Dimensional estimates for singular sets in geometric variational problems with free boundaries

TL;DR

This paper proves that the singular sets in certain free boundary problems are negligible in measure, leading to regularity results especially in three dimensions.

Contribution

It establishes measure estimates for singular sets in anisotropic free boundary problems, extending regularity results to capillarity and related problems.

Findings

Singular sets are $ ext{H}^{n-3}$-negligible in these problems.

Results imply regularity in three-dimensional cases.

Applicable to capillarity and anisotropic geometric variational problems.

Abstract

We show that singular sets of free boundaries arising in codimension one anisotropic geometric variational problems are $\mathcal H ^{n-3}$-negligible, where $n$ is the ambient space dimension. In particular our results apply to capillarity type problems, and establish everywhere regularity in the three-dimensional case.