The Faber-Krahn inequality

Among N-dimensional open sets with given measure, balls (uniquely) minimize the first eigenvalue of the Laplacian with homogeneous Dirichlet boundary conditions. We review this classical result and discuss some of its applications. Then we show how this can be enhanced by means of a quantitative stability estimate. The resulting inequality, first conjectured by Nadirashvili and Bhattacharya & Weitsman, is sharp. The results presented are contained in a paper in collaboration with Guido De Philippis and Bozhidar Velichkov.