Boundary correspondence, principle of
A principle formulated in the following way. One says that the principle of boundary correspondence holds for a mapping if the facts that
is a continuous mapping of the closure
of a domain
onto the closure
of a domain
and
is a homeomorphism of
onto
imply that
is a topological mapping of
onto
. Thus, the principle of boundary correspondence is in some sense converse to the boundary-correspondence principle (cf. Boundary correspondence (under conformal mapping)).
If and
are plane domains with Euclidean boundaries homeomorphic to a circle and
is bounded, then the principle of boundary correspondence holds for analytic functions
on
, i.e.
is a conformal mapping of
onto
. In addition to that given above, various other forms of the principle of boundary correspondence are commonly used for conformal mappings (see [1]). The principle of boundary correspondence has been verified for orientable mappings in Euclidean space (see [2]).
References
[1] | M.A. Lavrent'ev, B.V. Shabat, "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft. (1967) (Translated from Russian) |
[2] | L.D. Kudryavtsev, "On differentiable mappings" Dokl. Akad. Nauk SSSR , 95 : 5 (1954) pp. 921–923 (In Russian) |
[3] | S.I. Pinchuk, "Holomorphic equivalence of certain classes of domains in ![]() |
Boundary correspondence, principle of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boundary_correspondence,_principle_of&oldid=46128