On the dependence on p of the variational eigenvalues of the p-Laplace operator

TL;DR

This paper investigates how the variational eigenvalues of the p-Laplace operator change as the parameter p varies, providing a characterization of their continuity and addressing a question from prior research.

Contribution

It introduces an auxiliary problem to analyze the dependence of eigenvalues on p and characterizes their continuity, answering a longstanding question in the field.

Findings

Eigenvalues depend continuously on p

Characterization of the eigenvalues' behavior as p varies

Resolution of a question posed by Lindqvist in 1993

Abstract

We study the behavior of the variational eigenvalues of the p-Laplace operator, with homogeneous Dirichlet boundary condition, when p is varying. After introducing an auxiliary problem, we characterize the continuity answering, in particular, a question raised in [Lindqvist, 1993].