Namespaces
Variants
Actions

Difference between revisions of "Löwner equation"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (moved Löwner equation to Lowner equation: ascii title)
(TeX)
 
(One intermediate revision by one other user not shown)
Line 1: Line 1:
 +
{{TEX|done}}
 
A differential equation of the form
 
A differential equation of the form
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060950/l0609501.png" /></td> </tr></table>
+
$$\frac{dw}{dt}=-w\frac{1+e^{i\alpha(t)}w}{1-e^{i\alpha(t)}w},$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060950/l0609502.png" /> is a real-valued continuous function on the interval <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060950/l0609503.png" />. A generalization of the Löwner equation is the Kufarev–Löwner equation:
+
where $\alpha(t)$ is a real-valued continuous function on the interval $-\infty<t<\infty$. A generalization of the Löwner equation is the Kufarev–Löwner equation:
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060950/l0609504.png" /></td> </tr></table>
+
$$\frac{dw}{dt}=-wP(w,t),$$
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060950/l0609505.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060950/l0609506.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060950/l0609507.png" />, is a function measurable in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060950/l0609508.png" /> for fixed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060950/l0609509.png" /> and regular in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060950/l06095010.png" />, with positive real part, normalized by the condition <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060950/l06095011.png" />. The Löwner equation and the Kufarev–Löwner equation, which arise in the theory of univalent functions, are the basis of the [[Variation-parametric method|variation-parametric method]] of investigating extremal problems on conformal mapping.
+
where $P(w,t)$, $|w|<1$, $-\infty<t<\infty$, is a function measurable in $t$ for fixed $w$ and regular in $w$, with positive real part, normalized by the condition $P(0,t)=1$. The Löwner equation and the Kufarev–Löwner equation, which arise in the theory of univalent functions, are the basis of the [[Variation-parametric method|variation-parametric method]] of investigating extremal problems on conformal mapping.
  
The solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060950/l06095012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060950/l06095013.png" />, of the Kufarev–Löwner equation, regarded as a function of the initial value <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060950/l06095014.png" />, for any <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060950/l06095015.png" /> maps the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060950/l06095016.png" /> conformally onto a one-sheeted simply-connected domain belonging to the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060950/l06095017.png" />. From the formula
+
The solution $w(t,z,\tau)$, $w(\tau,z,\tau)=z$, of the Kufarev–Löwner equation, regarded as a function of the initial value $z$, for any $t>\tau$ maps the disc $|z|<1$ conformally onto a one-sheeted simply-connected domain belonging to the disc $|w|<1$. From the formula
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060950/l06095018.png" /></td> </tr></table>
+
$$f(z)=a+b\lim_{t\to\infty}e^tw(t,z,0),$$
  
by a suitable choice of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060950/l06095019.png" /> in the Kufarev–Löwner equation and complex constants <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060950/l06095020.png" /> one can obtain an arbitrary regular univalent function in the disc <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060950/l06095021.png" />. In this way the Löwner equation generates, in particular, the conformal mappings of the disc onto domains obtained from the whole plane by making a slit along some Jordan arc (see [[#References|[1]]]–[[#References|[4]]]).
+
by a suitable choice of $P(w,t)$ in the Kufarev–Löwner equation and complex constants $a,b$ one can obtain an arbitrary regular univalent function in the disc $|z|<1$. In this way the Löwner equation generates, in particular, the conformal mappings of the disc onto domains obtained from the whole plane by making a slit along some Jordan arc (see [[#References|[1]]]–[[#References|[4]]]).
  
 
The partial differential equation
 
The partial differential equation
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060950/l06095022.png" /></td> </tr></table>
+
$$\frac{\partial f(z,\tau)}{\partial\tau}=z\frac{\partial f(z,\tau)}{\partial z}P(z,\tau),$$
  
 
which is satisfied by the function
 
which is satisfied by the function
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l060/l060950/l06095023.png" /></td> </tr></table>
+
$$f(z,\tau)=\lim_{t\to\infty}e^tw(t,z,\tau),$$
  
 
is also called the Kufarev–Löwner equation.
 
is also called the Kufarev–Löwner equation.

Latest revision as of 08:57, 29 August 2014

A differential equation of the form

$$\frac{dw}{dt}=-w\frac{1+e^{i\alpha(t)}w}{1-e^{i\alpha(t)}w},$$

where $\alpha(t)$ is a real-valued continuous function on the interval $-\infty<t<\infty$. A generalization of the Löwner equation is the Kufarev–Löwner equation:

$$\frac{dw}{dt}=-wP(w,t),$$

where $P(w,t)$, $|w|<1$, $-\infty<t<\infty$, is a function measurable in $t$ for fixed $w$ and regular in $w$, with positive real part, normalized by the condition $P(0,t)=1$. The Löwner equation and the Kufarev–Löwner equation, which arise in the theory of univalent functions, are the basis of the variation-parametric method of investigating extremal problems on conformal mapping.

The solution $w(t,z,\tau)$, $w(\tau,z,\tau)=z$, of the Kufarev–Löwner equation, regarded as a function of the initial value $z$, for any $t>\tau$ maps the disc $|z|<1$ conformally onto a one-sheeted simply-connected domain belonging to the disc $|w|<1$. From the formula

$$f(z)=a+b\lim_{t\to\infty}e^tw(t,z,0),$$

by a suitable choice of $P(w,t)$ in the Kufarev–Löwner equation and complex constants $a,b$ one can obtain an arbitrary regular univalent function in the disc $|z|<1$. In this way the Löwner equation generates, in particular, the conformal mappings of the disc onto domains obtained from the whole plane by making a slit along some Jordan arc (see [1][4]).

The partial differential equation

$$\frac{\partial f(z,\tau)}{\partial\tau}=z\frac{\partial f(z,\tau)}{\partial z}P(z,\tau),$$

which is satisfied by the function

$$f(z,\tau)=\lim_{t\to\infty}e^tw(t,z,\tau),$$

is also called the Kufarev–Löwner equation.

The Löwner equation was set up by K. Löwner [1]; the Kufarev–Löwner equation was obtained by P.P. Kufarev (see [5]).

References

[1] K. Löwner, "Untersuchungen über schlichte konforme Abbildungen des Einheitskreises, I" Math. Ann. , 89 (1923) pp. 103–121
[2] P.P. Kufarev, "A theorem on solutions of a differential equation" Uchen. Zap. Tomsk. Gos. Univ. , 5 (1947) pp. 20–21 (In Russian)
[3] C. Pommerenke, "Ueber die Subordination analytischer Funktionen" J. Reine Angew. Math. , 218 (1965) pp. 159–173
[4] V.Ya. Gutlyanskii, "Parametric representation of univalent functions" Soviet Math. Dokl. , 11 (1970) pp. 1273–1276 Dokl. Akad. Nauk SSSR , 194 : 4 (1970) pp. 750–753
[5] P.P. Kufarev, "On one-parameter families of analytic functions" Mat. Sb. , 13 (1943) pp. 87–118 (In Russian)
[6] G.M. Goluzin, "Geometric theory of functions of a complex variable" , Transl. Math. Monogr. , 26 , Amer. Math. Soc. (1969) (Translated from Russian)


Comments

For more information see also Löwner method.

How to Cite This Entry:
Löwner equation. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=L%C3%B6wner_equation&oldid=22769
This article was adapted from an original article by V.Ya. Gutlyanskii (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article