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Difference between revisions of "Poincaré problem"

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To find a [[Harmonic function|harmonic function]] in a bounded simply-connected domain <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073080/p0730801.png" /> which, on the boundary <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073080/p0730802.png" /> of the domain, satisfies the condition
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{{DEF}}
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073080/p0730803.png" /></td> </tr></table>
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To find a [[Harmonic function|harmonic function]] in a bounded simply-connected domain $S^+$ which, on the boundary $L$ of the domain, satisfies the condition
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073080/p0730804.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073080/p0730805.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073080/p0730806.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073080/p0730807.png" /> are real-valued functions given on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073080/p0730808.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073080/p0730809.png" /> is the arc parameter and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073080/p07308010.png" /> is the normal to <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073080/p07308011.png" />. H. Poincaré (1910) arrived at this problem while working on the mathematical theory of fluid flow and gave an (incomplete) solution to the problem in case <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073080/p07308012.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073080/p07308013.png" /> and the contour <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073080/p07308014.png" /> and the functions <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073080/p07308015.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p073/p073080/p07308016.png" /> are analytic.
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$$A(s)\frac{du}{dn}+B(s)\frac{du}{ds}+c(s)u=f(s),$$
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where $A(s)$, $B(s)$, $c(s)$, and $f(s)$ are real-valued functions given on $L$, $s$ is the arc parameter and $n$ is the normal to $L$. H. Poincaré (1910) arrived at this problem while working on the mathematical theory of fluid flow and gave an (incomplete) solution to the problem in case $A(s)=1$, $c(s)=0$ and the contour $L$ and the functions $B(s)$ and $f(s)$ are analytic.
  
 
See also [[Boundary value problems of analytic function theory|Boundary value problems of analytic function theory]].
 
See also [[Boundary value problems of analytic function theory|Boundary value problems of analytic function theory]].

Latest revision as of 19:41, 14 August 2014


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To find a harmonic function in a bounded simply-connected domain $S^+$ which, on the boundary $L$ of the domain, satisfies the condition

$$A(s)\frac{du}{dn}+B(s)\frac{du}{ds}+c(s)u=f(s),$$

where $A(s)$, $B(s)$, $c(s)$, and $f(s)$ are real-valued functions given on $L$, $s$ is the arc parameter and $n$ is the normal to $L$. H. Poincaré (1910) arrived at this problem while working on the mathematical theory of fluid flow and gave an (incomplete) solution to the problem in case $A(s)=1$, $c(s)=0$ and the contour $L$ and the functions $B(s)$ and $f(s)$ are analytic.

See also Boundary value problems of analytic function theory.

How to Cite This Entry:
Poincaré problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poincar%C3%A9_problem&oldid=22937
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article