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.

On spectral and boundary properties of the volume potential for the Helmholtz equation

In this paper, we study boundary properties and some questions of spectral geometry for certain volume potential type operators (Bessel potential operators) in an open bounded Euclidean domains. In particular, the results can be valid for differential operators, which are related to a nonlocal boundary value problem for the Helmholtz equation, so we obtain isoperimetric inequalities for its eigenvalues as well, namely, analogues of the Rayleigh-Faber-Krahn inequality.

Reverse Faber–Krahn and Mahler inequalities for the Cheeger constant

We prove a reverse Faber–Krahn inequality for the Cheeger constant, stating that every convex body in ℝ2 has an affine image such that the product between its Cheeger constant and the square root of its area is not larger than the same quantity for the regular triangle. An analogous result holds for centrally symmetric convex bodies with the regular triangle replaced by the square. We also prove a Mahler-type inequality for the Cheeger constant, stating that every axisymmetric convex body in ℝ2 has a linear image such that the product between its Cheeger constant and the Cheeger constant of its polar body is not larger than the same quantity for the square.