 
Summary: On pharmonic mappings in the plane
Tomasz Adamowicz
November 18, 2007
Abstract
To every pharmonic vector field in the plane with p 2 there corre
sponds a quasilinear system of first order PDE's which couples the com
plex gradients of the coordinate functions of the field. The ellipticity of
such system is proved. A relation between planar quasiregular mappings
and pharmonic fields is discussed. The pharmonic conjugate problem is
stated.
Keywords: pharmonic mapping, complex gradient, quasilinear system,
quasiregular mapping
Mathematics Subject Classification (2000): 35J45, 35J60, 35J70, 30C99
1 Introduction and preliminaries.
One of the most challenging and difficult problems in the theory of nonlinear
PDEs is to understand the geometry of solutions. In that setting the profound
position is held by the question on the topological structure of the set of critical
points. Even for the simplest nonlinear equations pursuing the answer to this
question is difficult and often requires using advanced methods. Such is the case
of the pharmonic equation  a model differential equation of nonlinear analysis
