PlanetMath: Beltrami differential equation

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Beltrami differential equation

(Definition)

Suppose that $\mu : G \subset {\mathbb{C}} \rightarrow {\mathbb{C}}$ is a measurable function, then the partial differential equation

$\displaystyle f_{\bar{z}}(z) = \mu(z)f_z(z)$

is called the Beltrami differential equation.

If furthermore $\lvert \mu(z) \rvert < 1$ and in fact $\lvert \mu(z) \rvert$ has a uniform bound less then 1 over the domain of definition, then the solution is a quasiconformal mapping with complex dilation $\mu(z)$ and maximal small dilatation $d_f = \sup_z \lvert \mu(z) \rvert$ .

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

The partial derivatives and $f_{\bar{z}}$ (where $\bar{z}$ is the complex conjugate of ) can here be given in terms of the real and imaginary parts of as

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

Bibliography

1: L. V. Ahlfors. Lectures on Quasiconformal Mappings. Van Nostrand-Reinhold, Princeton, New Jersey, 1966

"Beltrami differential equation" is owned by jirka.

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AMS MSC:	35F20 (Partial differential equations :: General first-order equations and systems :: General theory of nonlinear first-order PDE)
	30C62 (Functions of a complex variable :: Geometric function theory :: Quasiconformal mappings in the plane)

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