# On the regularity of stationary points of a nonlocal isoperimetric problem

### Dorian Goldman

The New York Times Company, New York City, USA### Alexander Volkmann

Rocket Internet SE, Berlin, Germany

## 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 [4] in which the Euler–Lagrange equation was derived under the assumption of $C^2$-regularity of the topological boundary and the results in [27] 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.

## Cite this article

Dorian Goldman, Alexander Volkmann, On the regularity of stationary points of a nonlocal isoperimetric problem. Interfaces Free Bound. 17 (2015), no. 4, pp. 539–553

DOI 10.4171/IFB/353