Rectifiability and almost everywhere uniqueness of the blow-up for the vectorial Bernoulli free boundaries

TL;DR

This paper proves rectifiability and almost everywhere uniqueness of blow-ups for minimizers of the vectorial Alt-Caffarelli functional, advancing understanding of free boundary regularity and singular set structure.

Contribution

It establishes rectifiability of the singular set using recent techniques and proves almost everywhere uniqueness of blow-ups via a novel application of monotonicity formulas.

Findings

The two-phase singular set is rectifiable.

Blow-up is unique almost everywhere on the singular set.

The results extend regularity theory for vectorial free boundary problems.

Abstract

We prove that for minimizers of the vectorial Alt-Caffarelli functional the two-phase singular set of the free boundary is rectifiable and the blow-up is unique almost everywhere on it. While the first conclusion is an application of the recent techniques developed by Naber and Valtorta, the uniqueness part follows from the rectifiability and a new application of the Alt-Caffarelli-Friedman monotonicity formula.