Georgia Institute of Technology College of Sciences

In this talk I will present Hamiltonian identities for elliptic PDEs and systems of PDEs. I will also show some interesting applications of these identities to problems related to solutions of some nonlinear elliptic equations in the entire space or plane. In particular, I will give a rigorous proof to the Young's law in triple junction configuration for a vector-valued Allen Cahn model arising in phase transition; a necessary condition for the existence of certain saddle solutions for Allen-Cahn equation with asymmetric double well potential will be derived, and the structure of level sets of general saddle solutions will also be discussed.

Living systems are subject to constant evolution through the two processes of mutations and selection, a principle discovered by Darwin. In a very simple, general, and idealized description, their environment can be considered as a nutrient shared by all the population. This allows certain individuals, characterized by a 'phenotypical trait', to expand faster because they are better adapted to the environment. This leads to select the 'best fitted trait' in the population (singular point of the system). On the other hand, the new-born population undergoes small variance on the trait under the effect of genetic mutations. In these circumstances, is it possible to describe the dynamical evolution of the current trait? We will give a mathematical model of such dynamics, based on parabolic equations, and show that an asymptotic method allows us to formalize precisely the concepts of monomorphic or polymorphic population. Then, we can describe the evolution of the 'best fitted trait' and eventually compute various forms of branching points, which represent the cohabitation of two different populations. The concepts are based on the asymptotic analysis of the above mentioned parabolic equations, one appropriately rescaled. This leads to concentrations of the solutions and the difficulty is to evaluate the weight and position of the moving Dirac masses that describe the population. We will show that a new type of Hamilton-Jacobi equation, with constraints, naturally describes this asymptotic. Some additional theoretical questions as uniqueness for the limiting H.-J. equation will also be addressed. This work is based on collaborations with O. Diekmann, P.-E. Jabin, S. Mischler, S. Cuadrado, J. Carrillo, S. Genieys, M. Gauduchon and G. Barles.

The usual boundary condition adjoined to a second order elliptic equation is the Dirichlet problem, which prescribes the values of the solution on the boundary. In many applications, this is not the natural boundary condition. Instead, the value of some directional derivative is given at each point of the boundary. Such problems are usually considered a minor variation of the Dirichlet condition, but this talk will show that this problem has a life of its own. For example, if the direction changes continuously, then it is possible for the solution to be continuously differentiable up to a merely Lipschitz boundary. In addition, it's possible to get smooth solutions when the direction changes discontinuously as well.

We consider a system of hyperbolic-parabolic equations describing the material instability mechanism associated to the formation of shear bands at high strain-rate plastic deformations of metals. Systematic numerical runs are performed that shed light on the behavior of this system on various parameter regimes. We consider then the case of adiabatic shearing and derive a quantitative criterion for the onset of instability: Using ideas from the theory of relaxation systems we derive equations that describe the effective behavior of the system. The effective equation turns out to be a forward-backward parabolic equation regularized by fourth order term (joint work with Th. Katsaounis and Th. Baxevanis, Univ. of Crete).