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

这个美洲计划的合作研究项目将支持纽约市立大学H. H. Lehman学院的Nicholas Hanges和巴西圣保罗大学的Paulo Cordaro的研究，以研究低阶项对简并椭圆算子的影响。主要研究偏微分方程的解析正则性，重点研究退化椭圆算子以及这类算子的复一阶微扰。解析亚椭圆性将在通常意义上、全局意义上和细菌意义上进行研究。这些性质受到低阶项和双特征（特里夫斯曲线）的存在（或不存在）的影响。PI发现了一个算子，它在原点上有无限多条特里夫斯曲线，但在原点上是解析的半椭圆。然而，这个算子不是通常意义上的解析半椭圆算子。这表明特里夫斯的猜想，至少在细菌的意义上，是错误的。其中一个目标是理解低阶项在吉利奥利和特里夫斯传统工作中的影响。退化椭圆型偏微分方程出现在许多不同的数学领域，因此，在这个项目中解决的问题将在纯数学和应用数学的几个领域具有重要意义，在这些领域中，人们有兴趣建立解是解析的。它甚至可以应用于经济学和热力学等遥远的领域。巴西私家侦探和他的机构在这项调查领域非常强大。因此，美国学生不仅将有机会参与最前沿的问题，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.