One-Phase Free Boundary Problems on Riemannian Complexes

Yu. Peng, Hui-Chun Zhang, Xi-Ping Zhu

Department of Mathematics, Sun Yat-sen University, Guangzhou, 510275, China

Abstract: 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.

Keywords: Bernoulli functional, free boundary problem, Riemannian complex, Lipschitz regularity.

Received: May 5, 2025
Revised: June 2, 2025
Accepted: June 18, 2025

DOI: 10.4213/tm4468

 English version: Proceedings of the Steklov Institute of Mathematics, 2025, 330, 270–287

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