Nonlinear Elliptic Equations and Systems, and Applications

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，“完全非线性椭圆方程和系统”这一类特别重要。例如，这样的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，首席研究员培训博士生，其中许多人有望继续他们的教育事业。反过来，他们将向更年轻的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在项目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开放的问题，即全非线性Yamabe问题解的存在性和紧致性。这相当于在黎曼流形上求解一个完全非线性的二阶椭圆（但不是一致椭圆）偏微分方程。密切相关的工作包括一个完全非线性的尼伦堡问题和一个完全非线性的洛厄纳-尼伦堡问题。对这类方程的认识还不够充分，特别是与理论成熟得多的二阶完全非线性一致椭圆方程的认识相比。对开放问题的研究应该能使我们更好地理解这些椭圆型方程，而不是均匀椭圆型方程。这也将有助于更好地理解二阶退化椭圆型全非线性方程。这将为研究这一非线性偏微分方程以及其他由几何和物理产生的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，开发了新的工具来研究一个长期存在的开放问题，即不可压缩平稳Navier-Stokes方程在16维平面环面上的光滑解的存在性。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 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.