The nonlocal Almgren problem

TL;DR

This paper studies the convexity and stability of minimizers in a nonlocal free energy problem, establishing a quantitative stability theorem and extending crystal theory results to the nonlocal setting.

Contribution

It introduces a quantitative stability theorem for the nonlocal free energy with symmetric potentials and extends key crystal theory results to nonlocal contexts.

Findings

Proves a stability theorem for nonlocal free energy minimizers.

Establishes uniqueness and non-existence results in the nonlocal setting.

Provides moduli estimates related to crystal theory in nonlocal problems.

Abstract

In the nonlocal Almgren problem, the goal is to investigate the convexity of a minimizer under a mass constraint via a nonlocal free energy generated with some nonlocal perimeter and convex potential. In the paper, the main result is a quantitative stability theorem for the nonlocal free energy assuming symmetry on the potential. In addition, several results that involve uniqueness, non-existence, and moduli estimates from the theory for crystals are proven also in the nonlocal context.