Semilinear and nonlinear pdes in CR manifolds and complex variables

Shiferaw Berhanu的这一数学研究项目的主要内容涉及复半线性偏微分方程解的可解性，完全非线性一阶复偏微分方程解的正则性，以及CR流形的性质，其中CR函数的强极大值原理成立。半线性方程类似于柯西-黎曼算子的Vekua方程，由几何和物理问题产生。非线性方程需要研究的问题包括解的光滑性和实解析性。可用于非线性偏微分方程组的工具包括一类新的具有光滑性和实解析性的非线性傅立叶变换，以及在CR函数的正则性理论中发展的方法。对于半线性方程，工具包括复矢量场在各种函数空间中的可解性，该数学研究项目的结果在多复变函数论、几何以及所谓的半线性和非线性偏微分方程组理论中都有重要的应用。在这个项目中考虑的半线性方程在解决涉及某些类型的曲面弯曲的存在的几何问题时是至关重要的，这反过来又对薄壳的弹性有物理应用。所研究的非线性偏微分方程式与物理和几何应用有关，例如大气现象的模拟和使表面张力最小化的表面极限形状的研究。该项目将为研究生和年轻研究人员提供许多有趣的研究问题。

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.