In the small-mass regime, volume-constrained minimizers of the hybrid Ḣ¹ + Ḣ^{-1} energy exist and are C^{2,α} perturbations of a ball.
Maggi.Sets of Finite Perimeter and Geometric Variational Problems: An Introduction to Geometric Measure Theory, volume 135 ofCambridge Studies in Advanced Mathematics
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Existence and Regularity in the Small-Mass Regime for a Hartree--Ohta-Kawasaki Shape Optimization Problem
In the small-mass regime, volume-constrained minimizers of the hybrid Ḣ¹ + Ḣ^{-1} energy exist and are C^{2,α} perturbations of a ball.