A theorem of do Carmo-Zhou type: oscillations and estimates for the first eigenvalue of the p-Laplacian

TL;DR

This paper extends classical eigenvalue estimates from the Laplace-Beltrami operator to the p-Laplacian on manifolds, using oscillation theorems to derive new bounds for the first eigenvalue.

Contribution

It generalizes the do Carmo-Zhou eigenvalue estimates to the p-Laplacian, broadening their applicability to semi-elliptic singular operators on manifolds.

Findings

Extended eigenvalue estimates to p-Laplacian

Established oscillation-based bounds for first eigenvalue

Applicable to semi-elliptic singular operators

Abstract

It is shown that the estimates obtained by Manfredo P. do Carmo and Detang Zhou, in their paper "Eigenvalue estimate on complete noncompact Riemannian manifolds and applications", for the first eigenvalue of the Laplace-Beltrami operator on open manifolds, via an oscillation theorem, can be naturally extended for the semi-elliptic singular operator operator, p-Laplace on manifolds.