# Poincaré problem

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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.

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Poincaré problem.

*Encyclopedia of Mathematics.*URL: http://www.encyclopediaofmath.org/index.php?title=Poincar%C3%A9_problem&oldid=32942