Minimality of balls in the small volume regime for a general Gamow-type functional

Abstract

Abstract We consider functionals given by the sum of the perimeter and the double integral of some kernel g : ℝ N × ℝ N → ℝ + {g:\mathbb{R}^{N}\times\mathbb{R}^{N}\to\mathbb{R}^{+}} , multiplied by a “mass parameter” ε. We show that, whenever g is admissible, radial and decreasing, the unique minimizer of this functional among sets of given volume is the ball as soon as ε ≪ 1 {\varepsilon\ll 1} .