Sharp stability of higher order Dirichlet eigenvalues

par Mickaël Nahon

lundi 12 déc. 2022, 10:00 → 11:00 Europe/Paris

3L15

Let $\mathrm{\Omega }\subset {\mathbb{R}}^n$ be an open set with same area as the unit ball $B$ and call ${\lambda }_k(\mathrm{\Omega })$ the $k$-th eigenvalue of the Laplacian with Dirichlet condition on $\mathrm{\Omega }$. Suppose ${\lambda }_1(\mathrm{\Omega })-{\lambda }_1(B)$ is small, how large can $|{\lambda }_k(\mathrm{\Omega })-{\lambda }_k(B)|$ be ? We establish bounds with sharp exponents depending on the multiplicity of ${\lambda }_k(B)$ through the study of a perturbed shape optimization problem.

This is joint work with Dorin Bucur, Jimmy Lamboley and Raphaël Prunier.