Gibson 1st Floor SR

Γ-convergence is a variational convergence on functionals. The explicit characterization of the integrand of the Γ-limit of sequences of integral functionals with periodic integrands is by now well known. Here we focus on the explicit characterization of the limit energy density of a sequence of functionals with non-periodic integrands. Such characterization is achieved in terms of the Young measure associated with relevant sequences of functions. Interesting examples are considered.

We present a selection of recent results pertaining to Hessian

and Monge-Ampere equations, where the Hessian matrix is augmented by a

matrix valued lower order operator. Equations of this type arise in

conformal geometry, geometric optics and optimal transportation.In

particular we will discuss structure conditions, due to Ma,Wang and

myself, which imply the regularity of solutions.These conditions are a

refinement of a condition used originally by Pogorelev for general

equations of Monge-Ampere type in two variables and called strong

ellipticity by him.

In space dimension 2, it is well-known that the Smoluchowski-Poisson

system (also called the simplified or parabolic-elliptic Keller-Segel

chemotaxis model) exhibits the following phenomenon: there is a critical

mass above which all solutions blow up in finite time while all solutions

are global below that critical mass. We will investigate the case of the

critical mass along with the stability of self-similar solutions with

lower masses. We next consider a generalization to several space

dimensions which involves a nonlinear diffusion and show that a similar

phenomenon takes place but with some different features.

I will begin by talking briefly about the Lavrentiev phenomenon and its implications for computations. In short, if a minimization problem exhibits a Lavrentiev gap then `naive' numerical methods cannot be used to solve it. In the past, several regularization techniques have been used to overcome this difficulty. I will briefly mention them and discuss their strengths and weaknesses.

The main part of the talk will be concerned with a class of convex problems, and I will show that for this class, relatively simple numerical methods, namely (i) the Crouzeix--Raviart FEM and (ii) the P2-FEM with under-integration, can successfully overcome the Lavrentiev gap.

I will talk about recent work concerning the Onsager equation on metric

spaces. I will describe a framework for the study of equilibria of

melts of corpora -- bodies with finitely many

degrees of freedom, such as stick-and-ball models of molecules.