Current contact: Gerardo Mendoza
The seminar takes place Mondays 2:40 - 3:30 pm in Wachman 617. Click on title for abstract.
Marius Mitrea, University of Missouri
Jose Maria Martell, ICMAT, Madrid, Spain
Murat Akman, University of Connecticut
Atilla Yilmaz, Temple University
I will present joint work with Elena Kosygina and Ofer Zeitouni in which we prove the homogenization of a class of one-dimensional viscous Hamilton-Jacobi equations with random Hamiltonians that are nonconvex in the gradient variable. Due to the special form of the Hamiltonians, the solutions of these PDEs with linear initial conditions have representations involving exponential expectations of controlled Brownian motion in a random potential. The effective Hamiltonian is the asymptotic rate of growth of these exponential expectations as time goes to infinity and is explicit in terms of the tilted free energy of (uncontrolled) Brownian motion in a random potential. The proof involves large deviations, construction of correctors which lead to exponential martingales, and identification of asymptotically optimal policies.
Nestor Guillen, University of Massachusetts, Amherst
Francisco Villarroya, Temple University
I will introduce a Tb Theorem that characterizes all Calderón-Zygmund operators that extend compactly on $L^p(\mathbb R^n)$ by means of testing functions as general as possible. In the classical theory of boundedness, the testing functions satisfy a non-degeneracy property called accretivity, which essentially implies the existence of a positive lower bound for the absolute value of the averages of the testing functions over all dyadic cubes. However, in the the setting of compact operators, due to their better properties, the hypothesis of accretivity can be relaxed to a large extend. As a by-product, the results also describe those Calderon-Zygmund operators whose boundedness can be checked with non-accretive testing functions.
Alessia Elisabetta Kogoj, University of Urbino "Carlo Bo"
Several Liouville-type theorems are presented, related to evolution equations on Lie Groups and to their stationary counterpart. Our results apply in particular to the heat operator on Carnot groups, to linearized Kolmogorov operators and to operators of Fokker-Planck-type like the Mumford operator. An application to the uniqueness for the Cauchy problem is also shown.
These results are based on joint publications with A. Bonfiglioli, E. Lanconelli, Y. Pinchover and S. Polidoro.