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Asymptotics of the principal eigenvalue of an elliptic operator on closed and orientable Riemannian manifolds

Xin Xu, Kexin Zhang

Abstract

This paper investigates the asymptotic behavior of the principal eigenvalue $λ(s)$, as $s\to+\infty$, for the following elliptic eigenvalue problem \begin{equation*}\label{E} -Δ_{M}u-s\langle \nabla_M f, \nabla_M u\rangle_g +c u=λ(s)u, \end{equation*} defined on an orientable and closed Riemannian manifold $(M,g)$. Assuming $f$ is a Morse function defined on $M$, we find that the limit $\lim\limits_{s\to+\infty} λ(s)$ is determined by the minimum value of the function $c$ over the set of the maximum points of $f$, a result that is independent of the curvature of manifold.

Asymptotics of the principal eigenvalue of an elliptic operator on closed and orientable Riemannian manifolds

Abstract

This paper investigates the asymptotic behavior of the principal eigenvalue , as , for the following elliptic eigenvalue problem \begin{equation*}\label{E} -Δ_{M}u-s\langle \nabla_M f, \nabla_M u\rangle_g +c u=λ(s)u, \end{equation*} defined on an orientable and closed Riemannian manifold . Assuming is a Morse function defined on , we find that the limit is determined by the minimum value of the function over the set of the maximum points of , a result that is independent of the curvature of manifold.
Paper Structure (9 sections, 6 theorems, 60 equations)

This paper contains 9 sections, 6 theorems, 60 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Proof of Theorem \ref{['mmm']}
  4. Upper bound
  5. Lower bound
  6. The non-critical points
  7. Critical points which are not local maxima
  8. Proof of Theorem \ref{['mmm']}
  9. Appendix

Key Result

Theorem 1.1

Asumme that $M$ is a closed and orientable Riemannian manifold, and the function $f: M\rightarrow \mathbb{R}$ is a Morse function. Then, for the principal eigenvalue $\lambda(s)$ of the eigenvalue problem E, we have where $\mathcal{M}$ represents the maximum points set of the function $f$ on $M$.

Theorems & Definitions (14)

  • Theorem 1.1
  • Remark 1.1
  • Remark 1.2
  • Definition 2.1: See MT
  • Definition 2.2: Morse Function MT
  • Lemma 2.1: Morse's Lemma MT
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • proof
  • ...and 4 more