JANUARY 2013
Tuesday, January 22 - ROBERT JENSEN (Loyola University Chicago) - Singular Perturbations of Partial Differential Equations and Games (part I)
Abstract: In these lectures we will examine singular perturbations of Partial Differential Equations (PDEs) and their interpretation/motivation in the context of optimal control and differential games. We will consider a variety of results, including work due to E. N. Barron, L. C. Evans, A. Friedman, P. L. Lions, and P. E. Souganidis. Starting with a short review of the "value" of games, we will devote special attention to the dynamic programming principle and finding the PDEs (a.k.a. Hamilton-Jacobi-Bellman equations) satisfied by a "value" of these games. As we now know, a natural tool for studying solutions of these PDEs is the Crandall-Lions theory of viscosity solutions. Consequently, we also present a brief introduction to the topic, including basic definitions and results. Beginning our study of singular perturbations, we consider a simple example of differential games with Lipschitz controls, and the singular perturbation of its associated Hamilton-Jacobi-Bellman equation. Moving to deeper applications, we consider existence and $C^{1,1}$ regularity of solutions of convex, fully nonlinear, uniformly elliptic PDEs via games. Concluding our tour, we consider applications in homogenization and the approximation of second order parabolic equations by coupled systems of first order Hamilton-Jacobi equations.

Tuesday, January 29 - ROBERT JENSEN (Loyola University Chicago) - Singular Perturbations of Partial Differential Equations and Games (part II)

FEBRUARY 2013
Tuesday, February 5 - ROBERT JENSEN (Loyola University Chicago) - Singular Perturbations of Partial Differential Equations and Games (part III)

Tuesday, February 12 - ROBERT JENSEN (Loyola University Chicago) - Singular Perturbations of Partial Differential Equations and Games (part IV)

MARCH 2013
Tuesday, March 5 - No Seminar (Spring Break)

Tuesday, March 12 - ROBERT JENSEN (Loyola University Chicago) - Convex Solutions to Nonlinear PDEs

Tuesday, March 19 - RAFAL GOEBEL (Loyola University Chicago) - Duality Approach to the Hessian of the Convex Hull
Abstract: A recent result by Bob Jensen gives a formula for the Hessian of the convex hull of a collection of functions. The convex hull is the greatest convex function bounded above by each function in the collection. This talk will present an alternative approach to the result, which exploits the convex conjugacy between the convex hull operation and a pointwise maximum of a collection of functions.

Tuesday, March 26 - LUKASZ PIASECKI (Maria Curie - Sklodowska University, Lublin, Poland) - Some Results Related To The Stability Of The Fixed Point
Property For Nonexpansive Mappings
Abstract 
 
APRIL 2013
Tuesday, April 23 - ADINA CIOMAGA (University of Chicago) - Mixed Integro-Differential Equations: Regularity of Solutions and Large Time Behavior
Abstract: We introduce a new class of local-nonlocal equations that we term mixed integro-differential equations. These equations are particularly interesting, as they are degenerate both in the local and nonlocal term, but their overall behavior is driven by the local-nonlocal interaction, e.g. the fractional diffusion may give the ellipticity in one direction and the classical diffusion in thecomplementary one. We establish new Hölder and Lipschitz estimates for viscosity solutions of such equations,  by the Ishii-Lions’s method. As an application, we extend a classical result on the large time behavior for local equations, to mixed integro-differential equations, and pure nonlocal equations. Joint work with G. Barles, E. Chasseigne and C. Imbert.