preprint

Inserted: 21 jul 2025

Year: 2025

Abstract:

We study the regularity of quasi-minimal sets (in the sense of David and Semmes) with a boundary condition, which can be interpreted as quasi-minimizers of Plateau's problem in co-dimension one. For these Plateau-quasi-minimizers, we establish the optimal regularity, which is a characterization by bi-John domains with Ahlfors regular boundaries. This requires to investigate the Ahlfors regularity and also the uniform rectifiability of those sets, up to the boundary.