Mixed eigenvalues of p-Laplacian on trees

TL;DR

This paper provides quantitative estimates for the principal eigenvalue of the discrete p-Laplacian on rooted trees, linking it to Hardy inequalities and presenting variational formulas for better understanding.

Contribution

It introduces new variational formulas and estimates for the eigenvalues of p-Laplacian on trees, enhancing understanding of their spectral properties.

Findings

Basic estimate of the eigenvalue on trees

Variational formulas for mixed principal eigenvalue

Connection to weighted Hardy inequalities

Abstract

The purpose of the paper is to present quantitative estimates for the principal eigenvalue of discrete p-Laplacian on the set of rooted trees. Alternatively, it is studying the optimal constant of a class of weighted Hardy inequality. Three kinds of variational formulas in different formulation for the mixed principal eigenvalue of p-Laplacian on the set of trees with unique root as Dirichlet boundary are presented. As their applications, we obtain a basic estimate of the eigenvalue on trees.