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

这个项目涉及研究半线性偏微分方程在平面和系统的线性和非线性偏微分方程在更高的维度。半线性方程是由Cauchy-Riemann算子的Vekua型方程驱动的，而方程组是嵌入CR子流形上切向Cauchy-Riemann方程的推广。 所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取代时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向量场的可解性理论和周期系数常微分方程理论。 对于非线性方程，工具将包括CR函数的全纯可拓性理论，包括FBI变换，解析圆盘和粗糙系数向量场的Baouendi-Treves逼近定理。本项目的研究预计将应用于偏微分方程和几何。半线性方程产生于一个几何问题，该问题涉及给定曲面的非平凡无穷小弯曲的存在性。这个问题在薄壳的弹性问题上有物理上的应用。非线性方程出现在许多几何和物理应用中，包括在大气现象的建模中，以及在最小化表面张力的表面的极限形状的研究中。这个计画的研究活动将成为研究生和博士生有意义问题的来源。

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.