Frontiers in Mathematical Sciences
7th Conference University of Isfahan  January 13, 2020 
Title:
A comparison between Steklov and Laplace eigenvalues on a Riemannian manifold.
Speaker:
Asma Hassannezhad, University of Bristol
Date, Time, and Venue: Thursday, January 2  14:0014:45  Hall 1
Abstract:
The DirichlettoNeumann operator acts on smooth functions on the boundary of a Riemannian manifold and maps a function to the normal derivative of its harmonic extension. The eigenvalues of the DirichlettoNeumann map are also called Steklov eigenvalues. It has been known that the geometry of the boundary has a strong influence on the Steklov eigenvalues. We show that for every $k\in N$ the $k$th Steklov eigenvalue $\sigma_k$ is comparable to the square root of the $k$th eigenvalue $\sqrt{\lambda_k}$ of the Laplacian on the boundary. Our results, in particular, give interesting geometric lower and upper bounds for Steklov eigenvalues. This is joint work with Bruno Colbois and Alexandre Girouard.
