One-Phase Free Boundary Problems on Riemannian Complexes

We consider one-phase free boundary problems on admissible $$N$$ -dimensional piecewise smooth Riemannian complexes. We first obtain the existence of minimizers $$u$$ of the Bernoulli functional. Second, we prove the local Hölder continuity of these minimizers $$u$$ and the local Lipschitz continuity of $$u$$ away from the $$(N-2)$$ -skeleton. Finally, we get the nondegeneracy of the minimizers near the free boundary and show that the free boundaries are sets of locally finite perimeters.