The dual eigenvalue problems for $p$-Laplacian

TL;DR

This paper investigates eigenvalue optimization problems for the $p$-Laplacian, extending classical Sturm-Liouville results by identifying minimizers for eigenvalue gaps and ratios in specific potential and density problems.

Contribution

It introduces new minimization results for eigenvalue gaps and ratios in $p$-Laplacian problems, extending classical Sturm-Liouville theory to nonlinear cases.

Findings

Minimizer of eigenvalue gap for single-well potential identified

Eigenvalue ratio for single-barrier density problem determined

Results extend classical Sturm-Liouville eigenvalue theory

Abstract

In this paper, we find the minimizer of the eigenvalue gap for the single-well potential problem and the eigenvalue ratio for the single-barrier density problem and symmetric single-well (single-barrier)density problem for $p$-Laplacian. This extends the results of the classical Sturm-Liouville problem.