Some Recent Applications of Carleman Estimates

Carlos Kenig
Department of Mathematics
University of Chicago

Abstract: In this course we will discuss the method of Carleman estimates to prove uniqueness theorems. We will start out by recalling Carleman's classical unique continuation theorem for second order elliptic equations and then obtaining some new quantitative versions of it. These are instrumental in the recent proof by Bourgain-Kenig of Anderson localization for the Bernoulli model in higher dimensions. We will then move on to analogs for parabolic operators (joint work of Escauriaza, Kenig, Ponce and Vega) which settles a conjecture of Landis-Oleinik. This is in turn connected to the work of Escauriaza-Seregin-Sverak (2002) on regularity of a class of weak solutions to Navier-Stokes. Finally, we will discuss versions of these quantitative uniqueness results for dispersive equations, such as non-linear Schrodinger and KdV equations. (This is also joint work of Escauriaza. Kenig, Ponce and Vega.) These results provide a natural extension of Hardy's uncertainty principle to dispersive equations.