The asymptotic behavior of the first Robin eigenvalue with negative parameter as $p$ goes to $+\infty$

TL;DR

The paper analyzes the limit as $p\to\infty$ of the first Robin $p$-Laplacian eigenvalue with a negative boundary parameter on a bounded Lipschitz domain $\Omega$. They prove that $( -\lambda_{p,\beta}(\Omega) )^{1/p} \to \beta$ and that the associated eigenfunctions converge uniformly to a limit $u_\infty \in W^{1,\infty}(\Omega)$. The limit $u_\infty$ is shown to be a viscosity solution of the infinity-Laplacian type problem $-\min\{ |\nabla u| - \beta u, \Delta_\infty u\}=0$ in $\Omega$ with boundary condition $\min\{ |\nabla u| - \beta u, \partial u/\partial \nu \}=0$ on $\partial\Omega$, and the geometry of $\Omega$ no longer influences the limit. Additional results establish the uniqueness of the positive eigenfunction in the limit and provide explicit radial and one-dimensional illustrations of the limiting profile.

Abstract

In this paper, we want to study the asymptotic behavior of the first $p$-Laplacian eigenvalue, with Robin boundary conditions, with negative boundary parameter. In particular, we prove that the limit of the eigenfunctions is a viscosity solution for the infinity Laplacian eigenvalue problem.

Theorems & Definitions (13)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Definition 2.1
• Theorem 2.1
• Proposition 2.2
• proof
• proof : Proof of Theorem \ref{['existence_limit']}
• proof : Proof of Theorem \ref{['teo1.2']}
• Remark 3.1