Nonlinear Elliptic Equations and Systems and Applications

偏微分方程在物理学、工程学、几何学和许多其他领域中自然出现，它们构成了物理世界中许多现象建模的基础。 拟议的工作涉及非线性偏微分方程，这是特别重要的，由于非线性效应，他们被用来建模。例如，这种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，并最终提高复合材料制造的消费品的质量。作为项目的一部分，主要研究者将培训博士。学生，其中许多人预计将继续他们的职业生涯作为教育工作者。他们，反过来，将传达给年轻一代，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，社会。PI建议调查黎曼流形上的共形度量的紧致性，具有常数sigma-k曲率，k大于1，小于流形维数的一半。当k ≥流形维数的一半或局部共形平坦时，证明了紧性结果。在建立紧性结果的成功将导致新的存在性结果的共形度量与常数σ-k曲率。本文还提出了黎曼流形上常Q-曲率方程解的紧性的一个相关问题。PI还建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一类非线性椭圆型方程的爆破解进行精细分析。将努力推进进一步和更深入地了解共形不变方程的解决方案。

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.