Rigidity of critical points for a nonlocal Ohta--Kawasaki energy
- Dipierro, Serena
- Novaga, Matteo
- Valdinoci, Enrico
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.
- Nonlinearity, 30:4 (2017) pp. 1523--1535.