Nonlinear Elliptic Equations and Applications

该项目将继续由首席研究员对Yamabe问题的完全非线性版本的解的存在性和紧性的早期研究。这类方程已被广泛研究。该项目将特别关注与上述问题有关的紧凑性问题。这些问题的澄清是进一步研究椭圆方程的基础，它的研究将有助于我们更好地理解非线性椭圆方程，但不一定是一致椭圆方程，以及非线性退化椭圆方程。 主要研究者还将研究在复合材料建模中出现的椭圆系统。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。从这个角度来看，“完全非线性椭圆”方程的特殊类别尤其重要。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。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。

This project will continue earlier studies by the principal investigator on the existence and compactness of solutions to a fully nonlinear version of the Yamabe problem. Such equations have been studied extensively. The project will have a special focus on compactness issues related to the aforementioned problem. The clarification of these issues is needed for further progress to be made on it. The research should lead to a better understanding of nonlinear elliptic, but not necessarily uniformly elliptic, partial differential equations, as well as of nonlinear degenerate elliptic equations. The principal investigator will also study elliptic systems that arise in the modeling of composite materials.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 is especially important from this perspective. 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.