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.
