A spectral shape optimization problem with a nonlocal competing term

2020/09/16 by Mazzoleni, Dario, Ruffini, Berardo
#47A75 #49Q10 #49R05 #Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC) #Spectral Theory (math.SP)

paper · doi:10.48550/arxiv.2009.07699

Abstract

We study the minimization of a spectral functional made as the sum of the first eigenvalue of the Dirichlet Laplacian 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. Then we prove, by means of an expansion analysis, that the ball is a rigid minimizer when the Riesz repulsion is small enough. Eventually we show that for certain regimes of the Riesz repulsion, regular minimizers do not exist.

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