Colloquium Mathematics - Dr. Jean Lagacé University College London

Title: Homogenisation and shape optimisation: From Steklov to Neumann

Abstract:

The homogenisation limit consists in replacing a partial differential equation with rapidly oscillating coefficients by a limiting effective equation that should be in principle easier to study. While this approach, at 50 years of age, is comparatively new in mathematics, it has been used effectively for shape optimisation many times since : materials with many small holes often have better rigidity properties than a solid slab, and on top of it, they cost less material to make.

In this talk, will discuss homogenisation theory and how we recently used it to uncover links between the Neumann problem for the Laplacian on a domain, and the Steklov problem on its boundary. Between both lies an exotic boundary value problem, and in the process, we'll see how to obtain bounds for Neumann eigenvalues if we have bounds for Steklov eigenvalues.