Spectral optimization problems for potentials and measures

Abstract

In the present paper we consider spectral optimization problems involving the Schrödinger operator $-\Delta +\mu$ on ${\mathbb{R}}^d$, 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 ${\mathbb{R}}^d$ and that the optimal potentials or measures are equal to $+\infty$ outside a compact set.

30 pages, 1 figure