JournalsifbVol. 17, No. 4pp. 539–553

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
On the regularity of stationary points of a nonlocal isoperimetric problem cover
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Abstract

In this article we establish C3,αC^{3, \alpha}-regularity of the reduced boundary of stationary points of a nonlocal isoperimetric problem in a domain ΩRn\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 (n1)(n–1)-dimensional Hausdorff measure. This complements the results in [4] in which the Euler–Lagrange equation was derived under the assumption of C2C^2-regularity of the topological boundary and the results in [27] in which the authors assume local minimality. In case Ω\Omega has non-empty boundary, we show that stationary points meet the boundary of Ω\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