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

本课题主要研究平面上的半线性偏微分方程和高维上的线性和非线性偏微分方程组。半线性方程由Cauchy-Riemann算子的vekua型方程驱动，方程组是内嵌CR子流形上切向Cauchy-Riemann方程的推广。本文研究的问题包括非线性偏微分方程解的解析性，以及用更一般的复向量场代替Cauchy-Riemann算子时vequa型方程解的性质。可用于求解半线性方程的工具来自于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，工具将包括在CR函数的全纯可扩展性理论中发展的工具，包括FBI变换，解析盘，以及具有粗糙系数的向量场的Baouendi-Treves近似定理。本课题的研究有望应用于偏微分方程和几何。半线性方程起源于一个几何问题，该问题涉及给定曲面上的非平凡无穷小弯曲的存在性。这个问题在物理上适用于薄壳的弹性。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极限形状的研究中。该项目的研究活动将为研究生和新近的博士们提供有意义的问题。

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.