Xavier Cabré, ICREA and Universitat Politecnica de Catalunya
Stable solutions of semilinear elliptic equations: estimates and geometry of their level sets

Luis A. Caffarelli, University of Texas, Austin

Demetrios Christodoulou, ETH, Zurich
The formation of shocks in 3-dimensional fluids

Irene Gamba, University of Texas, Austin
Energy dissipative Boltzmann type equations
Abstract. We present the elastic and inelastic homogeneous Boltzman equation. We show the solutions that are self-similar, but cannot be Gaussian states.

Bernd Kawohl, Mathematisches Institut, Universitaet zu Koeln
On a class of overdetermined boundary value problems

Omar Lakkis, University of Sussex
Regularization and numerical approximation of the stochastic Allen-Cahn problem

Ricardo Nochetto, University of Maryland, College Park
Adaptive methods for parabolic PDE (mini course)
Contents: Error control via energy and duality methods. Evolution problems for subgradient and angle-bounded operators. Degenerate parabolic problems. Parabolic variational inequalities in finance. Mean curvature flow and Allen-Cahn equation. Shape optimization and applications.

Sandro Salsa, Politecnico di Milano, Milano

Yiorgos-Sokratis Smyrlis, Univeristy of Cyprus
Mathematical theory of the method of fundamental solutions

Mete Soner, Koc University, Istanbul
Stochastic optimal control and finance. (mini course)
Abstract. In these lectures I will first outline the general structure of a stochastic optimal control problem, the method of dynamic programming and the related partial differential equations. Then, I will concentrate on applications to finance including the classical Merton problem of optimal consumption and investment. I will consider the recently studied generalizations with transaction costs and taxes. Then, I will show how the PDE techniques apply to a new class of problems called target problems. These problems correspond to a class of pricing problems called super-replication price.

Panagiotis Souganidis, University of Texas, Austin
Stochastic homogenization for nonlinear first- and second-order pde and applications. (mini course)
Abstract. The theory of homogenization of first- and second-order fully nonlinear partial differential equations in random media has attracted lately a lot of attention. Averaging problems in random environments arise naturally in a variety of applications like front propagation, phase transitions, combustion, percolation and large deviation of diffusion processes.
Viscosity solutions have been employed successfully to study periodic/almost periodic homogenization. The key step is the analysis of an auxiliary macroscopic problem, known in this context as the cell problem, which defines the effective equation (nonlinearity). The fundamental difference between the periodic/almost periodic and random settings is that the latter lacks the necessary compactness to employ pde techniques. The macroscopic problem does not have, in general, a solution. There is therefore a need to develop an alternative methodology to identify the effective equation.
In these lectures I will present in detail the main difficulties and the new results, explain the new methodology and discuss the applications.

Thaleia Zariphopoulou, University of Texas, Austin

Contributed talks

Teitur Arnarson,  
Regularity near expiry of American options indifference setting

Genoveva Burca,  
Analysis of a solid particle behavior into vertical flow

Dragos-Patru Covei,  
Semilinear and Quasilinear Problem

Michael Filippakis,  
Degree theory and multiple positive solutions for periodic problems driven by the scalar SpS-Laplacian

Maria del Mar Gonzalez,  
A perturbation argument for a Monge-Ampere type question

Pablo Groisman,  
Optimization of the first Steklov eigenvalue in domains with holes: a shape derivative approach

Sandra Rita Martinez,  
A minimum problem with free boundary in Orlicz spaces

Salvador Moll,  
A characterization of convex calibrate sets with respect to an anisotropy

Adrian Muntean,  
The motion of an aggressive chemical reaction front in porous materials: modeling, analysis and simulation via a moving-boundary approach

Christos Sourdis,  
Analysis of a singularly perturbed problem

Catalina Spataru,  
An analysis of fluid mixtures in a vented room

Beata Stehlikova,  
Averaging in two-factor interest rate models

J. Ignacio Tello,  
On the stability of a model of chemotaxis

Jesper Tidblom,  
Hardy inequalities related to many particle systems