The spectrum of the Laplace operator with Robin boundary conditions has been studied extensively, with deep connections to physical models including heat flow, fluid dynamics, and wave propagation. Its nonlinear counterpart, the $p$-Laplacian, also plays a central role in modeling complex media, particularly non-Newtonian fluids.

 

We establish rigorous quantitative inequalities for the first eigenvalue of the generalized $p$-Robin problem, for both the classical diffusion absorption case and the superconducting regime, 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 boundary parameters $\alpha$, improving existing results in various directions, including requiring weaker boundary regularity for the case of the classical 2-Robin problem. 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. This is joint work with Robert Smits and Tiziana Giorgi.