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Is there a closed formula for the solution of Dirichlet problem ($\Delta u=0$) for annulus $r <|x| < R$, $x \in R^n$ (n>2), with two given boundary value functions, $f$ over $|x|=r$ and $g$ over $|x|=R$?

If the answer is yes, please give the formula or a reference for it.

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It seems that a formula could be derived using generalized cylindrical cordinates, seperation of variables and a lot of calculation, just like in the 2D case. – J.C. Ottem Nov 7 '10 at 17:35
It can also be established by using Fourier series. – Denis Serre Nov 7 '10 at 17:38
Of course you can. – timur May 13 '11 at 2:57

This is exercise 2.5 on page 29 in Gilbarg and Trudinger's book "Elliptic Partial Differential Equations of Second Order".

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