On certain nonlinear elliptic PDE and quasiconfomal mapps between Euclidean surfaces

TL;DR

This paper studies quasiconformal mappings between smooth 2D surfaces in Euclidean space that satisfy specific PDEs, demonstrating they are Lipschitz continuous, with applications to conformal parametrization of such surfaces.

Contribution

It extends the analysis of quasiconformal maps satisfying PDEs to $C^{2,eta}$ surfaces with smooth boundaries, showing these maps are Lipschitz continuous.

Findings

Quasiconformal mappings satisfying certain PDEs are Lipschitz.

Extension of conformal parametrization methods to $C^{2,eta}$ surfaces.

Application of PDE techniques to surface mappings in Euclidean space.

Abstract

We mainly investigate some properties of quasiconformal mappings between smooth 2-dimensional surfaces with boundary in the Euclidean space, satisfying certain partial differential equations (inequalities) concerning Laplacian, and in particular satisfying Laplace equation and show that that these mappings are Lipschitz. Conformal parametrization of such surfaces and the method developed in our paper \cite{km} have important role in this paper.dan curves and is extended to the case of $C^{2,\alpha}$ surfaces with smooth and compact boundary.