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A principle formulated in the following way. One says that the principle of boundary correspondence holds for a mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017270/b0172701.png" /> if the facts that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017270/b0172702.png" /> is a continuous mapping of the closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017270/b0172703.png" /> of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017270/b0172704.png" /> onto the closure <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017270/b0172705.png" /> of a domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017270/b0172706.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017270/b0172707.png" /> is a homeomorphism of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017270/b0172708.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017270/b0172709.png" /> imply that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017270/b01727010.png" /> is a topological mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017270/b01727011.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017270/b01727012.png" />. Thus, the principle of boundary correspondence is in some sense converse to the boundary-correspondence principle (cf. [[Boundary correspondence (under conformal mapping)|Boundary correspondence (under conformal mapping)]]).
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If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017270/b01727013.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017270/b01727014.png" /> are plane domains with Euclidean boundaries homeomorphic to a circle and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017270/b01727015.png" /> is bounded, then the principle of boundary correspondence holds for analytic functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017270/b01727016.png" /> on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017270/b01727017.png" />, i.e. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017270/b01727018.png" /> is a conformal mapping of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017270/b01727019.png" /> onto <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017270/b01727020.png" />. In addition to that given above, various other forms of the principle of boundary correspondence are commonly used for conformal mappings (see [[#References|[1]]]). The principle of boundary correspondence has been verified for orientable mappings in Euclidean space (see [[#References|[2]]]).
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A principle formulated in the following way. One says that the principle of boundary correspondence holds for a mapping  $  f $
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if the facts that  $  f $
 +
is a continuous mapping of the closure  $  \overline{G}\; $
 +
of a domain  $  G $
 +
onto the closure  $  \overline{D}\; $
 +
of a domain  $  D $
 +
and $  f $
 +
is a homeomorphism of  $  \overline{G}\; \setminus  G $
 +
onto  $  \overline{D}\; \setminus  D $
 +
imply that  $  f $
 +
is a topological mapping of  $  \overline{G}\; $
 +
onto  $  \overline{D}\; $.  
 +
Thus, the principle of boundary correspondence is in some sense converse to the boundary-correspondence principle (cf. [[Boundary correspondence (under conformal mapping)|Boundary correspondence (under conformal mapping)]]).
 +
 
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If  $  G $
 +
and  $  D $
 +
are plane domains with Euclidean boundaries homeomorphic to a circle and $  D $
 +
is bounded, then the principle of boundary correspondence holds for analytic functions $  f $
 +
on $  G $,  
 +
i.e. $  f $
 +
is a conformal mapping of $  G $
 +
onto $  D $.  
 +
In addition to that given above, various other forms of the principle of boundary correspondence are commonly used for conformal mappings (see [[#References|[1]]]). The principle of boundary correspondence has been verified for orientable mappings in Euclidean space (see [[#References|[2]]]).
  
 
====References====
 
====References====
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.A. Lavrent'ev,  B.V. Shabat,  "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft.  (1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.D. Kudryavtsev,  "On differentiable mappings"  ''Dokl. Akad. Nauk SSSR'' , '''95''' :  5  (1954)  pp. 921–923  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.I. Pinchuk,  "Holomorphic equivalence of certain classes of domains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017270/b01727021.png" />"  ''Math. USSR-Sb.'' , '''111 (153)''' :  1  (180)  pp. 67–94; 159  ''Mat. Sb.'' , '''111 (153)''' :  1  (1980)  pp. 67–94; 159</TD></TR></table>
 
<table><TR><TD valign="top">[1]</TD> <TD valign="top">  M.A. Lavrent'ev,  B.V. Shabat,  "Methoden der komplexen Funktionentheorie" , Deutsch. Verlag Wissenschaft.  (1967)  (Translated from Russian)</TD></TR><TR><TD valign="top">[2]</TD> <TD valign="top">  L.D. Kudryavtsev,  "On differentiable mappings"  ''Dokl. Akad. Nauk SSSR'' , '''95''' :  5  (1954)  pp. 921–923  (In Russian)</TD></TR><TR><TD valign="top">[3]</TD> <TD valign="top">  S.I. Pinchuk,  "Holomorphic equivalence of certain classes of domains in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b017/b017270/b01727021.png" />"  ''Math. USSR-Sb.'' , '''111 (153)''' :  1  (180)  pp. 67–94; 159  ''Mat. Sb.'' , '''111 (153)''' :  1  (1980)  pp. 67–94; 159</TD></TR></table>

Revision as of 06:28, 30 May 2020


A principle formulated in the following way. One says that the principle of boundary correspondence holds for a mapping $ f $ if the facts that $ f $ is a continuous mapping of the closure $ \overline{G}\; $ of a domain $ G $ onto the closure $ \overline{D}\; $ of a domain $ D $ and $ f $ is a homeomorphism of $ \overline{G}\; \setminus G $ onto $ \overline{D}\; \setminus D $ imply that $ f $ is a topological mapping of $ \overline{G}\; $ onto $ \overline{D}\; $. Thus, the principle of boundary correspondence is in some sense converse to the boundary-correspondence principle (cf. Boundary correspondence (under conformal mapping)).

If $ G $ and $ D $ are plane domains with Euclidean boundaries homeomorphic to a circle and $ D $ is bounded, then the principle of boundary correspondence holds for analytic functions $ f $ on $ G $, i.e. $ f $ is a conformal mapping of $ G $ onto $ D $. 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 " Math. USSR-Sb. , 111 (153) : 1 (180) pp. 67–94; 159 Mat. Sb. , 111 (153) : 1 (1980) pp. 67–94; 159
How to Cite This Entry:
Boundary correspondence, principle of. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boundary_correspondence,_principle_of&oldid=46128
This article was adapted from an original article by B.P. Kufarev (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article