One-phase free boundary problems on Riemannian complexes

In this paper, we consider the one-phase free boundary problems on admissible, $N$-dimensional piecewise smooth Riemannian complexes. We first obtain the existence of the minimizers $u$ of the Bernoulli functional. Secondly, we prove the local Hölder continuity of this minimizer $u$ 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 that the free boundaries are sets of locally finite perimeters.