# Beltrami differential equation

Suppose that $\mu :G\subset \u2102\to \u2102$ 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

${f}_{z}$ | $={\displaystyle \frac{1}{2}}({u}_{x}+{v}_{y})+{\displaystyle \frac{i}{2}}({v}_{x}-{u}_{y}),$ | ||

${f}_{\overline{z}}$ | $={\displaystyle \frac{1}{2}}({u}_{x}-{v}_{y})+{\displaystyle \frac{i}{2}}({v}_{x}+{u}_{y}).$ |

## References

- 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 |