Carleman estimates with nonconvex weights and Riesz rearrangement inequalities

University Of Wisconsin-Madison, Madison WI

Investigators

Abstract

DMS 0407090 Alexandru Ionescu University of Wisconsin Carleman estimates with nonconvex weights The research will focus in two main directions. The first objective is to understand the role of long-range perturbations in certain Carleman inequalities for Schr\"{o}dinger operators on Euclidean spaces. These inequalities have applications to questions concerning uniqueness of solutions of Schr\"{o}dinger equations. The second objective is to further develop real-variable methods that can be used in various aspects of analysis on semisimple Lie groups and symmetric spaces. In particular, the PI is interested in understanding Riesz rearrangement inequalities on semisimple Lie groups of high real rank. Many partial differential equations of the type the PI proposes to study originate from physics, chemistry or engineering. The role of these equations is to model certain phenomena, and their relevance is often verified numerically. The PI proposes to study these equations rigorously, and confirm the expected behavior of solutions, such as the infinite speed of propagation of solutions of large classes of nonlinear Schr\"{o}dinger equations, as mathematical theorems.