Theory of Nonlinear Elliptic Equations and Systems

偏微分方程自然而然地出现在物理、工程、几何和许多其他领域，它们构成了对物理世界中的许多现象进行建模的基础。从这个角度来看，一类特殊的完全非线性椭圆型方程和系统尤为重要。例如，这些方程和系统在复合材料的理论研究中起着重要的作用。该项目将有助于对完全非线性椭圆方程和系统的基本了解，从而为科学家和工程师提供对各种物理过程的更敏锐的洞察力，并最终提高由复合材料制造的消费产品的质量。作为该项目的一部分，首席研究人员将培训博士生，他们中的许多人预计将继续他们的教育工作者职业生涯，并帮助传递数学研究的长期价值，不仅是对科学和工程，而且最终对社会。在技术层面上，这个项目是首席研究员早期工作的自然延续，包括对这一研究领域的宝贵贡献。该项目的一部分涉及一个长期悬而未决的问题，即一个完全非线性的Yamabe问题解的存在性和紧性。这等价于在黎曼流形上求解一个完全非线性的二阶椭圆型偏微分方程，但不是一致椭圆型的。目前对这类方程的了解还不够充分，特别是与理论要成熟得多的完全非线性一致椭圆型二阶方程相比。对这个开放问题的研究应该会使我们更好地理解这些椭圆型方程，但不是一致椭圆型方程。它还将有助于更好地理解退化的二阶椭圆型完全非线性方程。这个项目将提供新的分析工具来研究这些和其他由几何和物理产生的重要的非线性偏微分方程式。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势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.