A variational method for functionals depending on eigenvalues

TL;DR

This paper introduces a variational approach for functionals depending on eigenvalues of Riemannian manifolds, utilizing a new Palais-Smale sequence concept and extending min-max methods to locally-Lipschitz functionals, with convergence results for Laplace and Steklov eigenvalues.

Contribution

It presents a novel variational framework and a new Palais-Smale sequence concept for eigenvalue-dependent functionals, extending classical methods to locally-Lipschitz functionals.

Findings

Convergence results for Palais-Smale sequences involving Laplace eigenvalues.

Convergence results for Palais-Smale sequences involving Steklov eigenvalues.

Extension of min-max methods to locally-Lipschitz functionals.

Abstract

We perform a systematic variational method for functionals depending on eigenvalues of Riemannian manifolds. It is based on a new concept of Palais Smale sequences that can be constructed thanks to a generalization of classical min-max methods on $C^1$ functionals to locally-Lipschitz functionals. We prove convergence results on these Palais-Smale sequences emerging from combinations of Laplace eigenvalues or combinations of Steklov eigenvalues in dimension 2.