Boundary Regularity for the Free Boundary in the One-phase Problem

TL;DR

This paper proves that the free boundary in a Bernoulli one-phase problem is $C^{1,1/2}$ regular near the fixed boundary by connecting it to a Signorini-type obstacle problem.

Contribution

It establishes boundary regularity of the free boundary in the Bernoulli problem using a novel connection to obstacle problems.

Findings

Free boundary is $C^{1,1/2}$ regular near the fixed boundary.

Relates free boundary behavior to Signorini-type obstacle problem.

Provides a new approach to boundary regularity in free boundary problems.

Abstract

We consider the Bernoulli one-phase free boundary problem in a domain $\Omega$ and show that the free boundary $F$ is $C^{1,1/2}$ regular in a neighborhood of the fixed boundary $\partial \Omega$. We achieve this by relating the behavior of $F$ near $\partial \Omega$ to a Signorini-type obstacle problem.