Optimal regularity up to the boundary for Plateau-quasi-minimizers

TL;DR

This paper proves optimal boundary regularity for Plateau-quasi-minimizers, showing they are characterized by bi-John domains with Ahlfors regular boundaries, advancing understanding of their geometric structure.

Contribution

It establishes the optimal boundary regularity of Plateau-quasi-minimizers and characterizes them via bi-John domains with Ahlfors regular boundaries.

Findings

Proves boundary regularity up to the boundary for Plateau-quasi-minimizers.

Shows these sets are characterized by bi-John domains.

Establishes Ahlfors regularity and uniform rectifiability up to the boundary.

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.