Videoconference: Homogenization of Steklov problems with applications to sharp isoperimetric bounds

Montreal Analysis Seminar (May 1, 2020)

Homogenization of Steklov problems with applications to sharp isoperimetric bounds, part I, by Alexandre Girouard (Université Laval)

Abstract: The question to find the best upper bound for the first nonzero Steklov eigenvalue of a planar domain goes back to Weinstock, who proved in 1954 that the first nonzero perimeter-normalized Steklov eigenvalue of a simply-connected planar domain is 2*pi, with equality iff the domain is a disk. In a recent joint work with Mikhail Karpukhin and and Jean Lagacé, we were able to let go of the simple connectedness assumption. We constructed a family of domains for which the perimeter-normalized first eigenvalue tends to 8π. In combination with Kokarev's bound from 2014, this solves the isoperimetric problem completely for the first nonzero eigenvalue. The domains are obtained by removing small geodesic balls that are asymptotically densely periodically distributed as their radius tends to zero. The goal of this talk will be to survey recent work on homogenisation of the Steklov problem which lead to the above result. On the way we will see that many spectral problems can be approximated by Steklov eigenvalues of perforated domains. A surprising consequence is the existence of free boundary minimal surfaces immersed in the unit ball by first Steklov eigenfunctions and with area strictly larger than 2*pi. This talk is based on joint work with Antoine Henrot (U. de Lorraine), Mikhail Karpukhin (UCI) and Jean Lagacé (UCL).

Homogenization of Steklov problems with applications to sharp isoperimetric bounds, part II, by Jean Lagacé (UCL) à partir de 51 min 32 sec.

Abstract: Traditionally, deterministic homogenisation theory uses the periodic structure of Euclidean space to describe uniformly distributed perturbations of a PDE. It has been known for years that it has many applications to shape optimization. In this talk, I will describe how the lack of periodic structure can be overcome to saturate isoperimetric bounds for the Steklov problem on surfaces. The construction is intrinsic and does not depend on any auxiliary periodic objects or quantities. Using these methods, we obtain the existence of free boundary minimal surfaces in the unit ball with large area. I will also describe how the intuition we gain from the homogenization construction allows us to actually construct some of them, partially verifying a conjecture of Fraser and Li. This talk is based on joint work with Alexandre Girouard (U. Laval), Antoine Henrot (U. de Lorraine) and Mikhail Karpukhin (UCI).