Generic properties in free boundary problems

TL;DR

This paper demonstrates that for a broad class of free boundary problems, minimizers are generically unique and have regular free boundaries in specific dimensions, with singular sets being smaller than expected for most boundary conditions.

Contribution

It establishes generic uniqueness and regularity of minimizers and free boundaries for the Alt-Caffarelli and Alt-Phillips functionals across various dimensions.

Findings

Generic uniqueness of minimizers for a large class of energies.

Almost all boundary data lead to smooth free boundaries in specified dimensions.

The singular set dimension is smaller than expected for most boundary conditions.

Abstract

In this work, we show the generic uniqueness of minimizers for a large class of energies, including the Alt-Caffarelli and Alt-Phillips functionals. We then prove the generic regularity of free boundaries for minimizers of the one-phase Alt-Caffarelli and Alt-Phillips functionals, for a monotone family of boundary data $\{\varphi_t\}_{t\in(-1,1)}$. More precisely, we show that for a co-countable subset of $\{\varphi_t\}_{t\in(-1,1)}$, minimizers have smooth free boundaries in $\mathbb{R}^5$ for the Alt-Caffarelli and in $\mathbb{R}^3$ for the Alt-Phillips functional. In general dimensions, we show that the singular set is one dimension smaller than expected for almost every boundary datum in $\{\varphi_t\}_{t\in(-1,1)}$.