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On the second Dirichlet eigenvalue of some nonlinear anisotropic elliptic operators

Francesco Della Pietra, Nunzia Gavitone, Gianpaolo Piscitelli

Abstract

Let $Ω$ be a bounded open set of $\mathbb R^{n}$, $n\ge 2$. In this paper we mainly study some properties of the second Dirichlet eigenvalue $λ_{2}(p,Ω)$ of the anisotropic $p$-Laplacian \[ -\mathcal Q_{p}u:=-\textrm{div} \left(F^{p-1}(\nabla u)F_ξ(\nabla u)\right), \] where $F$ is a suitable smooth norm of $\mathbb R^{n}$ and $p\in]1,+\infty[$. We provide a lower bound of $λ_{2}(p,Ω)$ among bounded open sets of given measure, showing the validity of a Hong-Krahn-Szego type inequality. Furthermore, we investigate the limit problem as $p\to+\infty$.

On the second Dirichlet eigenvalue of some nonlinear anisotropic elliptic operators

Abstract

Let be a bounded open set of , . In this paper we mainly study some properties of the second Dirichlet eigenvalue of the anisotropic -Laplacian where is a suitable smooth norm of and . We provide a lower bound of among bounded open sets of given measure, showing the validity of a Hong-Krahn-Szego type inequality. Furthermore, we investigate the limit problem as .

Paper Structure

This paper contains 7 sections, 17 theorems, 107 equations.

Table of Contents

  1. Introduction
  2. Notation and preliminaries
  3. The Dirichlet eigenvalue problem for TEXT
  4. The first eigenvalue
  5. Higher eigenvalues
  6. The second Dirichlet eigenvalue of TEXT
  7. The limit case TEXT

Key Result

Theorem 3.1

If $\Omega$ is a bounded open set in $\mathbb{R}^{n}$, $n\ge 2$, there exists a function $u_{1}\in C^{1,\alpha}(\Omega)\cap C(\overline{\Omega})$ which achieves the minimum in rayleigh, and satisfies the problem eigpb with $\lambda=\lambda_{1}(p,\Omega)$. Moreover, if $\Omega$ is connected, then $\l

Theorems & Definitions (41)

  • Remark 2.1
  • Definition 1
  • Definition 2
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • proof
  • Proposition 3.3
  • proof
  • Theorem 3.4
  • ...and 31 more