Nonlinear Elliptic Equations and Systems, and Applications

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，在复合材料的理论研究中就出现了这样的方程和系统。该项目有助于更好地了解完全非线性的椭圆方程和系统，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，首席调查员培训博士生，其中许多人预计将继续他们的教育工作者职业生涯。反过来，他们将把他们的数学知识和数学研究的长期价值传递给更年轻的一代，不仅对科学和工程，而且最终对社会。在技术层面上，国际数学奖在项目领域做出了宝贵的贡献，这一奖项支持的工作是他早期工作的自然延续。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(但不是一致椭圆型)偏微分方程。与此密切相关的工作包括完全非线性的Nirenberg问题和完全非线性的Loewner-Nirenberg问题。对这类方程的认识还不够深入，尤其是与理论比较成熟的完全非线性二阶一致椭圆型方程相比。对开放问题的研究应该有助于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。这将为研究几何和物理中产生的这个和其他重要的非线性偏微分方程组提供新的工具。该项目的另一部分涉及流体和复合材料研究中出现的椭圆型方程和系统。特别是，开发了新的工具来研究一个长期悬而未决的问题，即不可压缩定常Navier-Stokes方程在16维平面环面上的光滑解的存在性。这一奖项反映了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 turn up in the theoretical study of composite materials. This project contributes to a better 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 trains Ph.D. students, many of whom are expected to continue their careers as educators. They, in turn, will convey to even younger generations both their mathematical knowledge and the long-term value of mathematical research not only to science and engineering but also, in the end, to society.At a technical level, the PI has made valuable contributions in the areas of the project and the work supported by this award is a natural continuation of his earlier work. One part of the project concerns a long-standing 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. Closely related work includes a fully nonlinear Nirenberg problem and a fully nonlinear Loewner-Nirenberg problem. There has not been enough understanding for such type of equations, especially comparing to that available for fully nonlinear uniformly elliptic equations of second order where the theory is much more mature. The study of the open problem should lead to 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 will provide new tools in the study of this and other important nonlinear partial differential equations arising from geometry and physics. Another part of the project concerns elliptic equations and systems arising in the study of fluids and composite materials. In particular, new tools are developed to study a long-standing open problem on the existence of smooth solutions to the incompressible stationary Navier-Stokes equations on a flat torus of dimension sixteen.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.