View source for Quasi-regular mapping

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''mapping with bounded distortion''

Let $G$ be an open connected subset of the Euclidean space ${\bf R} ^ { n }$, $n \geq 2$, and let $f : G \rightarrow \mathbf{R} ^ { n }$ be a continuous mapping of Sobolev class $W _ { \text{loc} } ^ { 1 , n } ( G )$ (cf. also [[Sobolev space|Sobolev space]]). Then $f$ is said to be quasi-regular if there is a number $K \in [ 1 , \infty )$ such that the inequality

\begin{equation*} | f ^ { \prime } ( x ) | ^ { n } \leq K J _ { f } ( x ) \end{equation*}

is satisfied almost everywhere (a.e.) in $G.$ Here, $f ^ { \prime } ( x )$ denotes the formal derivative of $f$ at $x$, i.e. the $( n \times n )$-matrix of partial derivatives of the coordinate functions $f_j$ of $f = ( f _ { 1 } , \dots , f _ { n } )$. Further,

\begin{equation*} | f ^ { \prime } ( x ) | = \operatorname { max } \{ | f ^ { \prime } ( x ) h | : | h | = 1 \} \end{equation*}

and $J _ { f } ( x )$ is the Jacobian determinant (cf. also [[Jacobian|Jacobian]]) of $f$ at $x$. The smallest $K \geq 1$ in the above inequality is called the outer dilatation of $f$ and is denoted by $K_{\text{O}} ( f )$. If $f$ is quasi-regular, then the inner dilatation $K _ { \text{I} } ( f )$ is the smallest constant $K \geq 1$ in the inequality

\begin{equation*} J _ { f } ( x ) \leq K \text{l} ( f ^ { \prime } ( x ) ) ^ { n }, \end{equation*}

\begin{equation*} \operatorname { l(f } ^ { \prime } ( x ) ) = \operatorname { min } \{ | f ^ { \prime } ( x ) h | : | h | = 1 \}. \end{equation*}

The maximal dilatation is the number $K ( f ) = \operatorname { max } \{ K _ { \text{O} } (\, f ) , K _ { \text{I} } (\, f ) \}$, and one says that $f$ is $K$-quasi-regular if $K ( f ) \leq K$.

==Particular cases.==
For the dimension $n = 2$ and $K = 1$, the class of $K$-quasi-regular mappings agrees with that of the complex-analytic functions (cf. also [[Analytic function|Analytic function]]). Injective quasi-regular mappings in dimensions $n \geq 2$ are called quasi-conformal (see [[Quasi-conformal mapping|Quasi-conformal mapping]]; [[#References|[a4]]], [[#References|[a8]]], [[#References|[a10]]]). For $n = 2$ and $K \geq 1$, a $K$-quasi-regular mapping $f : G \rightarrow \mathbf{R} ^ { 2 }$ can be represented in the form $\varphi \circ w$, where $w : G \rightarrow G ^ { \prime }$ is a $K$-quasi-conformal [[Homeomorphism|homeomorphism]] and $\varphi : G ^ { \prime } \rightarrow \mathbf{R} ^ { 2 }$ is an analytic function (Stoilow's theorem). There is no such representation in dimensions $n \geq 3$. Note that for all dimensions $n \geq 2$ and $K &gt; 1$ the set of those points where a $K$-quasi-conformal mapping is not differentiable may be non-empty and the behaviour of the mapping may be very curious at the points of this set, which has also [[Lebesgue measure|Lebesgue measure]] zero. Thus, there is a substantial difference between the two cases $K &gt; 1$ and $K = 1$.

For higher dimensions $n \geq 3$ the theory of quasi-regular mappings is essentially different from the plane case $n = 2$. There are several reasons for this:

a) there are neither general existence theorems nor counterparts of power series expansions in higher dimensions;

b) the usual methods of function theory are not applicable in the higher-dimensional setup;

c) in the plane case the class of conformal mappings is very rich (cf. also [[Conformal mapping|Conformal mapping]]), while in higher dimensions it is very small (J. Liouville proved that for $n \geq 3$ and $K = 1$, sufficiently smooth $K$-quasi-conformal mappings are restrictions of Möbius transformations, see [[Quasi-conformal mapping|Quasi-conformal mapping]]);

d) for dimensions $n \geq 3$ the branch set (i.e. the set of those points at which the mapping fails to be a local homeomorphism) is more complicated than in the two-dimensional case; for instance, it does not contain isolated points (see also [[Zorich theorem|Zorich theorem]]).

