**Every wednesday** at **4pm** in **Eckhart 202**.

Mathematical challenges from nonlinear fiber optics.

The propagation of pulses through an dispersion managed glass fiber cable is described by the Gabitov-Turitsyn equation, which is a non-local version of the non-linear Schrödinger equation. This equation has been extensively studied numerically and on the level of theoretical physics due to its enormous practical relevance in the modeling of signal-transfer through ultra-high hight speed glass-fiber cables, but rigorous results are rare. As a test: google `dispersion management' and you'll get an overwhelming amount of hits (ca 628,000 at the moment on google scholar) but only very few are rigorous (I know of 7).

We describe recent work on the decay and regularity properties of stationary solutions of the Gabitov-Turitsyn equation (the so-called dispersion management solitons). Our results include a simple proof of existence of solutions of the dispersion management equation, regularity of weak solutions and, most recently, a proof of exponential decay of dispersion management solitons. This is joint work with Burak Erdoğan (UIUC) and Young-Ran Lee (Sogang University, Seoul, Korea).

Shock Reflection and Free Boundary Problems.

In this talk we will start with discussion of shock reflection phenomena. Then we will describe the results on the existence and stability of global solutions to regular shock reflection for all wedge angles up to the sonic angle. The approach is to reduce the shock reflection problem to a free boundary problem for a nonlinear elliptic equation, with ellipticity degenerate near a part of the boundary (the sonic arc). We will discuss techniques to handle such free boundary problems and degenerate elliptic equations. This talk is based on the joint work with Gui-Qiang Chen.

Best Dynamics for Continuous Two-Player Games

will present some joint work with E. Barron and R. Goebel in which we extend the convergence result of Hofbauer and Sorin for the best response differential inclusions coming from a continuous payoff function U(x,y) in a zero sum game. The original result was limited to functions which were convex in one variable and concave in the other. Our result extends the original to functions having convex level sets in each variable. I will also give a counter-example illustrating the obstruction of improving this to n-player non-zero sum games.

May 5th

Dispersive corrections to Navier-Stokes.

In this talk we will present some joint work with C.D. Levermore and K. Trivisa on a system which has dispersive corrections to Navier Stokes. This system is proposed by Levermore to unify classical fluid equations with ghost-effect systems which are beyond Navier-Stokes. We show some analysis of this system, including its local well-posedness and a low Mach number limit to a ghost-effect system.

Generalized principal eigenvalues and maximum principle for elliptic operators in unbounded domains.

It is well known that, for a linear elliptic operator in a bounded smooth domain, the maximum principle holds if and only if the principal eigenvalue is positive. This property was extended to non-smooth domains by Berestycki, Nirenberg and Varadhan in '94, by introducing the notion of generalized principal eigenvalue. That same notion can be defined in the case of unbounded domains, but it does not provide a characterization for the maximum principle to hold. Instead, we will show that a necessary and a sufficient condition can be expressed in terms of two other generalizations of the principal eigenvalue. We will further present a condition ensuring the equivalence of the three notions, which applies, in particular, when the zero-order term of the operator is negative at infinity. These results are in a joint work with H. Berestycki.

Large exit times of diffusions with incompressible drift.

Let Ω be a bounded domain in R^{n}. For x ∈ Ω, let τ(x) be the expected exit time from Ω of a diffusing particle starting at x and advected by an incompressible flow u. The question I will address is whichincompressibleflows maximise τ. That is, which flows u are most efficient in the creation of hot spots.

Surprisingly, among all simply connected domains in two dimensions, the discs are the only ones for which the zero flow u = 0 maximises ||τ||_{L∞}. A study of such flows leads very quickly to a peculiar non-linear PDE for which we are unable to prove anything. This is joint work with Alexei Novikov, Lenya Ryzhik, and Andrej Zlatos.

On De Giorgi Conjecture and Beyond.

A celebrated conjecture due to De Giorgi states that any bounded solution of the Allen-Cahn equation ∆u + (1−u^{2}) u = 0 in R^{N}with [(∂u)/(∂y_{N})] > 0 must be such that its level sets {u=λ} are all hyperplanes,for dimension N ≤ 8. Positve answers have been made by Ghoussoub-Gui (N=2), Ambrosio-Cabre (N=3) and Savin (4 ≤ N ≤ 8). In this talk, I will first discuss the counterexample in N = 9, based on Bombieri-De Giorgi-Giusti minimal graph. Then I will discuss an intricate relation between complete, embedded minimal surfaces with finite total curvature in Rat least^{3}, and bounded, entire solutions with finite Morse index of the Allen-Cahn equation. Our construction suggests parallels of De Giorgi conjecture for general bounded solutions of finite Morse index. (Joint work with M. del Pino and M. Kowalczyk)

New optimal microstructures and sharp restrictions on the attainable Hashin-Shtrikman bounds for multiphase composite materials.

The problem of the exact description of the set of effective conductivities of multiphase composite materials is addressed by deriving necessary conditions and sufficient conditions for the best known bounds --- the Hashin-Shtrikman's bounds --- to be attainable. The necessary condition is obtained by using a null lagrangian; the sufficient condition is achieved by constructing new optimal microstructures. Specialized to three-phase composites, these conditions yield a necessary and sufficient condition. For more general situations, parametric studies may be easily performed and examples are provided.

Modeling depolymerization-driven flow in single crawling cells and dipole stresses underlying collective migration during wound healing.

For questions, contact Luis Silvestre at