Calculus of Variations and Geometric Measure Theory
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Variational methods and applications

On an isoperimetric problem involving the competition between the standard perimeter and a nonlocal perimeter

Marc Pegon (Université de Lille, Laboratoire Paul Painlevé)

created by paolini on 08 Sep 2021

8 sep 2021 -- 14:50   [open in google calendar]

Abstract.

In this talk, I will present an isoperimetric problem where the perimeter P is replaced by P-γPε, with γ∈(0,1) and Pε a nonlocal energy which converges to the perimeter as ε vanishes. For small ε, this problem is related to the study of large mass (that is, volume) minimizers for Gamow's liquid drop model for the atomic nucleus, where the repulsive potential is given by a nonnegative, radial, integrable kernel with finite first moment. In arbitrary dimension, I will show that given γ ∊ (0,1), the problem admits minimizers for any ε small enough, and that any sequence of minimizers converges to the ball as ε vanishes. I will then discuss the planar case, where we show by a slicing argument that a minimizer is necessarily convex, and by perturbation, deduce that it can only be the disk for small ε. I will then give ideas on how to prove that the ball is the unique minimizer in higher dimension. This is joint work (in progress) with Michael Goldman and Benoît Merlet.

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