Keldysh-Lavrent'ev example
An example of a simply-connected domain in the complex
-plane bounded by a rectifiable Jordan curve but not belonging to the class
of Smirnov domains (cf. Smirnov domain).
Let be a function realizing a conformal mapping of the unit disc
onto a simply-connected domain
bounded by a rectifiable Jordan curve. It is known that
is continuous and one-to-one in the closed disc
and that the logarithm of the modulus of the derivative
can be represented in
by the Poisson–Stieltjes integral
![]() | (*) |
where is the normalized Borel measure on
,
. The class
consists of those closed domains
for which the measure
in the representation (*) is absolutely continuous with respect to the Lebesgue measure on
and the integral (*) becomes the Poisson–Lebesgue integral (cf. Poisson integral) of the boundary values
, which exist almost-everywhere on
.
M.V. Keldysh and M.A. Lavrent'ev [1] constructed for any ,
, a simply-connected domain
bounded by a rectifiable Jordan curve
, located in the disc
,
, such that under a conformal mapping of
onto
,
![]() |
and the image of any arc on the circle
is an arc of the same length. This domain
does not belong to the class
, since
almost-everywhere on
.
A complete solution of the problem of characterizing domains of class (domains of Smirnov type) has so far (1989) not been obtained (see [2], [3]).
References
[1] | M.V. Keldysh, M.A. Lavrent'ev, "Sur la répresentation conforme des domains limités par des courbes rectifiables" Ann. Ecole Norm. Sup. , 54 (1937) pp. 1–38 |
[2] | I.I. [I.I. Privalov] Priwalow, "Randeigenschaften analytischer Funktionen" , Deutsch. Verlag Wissenschaft. (1956) (Translated from Russian) |
[3] | A.J. Lohwater, "The boundary behaviour of analytic functions" Itogi Nauk. i Tekhn. Mat. Anal. , 10 (1973) pp. 99–259 (In Russian) |
Comments
The construction of Keldysh and Lavrent'ev is extraordinarily complicated. For a more accessible treatment, see [a1] and [a2].
References
[a1] | P.L. Duren, H.S. Shapiro, A.L. Shields, "Singular measures and domains not of Smirnov type" Duke Math. J. , 33 (1966) pp. 247–254 |
[a2] | G. Piranian, "Two monotonic, singular, uniformly almost smooth functions" Duke Math. J. , 33 (1966) pp. 254–262 |
Keldysh-Lavrent'ev example. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Keldysh-Lavrent%27ev_example&oldid=16860