Speaker
Description
In this talk, we discuss the following question: knowing that the first Dirichlet-Laplacian eigenvalue of an open set is close to the one of the ball of same volume (which is the minimizer due to Kaber-Krahn’s inequality), can we say that the other eigenvalues of this set are also close to the ones of the ball? More precisely we seek for quantitative estimates of the form
$$|\lambda_k(\Omega)-\lambda_k(B)|\leq C(\lambda_1(\Omega)-\lambda_1(B))^\alpha.$$
We show that such an estimate is valid with $\alpha=1/2$ and that this is sharp in general, though it can be improved to $\alpha=1$ if $\lambda_k(B)$ is simple. The proof of this last case requires the regularity analysis for minimizers of $\lambda_1\pm\varepsilon \lambda_k$, which involves a vectorial free boundary problem.
We also provide an improved result for multiple eigenvalues, and we observe that our analysis leads to a reverse Kohler-Jobin inequality.
This is a joint work with Dorin Bucur, Mickaël Nahon and Raphaël Prunier.
