On the regularity of stationary points of a nonlocal isoperimetric problem

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.