In one dimension, under g(0)=0, g>=0 and convex sub-level sets, every minimizer of the free energy with prescribed mass is an interval and a minimizer always exists.
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The one-dimensional equilibrium shape of a crystal
In one dimension, under g(0)=0, g>=0 and convex sub-level sets, every minimizer of the free energy with prescribed mass is an interval and a minimizer always exists.