One-phase Free Boundary Problems on RCD Metric Measure Spaces

In this paper, we consider a vector-valued one phase Bernoullitype free boundary problem on a metric measure space (X, d, μ) with Riemannian curvature-dimension condition RCD(K,N). We first prove the existence and the local Lipschitz regularity of the solutions, provided the space X is non collapsed, i.e. μ 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 6 N − 3.

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