Abstract
We prove the stability of the ball as global minimizer of an attractive shape functional under volume constraint, by means of mass transportation arguments. The stability exponent is 1∕2 and it is sharp. Moreover, we use such stability result together with the quantitative (possibly fractional) isoperimetric inequality to prove that the ball is a global minimizer of a shape functional involving both an attractive and a repulsive term with a sufficiently large fixed volume and with a suitable (possibly fractional) perimeter penalization.
Highlights
In recent times different works focused on shape functionals of the form
The authors prove that, up to a critical volume, the ball is the minimizer of the mixed energy described by Gamow’s liquid drop model
In [3] the same result is obtained for general Riesz potential in any dimension
Summary
Is the Riesz potential with α ∈ (0, N ). The most famous case is given by N = 3 and α = 2 It is linked with Gamow’s liquid drop model (see [29]) for the stability of atomic nuclei (see [9] for a review on the problem). Such problem has been studied independently in [32, 33] up to dimension 7 and in [31] for any dimension and α = 2. In [3] the same result is obtained for general Riesz potential in any dimension. Dipartimento di Matematica e Applicazioni “Renato Caccioppoli”, Universitadegli Studi di Napoli Federico II, 80126 Napoli, Italy.
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