Extremal Domains and Pólya-type Inequalities for the Robin Laplacian on Rectangles and Unions of Rectangles

We investigate the question of whether the eigenvalues of the Laplacian with Robin boundary conditions can satisfy inequalities of the same type as those in Pólya’s conjecture for the Dirichlet and Neumann Laplacians and, if so, what form these inequalities should take. Motivated in part by Pólya’s original approach and in part by recent analogous works treating the Dirichlet and Neumann Laplacians, we consider rectangles and unions of rectangles and show that for these two families of domains, for any fixed positive value $\alpha$ of the boundary parameter, Pólya-type inequalities do indeed hold, albeit with an exponent smaller than that of the corresponding Weyl asympotics for a fixed domain. We determine the optimal exponents in both cases, showing that they are different in the two situations. Our approach to proving these results includes a characterization of the corresponding extremal domains for the $k^{\textrm{}}$th eigenvalue in regions of the $(k,\alpha )$-plane, which in turn supports recent conjectures on the nature of the extrema among all bounded domains.