Rigidity of critical points for a nonlocal Ohta–Kawasaki energy

Abstract

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.

Lamellar phase solutions for diblock copolymers with nonlocal diffusions

“…For more details of this model and the associated parabolic problem, the readers may consult [4,14,22,27,30,31,41]. As with the nonlocal diffusion, Dipierro-Novaga-Valdinoci [21] considered a nonlocal energy involving the fractional perimeter functional and established a rigidity result for critical points (not just minimizers) provided the volume is small in a certain sense.…”