In spite of these difficulties, many basic properties of analytic functions have their counterparts for quasi-regular mappings. In a pioneering series of papers, Yu.G. Reshetnyak proved in 1966–1969 that these mappings share the fundamental topological properties of complex-analytic functions: non-constant quasi-regular mappings are discrete, open and sense-preserving. He also proved important convergence theorems for these mappings and several analytic properties: they preserve sets of zero Lebesgue $n$-measure, are differentiable almost everywhere and are Hölder continuous (cf. also [[Hölder condition|Hölder condition]]). The Reshetnyak theory (which uses the phrase mapping with bounded distortion for  "quasi-regular mapping" ) makes use of [[Sobolev space]]s, [[potential theory]], [[partial differential equation]]s, [[calculus of variations]], and differential geometry. These results led to the discovery of the connection between partial differential equations and quasi-regular mappings. One of the main results says that if $f$ is quasi-regular and non-zero, then $\operatorname{log}| f ( x ) |$ is a solution of a certain non-linear [[Elliptic partial differential equation|elliptic partial differential equation]]. For $n = 2$ and $K = 1$ these equations reduce to the familiar [[Laplace equation|Laplace equation]] and the corresponding potential theory is linear (see [[#References|[a5]]]).

The word  "quasi-regular"  was introduced in this meaning in 1969 by O. Martio, S. Rickman and J. Väisälä, who found another approach to quasi-regular mappings. Their approach uses techniques and tools from the theory of spatial quasi-conformal mappings and is based, in part, on Reshetnyak's fundamental results. The tools they use in their 1969–1972 work involve the modulus of a family of curves (the [[Extremal length|extremal length]]) and capacities of condensers. Their work showed the power of the method of the modulus of a family of curves, made these mappings more widely known, and led to a series of important results (cf. below).

Some of the topics considered for quasi-regular mappings include:

i) stability theory, which refers to the study of $K$-quasi-regular mappings with dilatation $K &gt; 1$ close to one, see [[#References|[a6]]];

ii) [[Value-distribution theory|value-distribution theory]], such as the analogue of the [[Picard theorem|Picard theorem]] and its refinements, see [[#References|[a7]]];

iii) non-linear potential theory (cf. also [[Potential theory|Potential theory]]) and connection of quasi-regular mappings to partial differential equations, see [[#References|[a2]]];

iv) geometric analysis, differential forms, and applications to elasticity theory, see [[#References|[a3]]];

v) conformal invariants and quasi-regular mappings, see [[#References|[a9]]], [[#References|[a1]]].

====References====
<table><tr><td valign="top">[a1]</td> <td valign="top">  G.D. Anderson,   M.K. Vamanamurthy,   M. Vuorinen,   "Conformal invariants, inequalities and quasiconformal mappings" , Wiley  (1997)  (book with a software diskette)</td></tr><tr><td valign="top">[a2]</td> <td valign="top">  J. Heinonen,   T. Kilpeläinen,   O. Martio,   "Nonlinear potential theory of degenerate elliptic equations" , ''Oxford Math. Monographs'' , Oxford Univ. Press  (1993)</td></tr><tr><td valign="top">[a3]</td> <td valign="top">  T. Iwaniec,   "$p$-Harmonic tensors and quasiregular mappings"  ''Ann. of Math.'' , '''136'''  (1992)  pp. 39–64</td></tr><tr><td valign="top">[a4]</td> <td valign="top">  O. Lehto,   K.I. Virtanen,   "Quasiconformal mappings in the plane" , ''Grundl. Math. Wissenschaft.'' , '''126''' , Springer  (1973)  (Edition: Second)</td></tr><tr><td valign="top">[a5]</td> <td valign="top">  Yu.G. Reshetnyak,   "Space mappings with bounded distortion" , ''Transl. Math. Monogr.'' , '''73''' , Amer. Math. Soc.  (1989)  (In Russian)</td></tr><tr><td valign="top">[a6]</td> <td valign="top">  Yu.G. Reshetnyak,   "Stability theorems in geometry and analysis" , Kluwer Acad. Publ.  (1994)</td></tr><tr><td valign="top">[a7]</td> <td valign="top">  S. Rickman,   "Quasiregular mappings" , ''Ergeb. Math. Grenzgeb.'' , '''26''' , Springer  (1993)</td></tr><tr><td valign="top">[a8]</td> <td valign="top">  J. Väisälä,   "Lectures on $n$-dimensional quasiconformal mappings" , ''Lecture Notes in Mathematics'' , '''229''' , Springer  (1971)</td></tr><tr><td valign="top">[a9]</td> <td valign="top">  M. Vuorinen,   "Conformal geometry and quasiregular mappings" , ''Lecture Notes in Mathematics'' , '''1319''' , Springer  (1988)</td></tr><tr><td valign="top">[a10]</td> <td valign="top">  "Quasiconformal space mappings: A collection of surveys, 1960–1990"  M. Vuorinen (ed.) , ''Lecture Notes in Mathematics'' , '''1508''' , Springer  (1992)</td></tr></table>