On the second eigenvalue of the infinity Laplacian with Robin boundary conditions

TL;DR

The paper analyzes the asymptotic behavior, as $p\to\infty$, of the second Robin $p$-Laplacian eigenvalue and eigenfunctions, establishing that $(\lambda_{2,p}(\Omega))^{1/p}$ converges to $\lambda_{2,\infty}(\Omega)$, equal to $1/s(\Omega)$ where $s(\Omega)$ is a geometric quantity defined via boundary cones. It shows subsequential convergence of the second eigenfunctions to a Lipschitz limit $u_{\infty}$ that solves a mixed $\infty$-eigenproblem in viscosity sense, and proves a geometric characterization of the limit with sharp upper and lower bounds linked to $s(\Omega)$ and the inradius/diameter of $\Omega$. A variational and path-based framework provides a robust characterization of $\lambda_{2,\infty}(\Omega)$, with explicit expressions in terms of $\|\nabla w\|_{\infty}$ and boundary values, and the analysis includes concrete examples (stadium and square) illustrating the dependence on the Robin parameter $\beta$. The results interpolate between Neumann and Dirichlet extremals and extend the program of understanding $p$-Laplacian limits under boundary conditions.

Abstract

We study the behaviour, as $p \to +\infty$, of the second eigenvalues of the $p$-Laplacian with Robin boundary conditions and the limit of the associated eigenfunctions. We prove that, up to some regularity of the set, the limit of the second eigenvalues is actually the second eigenvalue of the so-called $\infty$-Laplacian.

Figures (3)

• Figure 1: Definition of $s(\Omega)$.
• Figure 2: The stadium.
• Figure 3: The square.

Theorems & Definitions (24)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Lemma 2.1
• Definition 2.1
• Definition 2.2
• Definition 2.3
• Proposition 3.1
• proof
• Proposition 3.2