Mixed Eigenvalues of {\LARGE $\pmb p$\,}-Laplacian

TL;DR

This paper investigates the mixed principal eigenvalue of the p-Laplacian, providing variational formulas, criteria for positivity, and explicit estimates, with applications to weighted Hardy inequalities.

Contribution

It introduces new variational formulas and an approximation method for the mixed eigenvalues of the p-Laplacian, enhancing understanding and estimation techniques.

Findings

Derived variational formulas for the eigenvalue

Established criteria for eigenvalue positivity

Provided explicit estimates and an illustrative example

Abstract

The mixed principal eigenvalue of $p\,$-Laplacian (equivalently, the optimal constant of weighted Hardy inequality in $L^p$ space) is studied in this paper. Several variational formulas for the eigenvalue are presented. As applications of the formulas, a criterion for the positivity of the eigenvalue is obtained. Furthermore, an approximating procedure and some explicit estimates are presented case by case. An example is included to illustrate the power of the results of the paper.