Semilinear and nonlinear pdes in CR manifolds and complex variables

Shiferaw Berhanu的这一数学研究项目主要涉及复半线性偏微分方程的可解性、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。半线性方程是Cauchy Riemann算子的Vekua方程的类似物，起源于几何和物理问题。非线性方程的研究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的工具包括表征光滑性和实解析性的新的非线性傅立叶变换族，以及在CR函数的正则性理论中发展起来的方法。对于半线性方程，工具包括复向量场在各种函数空间中的可解性。这个数学研究项目的成果在几个复杂变量的函数理论、几何以及所谓的半线性和非线性偏微分方程理论中有重要的应用。在这个项目中考虑的半线性方程对于解决一个几何问题是至关重要的，这个几何问题涉及到某些类型的表面弯曲的存在，而这些弯曲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与物理和几何应用有关，例如大气现象的建模和表面极限形状最小化表面张力的研究。本项目将为研究生和青年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.