One-phase Free Boundary Problems on RCD Metric Measure Spaces

Abstract

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 that 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 N − 3. Contents 1. Introduction 1.1. The Bernoulli-type free boundary problems on Euclidean spaces 1.2. Free boundary problems in RCD-spaces and the main results 2. Preliminaries 2.1. RCD(K, N ) metric measure spaces and their calculus 2.2. Non-collapsed RCD(K, N ) metric measure spaces 2.3. Sets of finite perimeter and the reduced boundary 3. Existence of a minimizer 4. Hölder continuity of local minimizers 5. Lipschitz continuity of local minimizers 5.1. Mean value inequality 5.2. Lipschitz continuity of local minimizers of J Q 6. Local finiteness of perimeter for the free boundary 6.1. Nondegeneracy 6.2. Density estimates near the free boundary 6.3. Local finiteness of perimeter 7. Compactness and the Euler-Lagrange equation 8. Regularity of the free boundary Appendix A. Weiss-type monotonicity on cones References