Non-Local Isoperimetric Problems

TL;DR

This paper studies nonlocal isoperimetric problems involving differences of fractional perimeters, establishing conditions for minimizers, especially balls, and analyzing existence, regularity, and open questions for large volumes.

Contribution

It characterizes volume-constrained minimizers for a difference of fractional perimeters and proves existence and regularity results, extending known results for the fractional isoperimetric problem.

Findings

Balls are unique minimizers for small volumes when $s<t$.

Existence and regularity of minimizers are established for all $s,t$.

Open problem remains for large volume minimizers.

Abstract

We characterize the volume-constrained minimizers of a nonlocal free energy given by the difference of the $t$-perimeter and the $s$-perimeter, with $s$ smaller than $t$. Exploiting the quantitative fractional isoperimetric inequality, we show that balls are the unique minimizers if the volume is sufficiently small, depending on $t-s$, while the existence vs. nonexistence of minimizers for large volumes remains open. We also consider the corresponding isoperimetric problem and prove existence and regularity of minimizers for all $s,\,t$. When $s=0$ this problem reduces to the fractional isoperimetric problem, for which it is well known that balls are the only minimizers.