Measurable Riemann mapping theorem
Encyclopedia
In the mathematical theory of quasiconformal mapping
s in two dimensions, the measurable Riemann mapping theorem, proved by , generalizes the Riemann mapping theorem from conformal to quasiconformal homeomorphisms, and is stated as follows. Suppose that D is a simply connected domain in C that is not equal to C, and suppose that
is Lebesgue measurable and satisfies 
. Then there is a quasiconformal homeomorphism ƒ from D to the unit disk which is in the Sobolev space W1,2(D) and satisfies the corresponding Beltrami equation in the distributional sense
. As with Riemann's mapping theorem, this ƒ is unique up to 3 real parameters.
Quasiconformal mapping
In mathematical complex analysis, a quasiconformal mapping, introduced by and named by , is a homeomorphism between plane domains which to first order takes small circles to small ellipses of bounded eccentricity....
s in two dimensions, the measurable Riemann mapping theorem, proved by , generalizes the Riemann mapping theorem from conformal to quasiconformal homeomorphisms, and is stated as follows. Suppose that D is a simply connected domain in C that is not equal to C, and suppose that
is Lebesgue measurable and satisfies . Then there is a quasiconformal homeomorphism ƒ from D to the unit disk which is in the Sobolev space W1,2(D) and satisfies the corresponding Beltrami equation in the distributional senseWeak solution
In mathematics, a weak solution to an ordinary or partial differential equation is a function for which the derivatives may not all exist but which is nonetheless deemed to satisfy the equation in some precisely defined sense. There are many different definitions of weak solution, appropriate for...
. As with Riemann's mapping theorem, this ƒ is unique up to 3 real parameters.