Generic properties in free boundary problems

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 $\{\ddot{I}{†}_t{\}}_{t\in (-1,1)}$. More precisely, we show that for a co-countable subset of $\{\ddot{I}{†}_t{\}}_{t\in (-1,1)}$, 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 $\{\ddot{I}{†}_t{\}}_{t\in (-1,1)}$.