Calculus of Variations and Geometric Measure Theory

M. Bonacini - R. Cristoferi - I. Topaloglu

Minimality of polytopes in a nonlocal anisotropic isoperimetric problem

created by topaloglu1 on 21 Apr 2020
modified on 01 Apr 2021


Published Paper

Inserted: 21 apr 2020
Last Updated: 1 apr 2021

Journal: Nonlinear Analysis
Volume: 205
Pages: 112223
Year: 2021
Doi: 10.1016/

This is a post-peer-review, pre-copyedit version of an article published in Nonlinear Analysis.


We consider the minimization of an energy functional given by the sum of a crystalline perimeter and a nonlocal interaction of Riesz type, under volume constraint. We show that, in the small mass regime, if the Wulff shape of the anisotropic perimeter has certain symmetry properties, then it is the unique global minimizer of the total energy. In dimension two this applies to convex polygons which are reflection symmetric with respect to the bisectors of the angles. We further prove a rigidity result for the structure of (local) minimizers in two dimensions.