Speaker
Jean Lagace, King’s College London
Abstract
The Steklov problem consists in finding the eigenvalues and eigenfunctions of the Dirichlet-to-Neumann map, a first order elliptic pseudodifferential operator on the boundary of a bounded domain which is a square root of the boundary Laplacian to first order when the boundary is smooth. As such, its eigenvalues satisfy a Weyl law in accordance with the general theory. However, when for surfaces with smooth boundary, the eigenvalues are distributed much more regularly than predicted by Weyl’s law, a phenomenon first observed by Grisha Rozenblum for simply connected domains.
In this talk, I will present asymptotic upper bounds for the deviation of the eigenvalues of the Steklov problem from those of disks, depending explicitly on the regularity of the boundary. Somewhat surprisingly, it appears that these upper bounds are generically (but not always) saturated, meaning that we can generically detect the regularity of the boundary from the eigenvalue asymptotics.
This is joint work with Alix Deleporte (Paris-Saclay) and Leonid Parnovski (University College London).