A note on Bernoulli type free boundary problem on collapsed RCD(K,N)-spaces

In this paper, we investigate Bernoulli type free boundary problem on collapsed RCD(K,N)-spaces. We prove the existence of minimizers and prove the local Lipschitz continuity of minimizers provided that the negative part is locally Lipschitz continuous. In particular, we prove the local Lipschitz continuity of minimizers for the one-phase problem (i.e. when the solution is non-negative). And then we prove that the free boundaries of minimizers have locally finite perimeter. We emphasize that the proof in this paper applies to collapsed RCD(K,N)-spaces and does not rely on the non-collapsed condition.