Distortion (mathematics)

In mathematics, the distortion is a measure of the amount by which a function from the Euclidean plane to itself distorts circles to ellipses. If the distortion of a function is equal to one, then it is conformal; if the distortion is bounded and the function is a homeomorphism, then it is quasiconformal. The distortion of a function ƒ of the plane is given by

H(z,f)=\limsup _{{r\to 0}}{\frac {\max _{{|h|=r}}|f(z+h)-f(z)|}{\min _{{|h|=r}}|f(z+h)-f(z)|}}

which is the limiting eccentricity of the ellipse produced by applying ƒ to small circles centered at z. This geometrical definition is often very difficult to work with, and the necessary analytical features can be extrapolated to the following definition. A mapping ƒ : Ω → R² from an open domain in the plane to the plane has finite distortion at a point x ∈ Ω if ƒ is in the Sobolev space W1,1
loc(Ω, R²), the Jacobian determinant J(x,ƒ) is locally integrable and does not change sign in Ω, and there is a measurable function K(x) ≥ 1 such that

|Df(x)|^{2}\leq K(x)|J(x,f)|

almost everywhere. Here Df is the weak derivative of ƒ, and |Df| is the Hilbert–Schmidt norm.

For functions on a higher-dimensional Euclidean space Rⁿ, there are more measures of distortion because there are more than two principal axes of a symmetric tensor. The pointwise information is contained in the distortion tensor

G(x,f)={\begin{cases}|J(x,f)^{{-2/n}}D^{T}f(x)Df(x)&{\text{if }}J(x,f)\not =0\\I&{\text{if }}J(x,f)=0.\end{cases}}

The outer distortion K_O and inner distortion K_I are defined via the Rayleigh quotients

K_{O}(x)=\sup _{{\xi \not =0}}{\frac {\langle G(x)\xi ,\xi \rangle ^{{n/2}}}{|\xi |^{n}}},\quad K_{O}(x)=\sup _{{\xi \not =0}}{\frac {\langle G^{{-1}}(x)\xi ,\xi \rangle ^{{n/2}}}{|\xi |^{n}}}.

The outer distortion can also be characterized by means of an inequality similar to that given in the two-dimensional case. If Ω is an open set in Rⁿ, then a function ƒ ∈ W1,1
loc(Ω,Rⁿ) has finite distortion if its Jacobian is locally integrable and does not change sign, and there is a measurable function K_O (the outer distortion) such that

|Df(x)|^{n}\leq K_{O}(x)|J(x,f)|

almost everywhere.

References

Iwaniec, Tadeusz; Martin, Gaven (2001), Geometric function theory and non-linear analysis, Oxford Mathematical Monographs, The Clarendon Press Oxford University Press, ISBN 978-0-19-850929-5, MR 1859913 .
Reshetnyak, Yu. G. (1989), Space mappings with bounded distortion, Translations of Mathematical Monographs, 73, Providence, R.I.: American Mathematical Society, ISBN 978-0-8218-4526-4, MR 994644 .

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Distortion (mathematics)

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References