One-phase Free Boundary Problems on RCD Metric Measure Spaces

Abstract

In this paper, we consider a vector-valued one-phase Bernoulli-type free boundary problem on a metric measure space $(X,d,\hat{I}¼)$ 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. $\hat{I}¼$ 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 $⩽N-3$.

58 pages. Comments are welcome