Beltrami differential equation

Suppose that μ:G is a measurable functionMathworldPlanetmath, then the partial differential equationMathworldPlanetmath


is called the Beltrami differential equation.

If furthermore |μ(z)|<1 and in fact |μ(z)| has a uniform bound less then 1 over the domain of definition, then the solution is a quasiconformal mapping with complex dilation ( μ(z) and maximal small dilatation ( df=supz|μ(z)|.

A conformal mappingMathworldPlanetmathPlanetmath has fz¯0 and so the solution can be conformal if and only if μ0.

The partial derivativesMathworldPlanetmath fz and fz¯ (where z¯ is the complex conjugateMathworldPlanetmath of z) can here be given in terms of the real and imaginary parts of f=u+iv as

fz =12(ux+vy)+i2(vx-uy),
fz¯ =12(ux-vy)+i2(vx+uy).


  • 1 L. V. Ahlfors. . Van Nostrand-Reinhold, Princeton, New Jersey, 1966
Title Beltrami differential equation
Canonical name BeltramiDifferentialEquation
Date of creation 2013-03-22 14:08:34
Last modified on 2013-03-22 14:08:34
Owner jirka (4157)
Last modified by jirka (4157)
Numerical id 8
Author jirka (4157)
Entry type Definition
Classification msc 35F20
Classification msc 30C62
Related topic QuasiconformalMapping