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。所提出的工作涉及非线性偏微分方程组，这是特别重要的，因为它们被用来建模的非线性效应。例如，在复合材料的研究中出现了这样的方程。该项目将有助于对完全非线性椭圆型方程的基本理解，从而为科学家和工程师提供对各种物理过程的更敏锐的洞察力，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，首席调查员将培训博士生，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。PI建议研究具有常数sigma-k曲率的黎曼流形上的共形度量的紧性，其中k大于1且小于流形维度的一半。对于k大于或等于流形维度的一半，或者当流形局部共形平坦时，证明了紧性结果。若能成功地建立紧性结果，将会得到关于具有常Sigma-k曲率的共形度量的新的存在性结果。还提出了关于黎曼流形上常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。研究解的紧性的方法是给出流形上一类非线性椭圆型方程的爆破解的精细分析。将努力推进对共形不变方程解的进一步和深入的理解。

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.