International: U. S. Brazil Cooperative Research: Impact of the Lower Order Terms On Degenerate Elliptic Operators

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This Americas Program collaborative research project will support investigations by Nicholas Hanges of CUNY H. H. Lehman College together with Paulo Cordaro of the University of Sao Paulo in Brazil to study the impact of the lower order terms on degenerate elliptic operators. The main interest is in analytic regularity for partial differential equations with a focus on degenerate elliptic operators as well as on complex first order perturbations of such operators. Analytic hypoellipticity will be studied in the usual sense, in the global sense and in the sense of germs. These properties are influenced by the lower order terms and by the existence (or absence) of bicharacteristics (Treves curves). The PI has discovered an operator that has an infinite number of Treves curves over the origin, yet is analytic hypoelliptic at the origin in the sense of germs. However, this operator is not analytic hypoelliptic in the usual sense. This shows that Treves' conjecture, at least in the sense of germs, is false. One of the goals is to understand the influence of the lower order terms in the tradition of the work of Gilioli and Treves.Degenerate elliptic partial differential equations arise in many different areas of mathematics, and thus the resolution of problems tackled in this project will have significance in several areas of pure and applied mathematics in which one is interested in establishing that solutions are analytic. There may even be applications in such distant fields as economics and thermodynamics. The Brazilian PI and his institution are very strong in the area of this investigation. As a result, U.S. students will have the opportunity not only to become involved with cutting-edge problems, but through these exchanges they will also be exposed to and begin to participate in international collaborations.