Abstract
We establish sufficient conditions for a function on the torus to be equal to its Steiner symmetrization and apply the result to so-called volume-constrained minimizers of the Cahn–Hilliard energy. The resulting connectedness of superlevel sets is used in two dimensions together with the Bonnesen inequality to quantitatively estimate the sphericity of minimizers. We also show how two-point rearrangements can be used to give an alternate proof of symmetry for the constrained minimizers of the Cahn–Hilliard model.
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