Calculus of Variations and Geometric Measure Theory

D. Bucur - G. Buttazzo - B. Velichkov

Spectral optimization problems for potentials and measures

created by buttazzo on 03 Oct 2013
modified by velichkov on 21 Apr 2018


Published Paper

Inserted: 3 oct 2013
Last Updated: 21 apr 2018

Journal: SIAM J. Math. Anal.
Year: 2014


In the present paper we consider spectral optimization problems involving the Schr\"odinger operator $-\Delta +\mu$ on ${\bf R}^d$, 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 $\int V^{-p}dx \le m$ with $0<p<1$. We prove the existence of global solutions in ${\bf R}^d$ and that the optimal potentials or measures are equal to $+\infty$ outside a compact set.

Keywords: shape optimization, eigenvalues, Torsional rigidity, Kohler-Jobin problems, Schr\"odinger operator