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Distortion (mathematics)

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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 ) = lim sup r → 0 max | h | = r | f ( z + h ) − f ( z ) | min | h | = r | f ( z + h ) − f ( z ) | {\displaystyle 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 → R 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 W
loc(Ω, R), 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

| D f ( x ) | 2 ≤ K ( x ) | J ( x , f ) | {\displaystyle |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 R, 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 ) = { | J ( x , f ) | − 2 / n D T f ( x ) D f ( x ) if  J ( x , f ) ≠ 0 I if  J ( x , f ) = 0. {\displaystyle 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 KO and inner distortion KI are defined via the Rayleigh quotients

K O ( x ) = sup ξ ≠ 0 ⟨ G ( x ) ξ , ξ ⟩ n / 2 | ξ | n , K I ( x ) = sup ξ ≠ 0 ⟨ G − 1 ( x ) ξ , ξ ⟩ n / 2 | ξ | n . {\displaystyle K_{O}(x)=\sup _{\xi \not =0}{\frac {\langle G(x)\xi ,\xi \rangle ^{n/2}}{|\xi |^{n}}},\quad K_{I}(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 R, then a function ƒ ∈ W
loc(Ω,R) has finite distortion if its Jacobian is locally integrable and does not change sign, and there is a measurable function KO (the outer distortion) such that

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

almost everywhere.

See also

References

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