Nonlinear Elliptic Equations and Systems, and Applications

偏微分方程自然出现在物理学、工程学、几何学和许多其他领域，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，“完全非线性椭圆方程和系统”这一特殊类别尤其重要。例如，此类方程和系统出现在复合材料的理论研究中。该项目有助于更好地理解完全非线性椭圆方程和系统，从而为科学家和工程师提供对各种物理过程的深入了解，并最终提高复合材料制造的消费品的质量。作为该项目的一部分，首席研究员培训博士。学生，其中许多人预计将继续担任教育工作者。反过来，他们将向年轻一代传达他们的数学知识和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，PI在该项目的领域做出了宝贵的贡献，该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，即完全非线性 Yamabe 问题的解的存在性和紧凑性。这相当于在黎曼流形上求解二阶完全非线性椭圆（但不是一致椭圆）偏微分方程。密切相关的工作包括完全非线性尼伦伯格问题和完全非线性勒纳-尼伦伯格问题。对此类方程的理解还不够，特别是与理论更加成熟的二阶完全非线性一致椭圆方程相比。对开放问题的研究应该可以更好地理解这些椭圆方程，但不是均匀椭圆方程。它还将有助于更好地理解二阶简并椭圆完全非线性方程。这将为研究该方程以及几何和物理学中产生的其他重要非线性偏微分方程提供新的工具。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方程和系统。特别是，开发了新工具来研究一个长期悬而未决的问题，即十六维平面圆环上不可压缩平稳纳维-斯托克斯方程是否存在平滑解。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力优点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.