On the regularity of stationary points of a nonlocal isoperimetric problem

In this article we establish C-regularity of the reduced boundary of stationary points of a nonlocal isoperimetric problem in a domain Ω ⊂ R. 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-regularity of the topological boundary and the results in [27] in which the authors assume local minimality. In case Ω has non-empty boundary, we show that stationary points meet the boundary of Ω orthogonally in a weak sense, unless they have positive distance to it.