Comparison estimates on the first eigenvalue of a quasilinear elliptic system

TL;DR

This paper investigates the first eigenvalue of a quasilinear elliptic system on Riemannian manifolds, deriving comparison estimates, inequalities, and bounds that connect eigenvalues with geometric constants, and explores their limits as parameters approach 1.

Contribution

It extends comparison estimates and inequalities for the first eigenvalue of a $(p,q)$-Laplacian on manifolds, including new bounds and convergence results.

Findings

Recovered Cheng comparison estimates for the first eigenvalue.

Established Faber-Krahn inequality for the $(p,q)$-Laplacian.

Derived Cheeger-type lower bounds and convergence of eigenvalues to Cheeger's constant.

Abstract

We study a system of quasilinear eigenvalue problems with Dirichlet boundary conditions on complete compact Riemannian manifolds. In particular, Cheng comparison estimates and inequality of Faber-Krahn for the first eigenvalue of a $(p,q)$-Laplacian are recovered. Lastly, we reprove a Cheeger type estimates for $p$-Laplacian, $1<p<\infty$, from where a lower bound estimate in terms of Cheeger's constant for the first eigenvalue of a $(p,q)$-Laplacian is built. As a corollary, the first eigenvalue converges to Cheeger's constant as $p,q\to 1,1.$