Theory of Nonlinear Elliptic Equations and Systems

偏微分方程在物理学、工程学、几何学和许多其他领域中自然出现，它们构成了物理世界中许多现象建模的基础。从这个角度来看，完全非线性椭圆方程和系统的特殊类别尤其重要。例如，这样的方程和系统在复合材料的理论研究中发挥作用。该项目将有助于对完全非线性椭圆方程和系统的基本理解，从而为科学家和工程师提供对各种物理过程的敏锐洞察力，并最终提高由复合材料制造的消费品的质量。作为项目的一部分，主要研究者将培训博士。学生，其中许多人预计将继续他们的职业生涯作为教育工作者，并帮助传达数学研究的长期价值，不仅科学和工程，而且最终，社会。在技术层面上，该项目是主要研究者早期工作的自然延续，其中包括对该研究领域的宝贵贡献。该项目的一部分涉及一个长期的开放问题的存在性和紧凑性的解决方案，一个完全非线性Yamabe问题。这等价于在黎曼流形上求解一个完全非线性椭圆型但不是一致椭圆型的二阶偏微分方程。目前对这类方程的理解是不够的，特别是与已知的完全非线性一致椭圆二阶方程相比，其中理论要成熟得多。这个开放的问题的研究，应导致更好地了解这些椭圆，但不是一致椭圆，方程。这也将导致更好地理解退化椭圆型完全非线性方程的二阶。该项目将为这些和其他重要的非线性偏微分方程的研究提供新的分析工具，这些方程和其他重要的非线性偏微分方程产生的几何和physics.This奖项反映了NSF的法定使命，并已被认为是值得通过评估使用基金会的智力价值和更广泛的影响审查标准的支持。

Partial differential equations arise naturally in physics, engineering, geometry, and many other fields, and they form the basis for modeling many phenomena in the physical world. The particular class of fully nonlinear elliptic equations and systems is especially important from this perspective. For instance, such equations and systems play a role in the theoretical study of composite materials. This project will contribute to a basic understanding of fully nonlinear elliptic equations and systems, thereby providing scientists and engineers with sharpened insight into various physical processes and ultimately enhancing the quality of consumer products manufactured from composites. As part of the project, the principal investigator will train Ph.D. students, many of whom are expected to continue their careers as educators and help convey the long-term value of mathematical research not only to science and engineering but also, in the end, to society. At a technical level, this project is a natural continuation of the principal investigator's earlier work, which includes valuable contributions to this research area. One part of the project concerns a longstanding open problem on the existence and compactness of solutions to a fully nonlinear Yamabe problem. This is equivalent to solving, on a Riemannian manifold, a fully nonlinear elliptic, but not uniformly elliptic, partial differential equation of second order. The current understanding of this type of equations is insufficient, especially compared to what is known for fully nonlinear uniformly elliptic second order equations, where theories are much more mature. The study of this open problem should lead to a better understanding of these elliptic, but not uniformly elliptic, equations. It will also lead to a better understanding of degenerate elliptic fully nonlinear equations of second order. This project will provide new analytical tools for the study of these and other important nonlinear partial differential equations arising from geometry and physics.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.