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

TL;DR

This paper provides detailed quantitative estimates and variational formulas for the principal eigenvalue of the discrete p-Laplacian on nonnegative integers, with applications to Hardy inequalities and boundary conditions.

Contribution

It introduces multiple variational formulas and explicit bounds for the eigenvalue of the discrete p-Laplacian, expanding understanding of boundary conditions and approximation methods.

Findings

Derived explicit lower and upper bounds for the eigenvalue

Presented variational formulas in different forms

Provided examples illustrating the estimates and methods

Abstract

This paper deals with the principal eigenvalue of discrete $p$-Laplacian on the set of nonnegative integers. Alternatively, it is studying the optimal constant of a class of weighted Hardy inequalities. The main goal is the quantitative estimates of the eigenvalue. The paper begins with the case having reflecting boundary at origin and absorbing boundary at infinity. Several variational formulas are presented in different formulation: the difference form, the single summation form and the double summation form. As their applications, some explicit lower and upper estimates, a criterion for positivity (which was known years ago), as well as an approximating procedure for the eigenvalue are obtained. Similarly, the dual case having absorbing boundary at origin and reflecting boundary at infinity is also studied. Two examples are presented at the end of Section 2 to illustrate the value of the investigation.