  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.
