Spectral optimization problems for potentials and measures

TL;DR

This paper investigates spectral optimization problems involving Schrödinger operators with measures or potentials, proving the existence of solutions and characterizing their support in Euclidean space.

Contribution

It establishes the existence of global solutions for spectral optimization problems with measures or potentials and describes their compact support properties.

Findings

Existence of solutions in  space

Optimal potentials/measures are compactly supported

Solutions are infinite outside a compact set

Abstract

In the present paper we consider spectral optimization problems involving the Schr\"odinger operator $-\Delta +\mu$ on $\R^d$, the prototype being the minimization of the $k$ the eigenvalue $\lambda_k(\mu)$. Here $\mu$ may be a capacitary measure with prescribed torsional rigidity (like in the Kohler-Jobin problem) or a classical nonnegative potential $V$ which satisfies the integral constraint $\ds \int V^{-p}dx \le m$ with $0<p<1$. We prove the existence of global solutions in $\R^d$ and that the optimal potentials or measures are equal to $+\infty$ outside a compact set.