Löwner equation

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.