A note on the dimension of the singular set in free interface problems

TL;DR

This paper investigates the size of the singular set in free interface problems, demonstrating its porosity and establishing that its Hausdorff and Minkowski dimensions are less than n-1.

Contribution

It proves the porosity of the singular set and bounds its Hausdorff and Minkowski dimensions below n-1, advancing understanding of free interface regularity.

Findings

Singular set is porous in free interface problems.

Hausdorff and Minkowski dimensions of the singular set are less than n-1.

Provides geometric measure theory bounds on singular set size.

Abstract

The aim of this note is to investigate the size of the singular set of a general class of free interface problems. We show porosity of the singular set, obtaining as a corollary that both its Hausdorff and Minkowski dimensions are strictly smaller than $n-1$.