Calculus in infinite dimensions

Abstract

Have you ever wondered how to take a derivative in infinite dimensions? How about optimising a function with an uncountable number of variables? These questions and more can be addressed by the ‘calculus of variations’, an area of analysis that establishes techniques to optimise functions by considering small changes of variables (called variations).

The talk shall begin with an introduction to the calculus of variations, summarising some of the key concepts and results, with particular focus on the contrast with ‘standard’ calculus. We shall explore various optimisation tools, such as the Euler-Lagrange equation and the direct method, discussing the advantages and limitations of each. Here we will note the importance of the various definition of convexity, which will also go over.

We will finish with an overview of some results in current research. In particular, these results will relate to the minimisation of a parameterised family of polyconvex functionals that are related with some open problems in the area of elasticity. Here we will also introduce partial differential inclusions, a generalisation of partial differential inclusions, and the problem of potential wells.