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

这个美洲计划合作研究项目将支持纽约市立大学的尼古拉斯·汉斯的调查。H. Lehman学院与巴西圣保罗大学的Paulo Cordaro一起研究了低阶项对退化椭圆算子的影响。 主要的兴趣是在分析正则性偏微分方程的重点是退化椭圆算子以及复杂的一阶扰动等运营商。 分析亚椭圆性将在通常意义上，在整体意义上和芽的意义上进行研究。这些属性的影响，低阶项和存在（或不存在）的双特征（Treves曲线）。 PI发现了一个算子，它在原点上有无穷多条Treves曲线，但在原点处是芽意义下的解析亚椭圆。 然而，该算子不是通常意义下的解析亚椭圆算子。 这表明，特里夫斯的猜想，至少在细菌的意义上，是错误的。 其中一个目标是了解的影响，低阶项的传统工作的Gilioli和Treves.Degenerate椭圆型偏微分方程出现在许多不同领域的数学，因此解决问题的解决在这个项目将有意义的几个领域的纯数学和应用数学，其中一个是有兴趣建立的解决方案是解析的。 甚至在经济学和热力学等遥远的领域也有应用。 巴西PI及其机构在该调查领域非常强大。 因此，美国学生不仅有机会参与前沿问题，而且通过这些交流，他们还将接触并开始参与国际合作。

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.