Difference between revisions of "Poincaré problem"
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| − | To find a [[Harmonic function|harmonic function]] in a bounded simply-connected domain | + | 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 |
| − | + | $$A(s)\frac{du}{dn}+B(s)\frac{du}{ds}+c(s)u=f(s),$$ | |
| − | where | + | 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=32942
Poincaré problem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Poincar%C3%A9_problem&oldid=32942
This article was adapted from an original article by A.B. Ivanov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article