Calculus of Variations and Geometric Measure Theory

S. Dipierro - M. Novaga - E. Valdinoci

Rigidity of critical points for a nonlocal Ohta-Kawasaki energy

created by novaga on 25 Apr 2016
modified on 31 Aug 2017


Published Paper

Inserted: 25 apr 2016
Last Updated: 31 aug 2017

Journal: Nonlinearity
Volume: 30
Number: 4
Pages: 1523-1535
Year: 2017


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.