One-phase Free Boundary Problems on RCD Metric Measure Spaces

TL;DR

This paper studies a one-phase Bernoulli-type free boundary problem on RCD metric measure spaces, establishing existence, regularity, and geometric properties of solutions and their free boundaries.

Contribution

It proves existence and Lipschitz regularity of solutions on non-collapsed RCD spaces and characterizes the free boundary as a topological manifold with controlled singular set.

Findings

Solutions exist and are locally Lipschitz continuous.

Free boundary is an (N-1)-dimensional topological manifold outside a small singular set.

The singular set has Hausdorff dimension at most N-3.

Abstract

In this paper, we consider a vector-valued one-phase Bernoulli-type free boundary problem on a metric measure space $(X,d,\mu)$ with Riemannian curvature-dimension condition $RCD(K,N)$. We first prove the existence and the local Lipschitz regularity of the solutions, provided that the space $X$ is non collapsed, i.e. $\mu$ is the $N$-dimensional Hausdorff measure of $X$. And then we show that the free boundary of the solutions is an $(N-1)$-dimensional topological manifold away from a relatively closed subset of Hausdorff dimension $\leqslant N-3$.