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How to solve the exterior problem on a ball with radius $r$ in the 3d space? I have to found u such that:

$\Delta u = 0$ in $B(0,r)^C$

Thanks!

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    You should clarify all of that in the body of the question, itself. That will "bump" the question up on the list, and someone should see it and hopefully be able to answer it.2012-11-02

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The solution to the Dirichlet problem uses the Green's function for the Laplacian. In 3D, this is $H(r) = -1/4\pi|r|$ (for vector $r$), such that $\nabla^2 H(r) = \delta(r)$, per the definition of the Green's function.

Some use of the various generalized Stokes theorems gives the following (when $\nabla^2 u = 0$):

$u(r) = \oint \nabla'H(r-r') \cdot dS' \; u(r') - \oint H(r-r') \; dS' \cdot \nabla' u(r')$

The choice of the Dirichlet Green's function enforces that $H(r-r') = 0$ on the boundary. The second integral vanishes, leaving only the first. You'll have to be careful about the orientation of $dS'$ (since this is an exterior problem), but otherwise, you should be able to exploit the symmetries of the problem to reduce the surface integral to something tractable. Good luck!

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    Now the question is complete!2012-11-02