Boundary Behaviour of Partial Derivatives for Solutions to Certain Laplacian-Gradient Inequalities and Spatial QC Maps

The chapter investigates quasiconformal mappings between smooth domains in the plane and space, focusing on solutions to Laplacian-gradient inequalities. It introduces definitions and concepts such as Laplacian-gradient inequality, quasiconformal mappings, and harmonic maps. The study outlines results related to spatial versions of Kellogg’s theorem and discusses the boundary behavior of partial derivatives for functions satisfying these inequalities. Notably, it presents a local C²-coordinate method flattering the boundary and demonstrates its application to real-valued functions, offering insights into their gradient behavior near the boundary.