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

TL;DR

This paper rigorously analyzes the behavior of the first eigenvalue in generalized $p$-Robin problems, providing new asymptotic results, inequalities, and characterizations for both bounded and exterior domains across all $p$ and small boundary parameters.

Contribution

It introduces a unified approach to derive asymptotics and inequalities for $p$-Robin eigenvalues, improving regularity requirements and extending results to exterior domains.

Findings

Asymptotic behavior of eigenvalues as boundary parameter approaches zero.

New inequalities and geometric bounds for convex domains.

Characterization of eigenvalue existence in exterior domains.

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 $\alpha$ is positive, and the superconducting generation regime ($\alpha<0$), 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 $\alpha$, 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 Ren\'e Sperb. In exterior domains, we characterize the existence of eigenvalues, establish general inequalities and asymptotics as $\alpha\to 0$ for the first eigenvalue of the exterior of a ball, and obtain some sharp geometric inequalities for convex domains in two dimensions.