Uniqueness of critical points of the anisotropic isoperimetric problem for finite perimeter sets

TL;DR

This paper proves that, for a smooth elliptic integrand, the only volume-critical points of the anisotropic surface energy are finite unions of disjoint Wulff shapes with equal radii, establishing a uniqueness result.

Contribution

It establishes the uniqueness of critical points as unions of disjoint Wulff shapes with equal radii for the anisotropic isoperimetric problem with smooth integrands.

Findings

Finite unions of disjoint Wulff shapes are the only critical points.

Critical points occur under volume constraint.

Results hold for integrands of class C^3.

Abstract

Given an elliptic integrand of class $ \mathscr{C}^{3} $, we prove that finite unions of disjoint open Wulff shapes with equal radii are the only volume-constrained critical points of the anisotropic surface energy among all sets with finite perimeter and reduced boundary almost equal to its closure.