Georgia Institute of Technology College of Sciences

We address the deterministic homogenization, in the general context of ergodic algebras, of a doubly nonlinear problem whichgeneralizes the well known Stefan model, and includes the classical porous medium equation. It may be represented by the differential inclusion, for a real-valued function $u(x,t)$, $$0\in \frac{\partial}{\partial t}\partial_u \Psi(x/\ve,x,u)+\nabla_x\cdot \nabla_\eta\psi(x/\ve,x,t,u,\nabla u) - f(x/\ve,x,t, u), $$ on a bounded domain $\Om\subset \R^n$, $t\in(0,T)$, together with initial-boundary conditions, where $\Psi(z,x,\cdot)$ is strictly convex and $\psi(z,x,t,u,\cdot)$ is a $C^1$ convex function, both with quadratic growth,satisfying some additional technical hypotheses. As functions of the oscillatory variable, $\Psi(\cdot,x,u),\psi(\cdot,x,t,u,\eta)$ and $f(\cdot,x,t,u)$ belong to the generalized Besicovitch space $\BB^2$ associated with an arbitrary ergodic algebra $\AA$. The periodic case was addressed by Visintin (2007), based on the two-scale convergence technique. Visintin's analysis for the periodic case relies heavily on the possibility of reducing two-scale convergence to usual $L^2$ convergence in the Cartesian product $\Om\X\Pi$, where $\Pi$ is the periodic cell. This reduction is no longer possible in the case of a general ergodic algebra. To overcome this difficulty, we make essential use of the concept of two-scale Young measures for algebras with mean value, associated with uniformly bounded sequences in $L^2$.

I will describe a joint work with Vincent Millot (Paris 7) where we investigate the singular limit of a fractional GL equation towards the so-called boundary harmonic maps.

In this talk we first present some applied examples (coming from Economics and Finance) of Optimal Control Problems for Dynamical Systems with Delay (deterministic and stochastic). To treat such problems with the so called Dynamic Programming Approach one has to study a class of infinite dimensional HJB equations for which the existing theory does not apply due to their specific features (presence of state constraints, presence of first order differential operators in the state equation, possible unboundedness of the control operator). We will present some results on the existence of regular solutions for such equations and on existence of optimal control in feedback form.

We prove via explicitly constructed initial data that solutionsto the gravity-capillary wave system in R^3 representing a 2d air-waterinterface immediately fail to be C^3 with respect to the initial data ifthe initial (h_0, \psi_0) \in H^{s + 1/2} \times H^s for s<3, where h isthe free surface and \psi is the velocity potential.