Calculus of Variations and Geometric Measure Theory

D. Mazzoleni - B. Ruffini

A spectral shape optimization problem with a nonlocal competing term

created by ruffini on 07 Sep 2020
modified on 12 Oct 2021


Published Paper

Inserted: 7 sep 2020
Last Updated: 12 oct 2021

Journal: Calc. Var. PDE
Year: 2020

32 pages


We study a spectral optimization problem made as the sum of the first Dirichlet Laplacian eigenvalue, and the relative strength of a Riesz-type interaction functional. We show that when the Riesz repulsion strength is below a critical value, existence of minimizers occurs and they are $C^{1,\alpha}-$regular. This allows to show by means of an expansion analysis that the ball is a rigid minimizer as the Riesz repulsion is small enough. Eventually we show that for certain regimes of the Riesz repulsion, regular minimizers do not exist.