A quantitative reverse Faber-Krahn inequality for the first Robin eigenvalue with negative boundary parameter

The aim of this paper is to prove a quantitative form of a reverse Faber-Krahn type inequality for the first Robin Laplacian eigenvalue λ β with negative boundary parameter among convex sets of prescribed perimeter. In that framework, the ball is the only maximizer for λ β and the distance from the optimal set is considered in terms of Hausdorff distance. The key point of our stategy is to prove a quantitative reverse Faber-Krahn inequality for the first eigenvalue of a Steklov-type problem related to the original Robin problem.

Counterexample to Strong Diamagnetism for the Magnetic Robin Laplacian

We determine a counterexample to strong diamagnetism for the Laplace operator in the unit disc with a uniform magnetic field and Robin boundary condition. The example follows from the accurate asymptotics of the lowest eigenvalue when the Robin parameter tends to $-\infty $ − ∞ .

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.