On the regularity of stationary points of a nonlocal isoperimetric problem

TL;DR

This paper proves the $C^{3,eta}$ regularity of stationary points in a nonlocal isoperimetric problem, showing they satisfy Euler-Lagrange equations classically and have negligible singular sets, extending previous regularity results.

Contribution

It establishes higher regularity of stationary points and their boundary behavior in nonlocal isoperimetric problems, relaxing previous regularity assumptions.

Findings

Stationary points are $C^{3,eta}$ regular.

Euler-Lagrange equations hold classically on the reduced boundary.

Singular set has zero $(n-1)$-dimensional Hausdorff measure.

Abstract

In this article we establish $C^{3,\alpha}$-regularity of the reduced boundary of stationary points of a nonlocal isoperimetric problem in a domain $\Omega \subset \mathbb{R}^n$. In particular, stationary points satisfy the corresponding Euler-Lagrange equation classically on the reduced boundary. Moreover, we show that the singular set has zero $(n-1)$-dimensional Hausdorff measure. This complements the results in Choksi & Sternberg, in which the Euler-Lagrange equation was derived under the assumption of $C^2$-regularity of the topological boundary and the results in Sternberg & Topaloglu in which the authors assume local minimality. In case $\Omega$ has non-empty boundary, we show that stationary points meet the boundary of $\Omega$ orthogonally in a weak sense, unless they have positive distance to it.