Barta Theorem for the $p$-Laplacian and Geometric Applications

TL;DR

This paper extends Barta's theorem to the nonlinear $p$-Laplacian on Riemannian manifolds, providing sharp spectral bounds and geometric applications including eigenvalue comparisons and stability criteria.

Contribution

It develops a nonlinear Barta-type formulation for the $p$-Laplacian, leading to new spectral bounds and geometric insights in the context of minimal immersions and curvature conditions.

Findings

Sharp lower bounds for the $p$-fundamental tone without boundary regularity assumptions

Nonlinear extensions of Cheng's eigenvalue comparison theorem

Lower bounds for the $p$-fundamental tone with bounded mean curvature

Abstract

In this article, we develop a Barta-type formulation for the $p$-Laplacian on Riemannian manifolds, extending the approach of Cheung-Leung and Bessa-Montenegro from the linear to the nonlinear setting. This framework yields sharp lower bounds for the $p$-fundamental tone without any assumptions on boundary regularity. As applications, we obtain nonlinear extensions of Cheng's eigenvalue comparison theorem and the Cheng-Li-Yau estimate for $p \geq 2$ in the context of minimal immersions. In particular, under the above assumptions, the domain $\Omega$ is $p$-stable for the Schr\"odinger-type operator associated with the potential $\mathcal{V} = \|A\|^{p}$, where $A$ denotes the second fundamental form of the minimal immersion. In addition, we establish a lower bound for the $p$-fundamental tone in the setting where the immersion has locally bounded mean curvature. Finally, we provide a Kazdan-Kramer type characterization of the $p$-fundamental tone, offering a unified and geometric perspective on spectral bounds for the operator $p$-Laplacian.