Theory of Nonlinear Elliptic Equations and Systems

偏微分方程自然地出现在物理、工程、几何和许多其他领域，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，一类特殊的完全非线性椭圆方程和系统尤为重要。例如，这样的方程和系统在复合材料的理论研究中发挥了作用。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。作为该项目的一部分，首席研究员将培养博士生，其中许多人有望继续他们的教育事业，并帮助传达数学研究的长期价值，不仅对科学和工程，而且最终对社会。在技术层面上，该项目是首席研究员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开放的问题，即完全非线性Yamabe问题的解的存在性和紧致性。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均匀椭圆二阶方程相比，后者的理论要成熟得多。对这一开放问题的研究将有助于更好地理解这些椭圆方程，但不是均匀椭圆方程。这也将有助于更好地理解二阶退化椭圆型全非线性方程。该项目将为研究这些和其他重要的几何和物理非线性偏微分方程提供新的分析工具。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.