Semilinear and nonlinear pdes in CR manifolds and complex variables

CR 流形和复变量中的半线性和非线性偏微分方程

• 批准号: 1300026
• 负责人: Shiferaw Berhanu
• 金额: $ 16.59万
• 依托单位: Temple University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2013
• 资助国家: 美国
• 起止时间: 2013-07-01 至 2016-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1300026&HistoricalAwards=false
• 关键词: Semilinear; nonlinear; pdes; CR; manifolds

The main parts of this mathematics research project by Shiferaw Berhanu involve the solvability of complex semi-linear partial differential equations, the regularity of the solutions of fully nonlinear, first order complex partial differential equations, and the properties of CR manifolds where the strong maximum principle holds for CR functions. The semi-linear equations are analogues of Vekua's equation for the Cauchy Riemann operator and arise from geometrical and physical problems. The questions to be investigated for the nonlinear equations include the smoothness and real analyticity of the solutions. The tools that may be used for the nonlinear partial differential equations include a new family of nonlinear Fourier transforms that characterize smoothness and real analyticity and the methods developed in the regularity theory of CR functions. For the semi-linear equations, the tools include the solvability of complex vector fields in various function spaces.Results from this mathematics research project have important applications to function theory of several complex variables, geometry, and the theory of so-called semi-linear and non-linear partial differential equations. The semi-linear equations considered in this project are crucial in solving a geometric problem that involves the existence of certain types of bending of surfaces which in turn has physical applications to the elasticity of thin shells. The non-linear partial differential equations under study are relevant to physical and geometrical applications such as the modeling of atmospheric phenomena and the study of limit shapes of surfaces that minimize surface tension. This project will provide many interesting research problems to graduate students and young researchers.

这个数学研究项目的主要部分Shiferaw Berhanu涉及复杂的半线性偏微分方程的可解性，完全非线性，一阶复杂偏微分方程的解决方案的规律性，以及CR流形的性质，其中强极大值原理适用于CR函数。半线性方程是柯西黎曼算子的Vekua方程的类似物，并且产生于几何和物理问题。非线性方程组的研究问题包括解的光滑性和真实的解析性。可用于非线性偏微分方程的工具包括一个新的非线性傅立叶变换族，其特征在于光滑性和真实的解析性，以及在CR函数的正则性理论中开发的方法。 对于半线性方程组，其工具包括复向量场在各种函数空间中的可解性，这一数学研究项目的结果对多复变函数论、几何学以及所谓的半线性和非线性偏微分方程理论有重要的应用。在这个项目中考虑的半线性方程组是至关重要的几何问题，涉及存在的某些类型的弯曲的表面，这反过来又有物理应用的弹性薄壳。研究中的非线性偏微分方程与物理和几何应用有关，例如大气现象的建模和研究使表面张力最小化的表面的极限形状。这个项目将为研究生和年轻的研究人员提供许多有趣的研究问题。