Existence, Lipschitz regularity, and almost-(N-1)-manifold structure of free boundaries are proved for one-phase Bernoulli problems on non-collapsed RCD(K,N) spaces.
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Minimizers of the multiphase vectorial Bernoulli functional exist, are locally Lipschitz, avoid triple points on the free boundary, and have C^{1,η} regularity near two-phase and branching points.
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One-phase Free Boundary Problems on RCD Metric Measure Spaces
Existence, Lipschitz regularity, and almost-(N-1)-manifold structure of free boundaries are proved for one-phase Bernoulli problems on non-collapsed RCD(K,N) spaces.
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On a Multiphase Vectorial Bernoulli Free Boundary Problem
Minimizers of the multiphase vectorial Bernoulli functional exist, are locally Lipschitz, avoid triple points on the free boundary, and have C^{1,η} regularity near two-phase and branching points.