Beltrami differential equation

Suppose that $\mu :G\subset ℂ\to ℂ$ is a measurable function, then the partial differential equation

$$f_{\overline{z}}(z)=\mu (z)f_z(z)$$

is called the Beltrami differential equation.

If furthermore and in fact $|\mu (z)|$ has a uniform bound less then 1 over the domain of definition, then the solution is a quasiconformal mapping with complex dilation (http://planetmath.org/QuasiconformalMapping) $\mu (z)$ and maximal small dilatation (http://planetmath.org/QuasiconformalMapping) $d_f={sup}_z|\mu (z)|$.

A conformal mapping has $f_{\overline{z}}\equiv 0$ and so the solution can be conformal if and only if $\mu \equiv 0$.

The partial derivatives $f_z$ and $f_{\overline{z}}$ (where $\overline{z}$ is the complex conjugate of $z$) can here be given in terms of the real and imaginary parts of $f=u+iv$ as

$f_z$ $={\displaystyle \frac{1}{2}}(u_x+v_y)+{\displaystyle \frac{i}{2}}(v_x-u_y),$

$f_{\overline{z}}$ $={\displaystyle \frac{1}{2}}(u_x-v_y)+{\displaystyle \frac{i}{2}}(v_x+u_y).$

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