Nonlinear Elliptic Equations and Systems and Applications

非线性椭圆方程和系统及应用

• 批准号: 1501004
• 负责人: Yanyan Li
• 金额: $ 59.04万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-06-01 至 2022-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1501004&HistoricalAwards=false
• 关键词: Nonlinear; Elliptic; Equations; Systems; Applications

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 proposed work concern nonlinear partial differential equations, which are especially important due to the nonlinear effects they are used to model. For instance, such equations turn up in the study of composite materials. This project will contribute to a basic understanding of fully nonlinear elliptic equations, thereby providing scientists and engineers with sharpened insight into various physical processes and ultimately enhancing the quality of, say, 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. 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.The PI proposes to investigate the compactness of conformal metrics on a Riemannian manifold having constant sigma-k curvature for k larger than 1 and less than half of the dimension of the manifold. For k greater than or equal to half of the dimension of the manifold, or when the manifold is locally conformally flat, the compactness result has been proved. A success in establishing the compactness results would lead to new existence results on conformal metrics with constant sigma-k curvature. A related problem on compactness of solutions to the constant Q-curvature equations on Riemannian manifolds is also proposed. The PI has also proposed to study elliptic systems arising from composite material. The approach to the study of the compactness of solutions is to give a fine analysis of blow up solutions to the type of nonlinear elliptic equations on manifolds. Efforts will be made in advancing further and deeper understanding of solutions of conformally invariant equations.

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。 所提出的工作涉及非线性偏微分方程，由于它们用于建模的非线性效应，因此尤为重要。例如，此类方程出现在复合材料的研究中。该项目将有助于对完全非线性椭圆方程的基本理解，从而使科学家和工程师能够深入了解各种物理过程，并最终提高复合材料制造的消费品等产品的质量。作为该项目的一部分，首席研究员将培训博士。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。PI建议研究黎曼流形上的共形度量的紧性，当k大于1且小于流形维数的一半时，该流形具有恒定的sigma-k曲率。当k大于或等于流形维数的一半时，或者当流形局部共形平坦时，紧性结果已得到证明。成功建立紧致性结果将导致在具有恒定 sigma-k 曲率的共形度量上产生新的存在结果。还提出了黎曼流形上常Q曲率方程解的紧性问题。 PI 还提议研究由复合材料产生的椭圆系统。研究解的紧性的方法是对流形上的非线性椭圆方程类型的爆炸解进行精细分析。我们将努力进一步加深对共形不变方程解的理解。