Behavior of Absorbing and Generating $p$-Robin Eigenvalues in Bounded and Exterior Domains

TL;DR

This paper studies the generalized Robin problem for the $p$-Laplacian in bounded and exterior domains, focusing on the first eigenvalue $\lambda_1(\alpha,p,n,\mathcal{U})$ defined variationally. It develops a unified, quantitative framework that yields lower bounds for both absorbing $(\alpha>0)$ and generating $(\alpha<0)$ regimes on bounded domains, incorporating constants $C(\Omega,p)$ and $C_2(\Omega,p)$ to handle weaker boundary regularity. In exterior domains, the authors establish existence criteria for a variational first eigenvalue, characterize the negativity threshold via a Steklov-type quantity $\mu_1(p,n,\Omega^\mathrm{ext})$, and provide sharp asymptotics for the exterior of a ball as $\alpha\to 0$, including detailed behavior depending on $p$ relative to $n$. They also derive geometric inequalities and extremal-shape results, showing, for example, that in 2D the disk exterior can maximize $\lambda_1$ in certain convex classes, while in higher dimensions the ball is not universally extremal. Overall, the work advances understanding of $p$-Robin spectra in nonlinear diffusion and superconductivity contexts and informs shape optimization under nonlinear boundary conditions.

Abstract

We establish rigorous quantitative inequalities for the first eigenvalue of the generalized $p$-Robin problem, for both the classical diffusion absorption case, where the Robin boundary parameter $α$ is positive, and the superconducting generation regime ($α<0$), where the boundary acts as a source. In bounded domains, we use a unified approach to derive a precise asymptotic behavior for all $p$ and all small real $α$, improving existing results in various directions, including requiring weaker boundary regularity for the case of the classical 2-Robin problem, studied in the fundamental work by René Sperb. In exterior domains, we characterize the existence of eigenvalues, establish general inequalities and asymptotics as $α\to 0$ for the first eigenvalue of the exterior of a ball, and obtain some sharp geometric inequalities for convex domains in two dimensions.

Figures (3)

• Figure 1: Comparison of the lower bounds of $\lambda_1(\alpha,2,2,B_1)$ for $\alpha <0$
• Figure 2: Behavior of $\lambda_1(\alpha,2,2,B_1)$ for $\alpha < 0$
• Figure 3: $\Omega_{m}$ from Theorem \ref{['prop:bound']} for $p=2$

Theorems & Definitions (35)

• Theorem \ref{theo:lowbound1}
• Theorem \ref{theo:lowbound2}
• Theorem \ref{theo:asym}
• Theorem \ref{theo:cuteidea}
• Theorem \ref{prop:bound}
• proof : Proof of Theorem \ref{['theo:lowbound1']}
• Proposition 2.2
• proof
• proof : Proof of Theorem \ref{['theo:lowbound2']}
• Lemma 3.1