| Abstract |
Lord Rayleigh conjectured in 1877 that the disc has the lowest principal frequency among fixed membranes with a given area, where the principal frequency corresponds to the first eigenvalue of the Laplacian with Dirichlet boundary condition. This conjecture was proven independently by Faber and Krahn in the 1920s, a result now known as the Rayleigh-Faber-Krahn inequality. Such problem is known as an extremal eigenvalue problem or a shape optimisation problem, where one aims to minimise or maximise the k-th eigenvalue of a differential operator subject to a geometric constraint. In this talk, we will survey known results about extremal eigenvalues of the Laplacian for various boundary conditions. Time permitting, we will discuss a local shape optimisation problem for Steklov eigenvalues using perturbation methods. |
