Rectifiability and uniqueness of blow-ups for points with positive Alt-Caffarelli-Friedman limit

TL;DR

This paper proves the rectifiability and uniqueness of blow-ups at points with positive Alt-Caffarelli-Friedman limit, advancing understanding of interface regularity in free boundary problems.

Contribution

It establishes rectifiability and blow-up uniqueness for points with positive ACF limit, using Naber-Valtorta framework and a new quantitative ACF monotonicity estimate.

Findings

The interface where ACF is positive is rectifiable.

Almost every such point has unique blowups converging to linear functions.

Results have applications to free boundary problems.

Abstract

We study the regularity of the interface between the disjoint supports of a pair of nonnegative subharmonic functions. The portion of the interface where the Alt-Caffarelli-Friedman (ACF) monotonicity formula is asymptotically positive forms an $\mathcal{H}^{n-1}$-rectifiable set. Moreover, for $\mathcal{H}^{n-1}$-a.e. such point, the two functions have unique blowups, i.e. their Lipschitz rescalings converge in $W^{1,2}$ to a pair of nondegenerate truncated linear functions whose supports meet at the approximate tangent plane. The main tools used include the Naber-Valtorta framework and our recent result establishing a sharp quantitative remainder term in the ACF monotonicity formula. We also give applications of our results to free boundary problems.