Poisson integral formula

In mathematics, the Poisson integral formula gives an explicit solution to the Dirichlet problem for Laplace's equation in a ball in Euclidean space Rⁿ.

If u is a harmonic function in the ball in Rⁿ centered at the origin with radius R, then the formula states

$u(x) = \frac{R^2 - |x|^2}{\omega_n R} \int\limits_{\partial B_R} \frac{u(y)}{|x - y|^n}\,dS(y)$

where $ω n$ is the surface area of the unit sphere. The integration is performed over the surface of the ball, with unit surface area $d S (y)$ .

[edit] References

D. Gilbarg, N. Trudinger Elliptic Partial Differential Equations of Second Order. ISBN 3-540-41160-7.

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