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

Paulo Henryque C. Silva

Abstract

In this article, we develop a Barta-type formulation for the $p$-Laplacian on Riemannian manifolds, extending the approach of Cheung-Leung \cite{cheung} and Bessa-Montenegro \cite{BessaMontenegro} 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 $Ω$ is $p$-stable for the Schrödinger-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.

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

Abstract

In this article, we develop a Barta-type formulation for the -Laplacian on Riemannian manifolds, extending the approach of Cheung-Leung \cite{cheung} and Bessa-Montenegro \cite{BessaMontenegro} from the linear to the nonlinear setting. This framework yields sharp lower bounds for the -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 in the context of minimal immersions. In particular, under the above assumptions, the domain is -stable for the Schrödinger-type operator associated with the potential , where denotes the second fundamental form of the minimal immersion. In addition, we establish a lower bound for the -fundamental tone in the setting where the immersion has locally bounded mean curvature. Finally, we provide a Kazdan-Kramer type characterization of the -fundamental tone, offering a unified and geometric perspective on spectral bounds for the operator -Laplacian.
Paper Structure (9 sections, 17 theorems, 170 equations, 4 figures)

This paper contains 9 sections, 17 theorems, 170 equations, 4 figures.

Table of Contents

  1. Introduction
  2. Bartas’s Theorem for aLaplacian
  3. Proof of the Main Results
  4. Proof of Theorem \ref{['thm:Cheng']}
  5. Proof of Theorem \ref{['thm:pminsub_ball']}
  6. Proof of Corollary \ref{['radiuscor']}
  7. Proof of Theorem \ref{['thmlowerptone']}
  8. Proofs of the p-stability criteria
  9. Proof of Theorem \ref{['Thm:pKazdame']}

Key Result

Theorem 1.1

Let $\Omega$ be a bounded domain in a Riemannian $m$-manifold $M$ with piecewise smooth non-empty boundary $\partial \Omega$, and let $\eta \in \mathcal{C}^{2}(\Omega)\cap \mathcal{C}^{0}(\overline{\Omega})$ satisfy $\eta>0$ in $\Omega$ and $\eta|_{\partial \Omega}=0$. Let $\lambda_{1}^{\mathcal{D}} Moreover, equality holds if and only if $\eta$ is a first eigenfunction of $\Omega$.

Figures (4)

  • Figure 1: Examples of admissible test functions on the interval $[0,1]$.
  • Figure 2: Euclidian region of non-confinement for $\varphi(\Omega)$.
  • Figure 3: Typical behavior of the function $\Phi_0(s)$.
  • Figure 4: Quadratic barrier for $\Phi_1$.

Theorems & Definitions (34)

  • Theorem 1.1: Barta's inequality Barta1937
  • Theorem 1.2: $p$-Barta's inequality
  • Theorem 1.3: Cheng Cheng1975
  • Theorem 1.4
  • Theorem 1.5: Cheng--Li--Yau
  • Theorem 1.6
  • Remark 1.1
  • Remark 1.2
  • Definition 1.3: $p$-stability
  • Corollary 1.4: A $p$-stability criterion
  • ...and 24 more