In this article we establish -regularity of the reduced boundary of stationary points of a nonlocal isoperimetric problem in a domain . In particular, stationary points satisfy the corresponding Euler–Lagrange equation classically on the reduced boundary. Moreover, we show that the singular set has zero -dimensional Hausdorff measure. This complements the results in  in which the Euler–Lagrange equation was derived under the assumption of -regularity of the topological boundary and the results in  in which the authors assume local minimality. In case has non-empty boundary, we show that stationary points meet the boundary of 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. 539–553DOI 10.4171/IFB/353