Poisson integral formula

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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 ωn is the surface area of the unit sphere. The integration is performed over the surface of the ball, with unit surface area dS(y).

[edit] References

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

[edit] See also

[edit] External links


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