View source for Boundary correspondence, principle of

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)]]).

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]]]).

====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>