Sharp stability of some spectral inequalities

In this work we review two classical isoperimetric inequalities involving eigenvalues of the Laplacian, both with Dirichlet and Neumann boundary conditions. The first one is classically attribuited to Krahn and P. Szego and asserts that among sets of given measure, the disjoint union of two balls with the same radius minimizes the second eigenvalue of the Dirichlet-Laplacian, while the second one is due to G. Szeg\H{o} and Weinberger and deals with the maximization of the first non trivial eigenvalue of the Neumann-Laplacian. New stability estimates are provided for both of them