Semilinear and nonlinear pdes motivated by complex variables and CR manifolds and the Bochner extension phenomenon

由复变量和 CR 流形以及 Bochner 扩展现象驱动的半线性和非线性偏微分方程

• 批准号: 1001283
• 负责人: Shiferaw Berhanu
• 金额: $ 13.47万
• 依托单位: Temple University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2010
• 资助国家: 美国
• 起止时间: 2010-07-01 至 2013-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1001283&HistoricalAwards=false
• 关键词: Semilinear; nonlinear; pdes; motivated; complex

This project involves the study of semilinear partial differential equations in the plane and systems of linear and nonlinear partial differential equations in higher dimensions. The semilinear equations are motivated by Vekua-type equations for the Cauchy-Riemann operator while the systems of equations are generalizations of the tangential Cauchy-Riemann equations on embedded CR submanifolds. The problems to be investigated include the analyticity of solutions of nonlinear partial differential equations and the properties of the solutions of Vequa-type equations when the Cauchy-Riemann operator is replaced by more general complex vector fields. The tools that may be used for the semilinear equations come from the solvability theory for complex vector fields in various function spaces and the theory of ordinary differential equations with periodic coefficients. For the nonlinear equations, the tools will include those developed in the theory of holomorphic extendability of CR functions including the FBI transform, analytic discs, and the Baouendi-Treves approximation theorem for vector fields with rough coefficients.The research in this project is expected to have applications to partial differential equations and geometry. The semilinear equations arise from a geometrical problem that concerns the existence of nontrivial infinitesimal bendings for a given surface. This problem has physical applications to the elasticity of thin shells. The nonlinear equations arise in numerous geometrical and physical applications including in the modeling of atmospheric phenomena, and in the study of limit shapes of surfaces that minimize surface tension. The research activity on this project will be a source of meaningful problems for graduate students and recent Ph.D's.

该项目涉及平面中的半线性偏微分方程以及高维中的线性和非线性偏微分方程组的研究。半线性方程由柯西-黎曼算子的 Vekua 型方程驱动，而方程组是嵌入式 CR 子流形上切向柯西-黎曼方程的推广。 要研究的问题包括非线性偏微分方程解的解析性以及当Cauchy-Riemann算子被更一般的复向量场代替时Vequa型方程解的性质。可用于半线性方程的工具来自各种函数空间中复矢量场的可解性理论和具有周期系数的常微分方程理论。 对于非线性方程，这些工具将包括在 CR 函数的全纯可扩展性理论中开发的工具，包括 FBI 变换、解析盘和具有粗糙系数的向量场的 Bauuendi-Treves 近似定理。该项目的研究预计将应用于偏微分方程和几何。半线性方程源于一个几何问题，该问题涉及给定表面是否存在非平凡的无穷小弯曲。这个问题对于薄壳的弹性有物理应用。非线性方程出现在许多几何和物理应用中，包括大气现象的建模以及最小化表面张力的表面极限形状的研究。该项目的研究活动将为研究生和最近的博士生带来有意义的问题。