Mathematics

The aim of this course is an introduction to the analysis of variational problems depending on a parameter. Such problems appear in different ways, and the parameter may be constitutive, a geometric inhomogeneity scale, or a coefficient of a perturbative term. It may have different effects favoring oscillations, concentration, topological singularities, dimension-reduction, etc., some times a combination of these. Our scope is recognize of what kind is the asymptotic effect of the parameter and give a series of ways to describe this effect in the spirit of the direct methods of the Calculus of Variations. To that end we will apply our analysis to a series of prototypical problems: homogenization, relaxed Dirichlet problems, theories of thin objects, phase transitions, interfacial problems, free-discontinuity problems, Ginzburg-Landau vortices, discrete approximations, etc. The course is reasonably self-contained, and the level of prerequisites (elementary Calculus of Variations, Sobolev Spaces, Functional Analysis) will depend on the audience.