Eigenvalues for double phase variational integrals

TL;DR

This paper investigates the eigenvalue problem for double phase variational integrals, introducing a sequence of nonlinear eigenvalues, analyzing their continuity, growth, and establishing a Weyl-type law.

Contribution

It introduces a new sequence of nonlinear eigenvalues for double phase integrals and proves their continuity and growth properties, extending classical spectral laws.

Findings

Established continuity of eigenvalues with phase variations

Analyzed the growth rate of the eigenvalue sequence

Derived a Weyl-type law for the eigenvalues

Abstract

We study an eigenvalue problem in the framework of double phase variational integrals and we introduce a sequence of nonlinear eigenvalues by a minimax procedure. We establish a continuity result for the nonlinear eigenvalues with respect to the variations of the phases. Furthermore, we investigate the growth rate of this sequence and get a Weyl-type law consistent with the classical law for the $p$-Laplacian operator when the two phases agree.