Theoretical analysis finds quantitative stability in nonlocal free energy under symmetry, suggesting potential unique solutions.
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 a nonlocal perimeter and convex potential. In this paper, the main result is a quantitative stability theorem for the nonlocal free energy under a symmetry assumption on the potential. In addition, several results that involve uniqueness, non-existence, and moduli estimates from the theory of crystals are also proven in the nonlocal context.
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Emanuel Indrei (2025) studied this question.
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