the free boundary $F(u)=\p_{\R^n}\{u>0\}$ of a minimizer $u$ has locally finite $\mathcal{H}^{n-1}$ measure and is a $C^{2,\alpha}$ surface except on a small singular set of Hausdorff dimension $n-3$. We also obtain $C^{2,\alpha}$ regularity of Lipschitz free boundaries of viscosity solutions associated to this problem.