### Bounds for Steklov eigenvalues on submanifolds of Euclidean space with prescribed boundary

Analysis and Geometry Seminar

26th March 2020, 3:15 pm – 4:15 pm

Fry Building, 2.04

We consider the Steklov eigenvalues on a submanifold of Euclidean space with prescribed boundary. The relationship between the Steklov eigenvalues and the geometry of the boundary has received a great deal of attention in recent years. One way to gain insight into this relationship is to obtain bounds for the Steklov eigenvalues in terms of some of the geometric quantities of the boundary.

Several of the known upper bounds for the *k*-th Steklov eigenvalue involve a power of *k* which is not optimal with respect to the known asymptotics as *k → ∞*. They also tend to have a complicated dependence on the geometry of the boundary.

We discuss some geometric situations in which it is possible to obtain upper bounds for the Steklov eigenvalues which have the optimal power of *k* and we describe the dependence upon the geometry of the boundary.

This is based on an ongoing project with Bruno Colbois (Université de Neuchâtel).

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