WIAS Preprint No. 2252, (2016)

Rigidity of critical points for a nonlocal Ohta--Kawasaki energy


  • Dipierro, Serena
  • Novaga, Matteo
  • Valdinoci, Enrico
    ORCID: 0000-0001-6222-2272

2010 Mathematics Subject Classification

  • 49Q10 49Q20 35B38 58J70


  • Otha-Kawasaki functional, long-range interactions, symmetry results, critical point




We investigate the shape of critical points for a free energy consisting of a nonlocal perimeter plus a nonlocal repulsive term. In particular, we prove that a volume-constrained critical point is necessarily a ball if its volume is sufficiently small with respect to its isodiametric ratio, thus extending a result previously known only for global minimizers.

We also show that, at least in one-dimension, there exist critical points with arbitrarily small volume and large isodiametric ratio. This example shows that a constraint on the diameter is, in general, necessary to establish the radial symmetry of the critical points.

Appeared in

  • Nonlinearity, 30:4 (2017) pp. 1523--1535.

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