Centered spheres minimize the weighted free energy for all volumes only under extra monotonicity of both weights; otherwise the second-variation condition psi''+g' >= 0 can fail to select global minimizers.
The one-dimensional equilibrium shape of a crystal
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abstract
Optimizing the free energy under a mass constraint may generate a convex crystal subject to assumptions on the potential $g(0)=0$, $g \ge 0$. The general problem classically attributed to Almgren is to infer if this is the case assuming the sub-level sets of g are convex. The theorem proven in the paper is that in one dimension the answer is positive.
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Free energy minimizers with radial densities: classification and quantitative stability
Centered spheres minimize the weighted free energy for all volumes only under extra monotonicity of both weights; otherwise the second-variation condition psi''+g' >= 0 can fail to select global minimizers.