This project mainly studies on the following three aspects. First, we will use the theory of harmonic mapping and the theory of topological degree to discuss Landau-Bloch type theorems and distortion theorems on solutions to some nonlinear elliptic PDEs. Second, we will apply the theory of operator algebras and the theory of PDEs to study univalence criteria of pluriharmonic mappings. The obtained results will improve the Clunie–Sheil construction of planar harmonic mappings. At last, we will apply majorants and functional analysis theory to i...