Beltrami equation

In mathematics, the Beltrami equation, named after Eugenio Beltrami, is the partial differential equation In mathematics, the Beltrami equation, named after Eugenio Beltrami, is the partial differential equation 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. 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 Lp theory of the Beurling transform, a singular integral operator defined on LP(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. Consider a 2-dimensional Riemannian manifold, say with an (x, y) coordinate system on it. The curves of constant x on that surface typically don't intersect the curves of constant y orthogonally. A new coordinate system (u, v) is called isothermal when the curves of constant u do intersect the curves of constant v orthogonally and, in addition, the parameter spacing is the same — that is, for small enough h, the little region with a ≤ u ≤ a + h {displaystyle aleq uleq a+h} and b ≤ v ≤ b + h {displaystyle bleq vleq b+h} is nearly square, not just nearly rectangular. The Beltrami equation is the equation that has to be solved in order to construct isothermal coordinate systems. To see how this works, let S be an open set in C and let be a smooth metric g on S. The first fundamental form of g is a positive real matrix (E > 0, G > 0, EG − F2 > 0) that varies smoothly with x and y. The Beltrami coefficient of the metric g is defined to be