Calculus of Variations and Geometric Measure Theory
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D. Mazzoleni - B. Ruffini

A spectral shape optimization problem with a nonlocal competing term

created by ruffini on 07 Sep 2020
modified on 13 Oct 2020


Submitted Paper

Inserted: 7 sep 2020
Last Updated: 13 oct 2020

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. We also show that if the Riesz repulsion is large enough, then minimizers do not exist.


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