cvgmt: Optimal regularity for quasiminimal sets of codimension one in $\mathbb{R}^2$ and $\mathbb{R}^3$

Calculus of Variations and Geometric Measure Theory

C. Labourie - Y. Teplitskaya

Optimal regularity for quasiminimal sets of codimension one in $\mathbb{R}^2$ and $\mathbb{R}^3$

created by teplitskaya1 on 12 Nov 2024
modified by labourie on 18 Nov 2024

[BibTeX]

preprint

Inserted: 12 nov 2024
Last Updated: 18 nov 2024

Year: 2024

Abstract:

Quasiminimal sets are sets for which a pertubation can decrease the area but only in a controlled manner. We prove that in dimensions $2$ and $3$, such sets separate a locally finite family of local John domains. Reciprocally, we show that this condition is sufficient for quasiminimality in every dimension. In addition, we show that quasiminimal sets locally separate the space in two components, except at isolated points in $\mathbf{R}^2$ or out of a subset of dimension strictly less than $N-1$ in $\mathbf{R}^N$.