A differential equation of the form
where
is a real-valued continuous function on the interval
.
A generalization of the Löwner equation is the
Kufarev–Löwner equation:
where
,
,
,
is a function measurable in
for fixed
and regular in
,
with positive real part, normalized by the condition
.
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
,
,
of the Kufarev–Löwner equation, regarded as a function of the initial value
,
for any
maps the disc
conformally onto a one-sheeted simply-connected domain belonging to the disc
.
From the formula
by a suitable choice of
in the Kufarev–Löwner equation and complex constants
one can obtain an arbitrary regular univalent function in the disc
.
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
which is satisfied by the function
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]).