The calculus of variations and asymptotic behaviour of models from materials science. [MAT/05, 3 ECTS]

Whilst PDEs used in physical sciences often describe equilibrium behaviour by prescribing "force balance"-type relationships within a material, the calculus of variations takes a different perspective, which is that of a free energy minimisation principle, where the optimisation takes place over an infinite dimensional vector space which represents physically admissible states. Through the Euler-Lagrange equation, which is morally a critical point condition for minimisers, we can see that in many situations the two approaches are in fact equivalent, and thus the calculus of variations gives us an extra toolkit for dealing with problems in materials science. In particular, we have the tool of Gamma-convergence, which is a notion of convergence of energy functionals which allows us to rigorously understand the asymptotic behaviour of such problems in singular parameter regimes, which often arise in physical systems where certain material parameters may be orders of magnitude larger or smaller than others.  Within this mini-course, we will introduce some fundamental aspects of the calculus of variations, such as existence and uniqueness of minimisers, and the equivalence of the Euler-Lagrange equation. Later, we will introduce the notion of Gamma-convergence, and through several classical examples consider the asymptotic behaviour of various models that are related to problems in materials science. 

Weekly Course Schedule  (two weeks from May 2 to May 13, 2022)

Tue 17:30-19:30 Aula G

Thu 16:30-18:30 Aula G

Fri  17:30-19:30 Aula G

contact person: G. Orlandi