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 -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 [4] in which the Euler–Lagrange equation was derived under the assumption of -regularity of the topological boundary and the results in [27] 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. 539–553

DOI 10.4171/IFB/353