Beltrami equation

In mathematics, the Beltrami equation, named after Eugenio Beltrami, is the partial differential equation

${\displaystyle {\partial w \over \partial {\overline {z}}}=\mu {\partial w \over \partial z}.}$

for w a complex distribution of the complex variable z in some open set U, with derivatives that are locally L², and where μ is a given complex function in L^∞(U) of norm less than 1, called the Beltrami coefficient, and where ${\displaystyle \partial /\partial z}$ and ${\displaystyle \partial /\partial {\overline {z}}}$ are Wirtinger derivatives. Classically this differential equation was used by Gauss to prove the existence locally of isothermal coordinates on a surface with analytic Riemannian metric. Various techniques have been developed for solving the equation. The most powerful, developed in the 1950s, provides global solutions of the equation on C and relies on the L^p theory of the Beurling transform, a singular integral operator defined on L^p(C) for all 1 < p < ∞. The same method applies equally well on the unit disk and upper half plane and plays a fundamental role in Teichmüller theory and the theory of quasiconformal mappings. Various uniformization theorems can be proved using the equation, including the measurable Riemann mapping theorem and the simultaneous uniformization theorem. The existence of conformal weldings can also be derived using the Beltrami equation. One of the simplest applications is to the Riemann mapping theorem for simply connected bounded open domains in the complex plane. When the domain has smooth boundary, elliptic regularity for the equation can be used to show that the uniformizing map from the unit disk to the domain extends to a C^∞ function from the closed disk to the closure of the domain.