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

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.