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i have a question : Is there any machinery that takes me from the 3D green function of the laplacian differential operator to the 2D green function and vice versa ? That is , assume i have the poisson equation in 3D where the domain is a sphere and i have the Green function G, now i want to reduce the problem to the surface of the sphere only(2D), how can i get the green function now for the poisson in 2D

Thank you

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In ${\mathbb R}^3$ it is easy:

Use the three d green function to solve the problem with a delta-function source on the $z$ axis. In other words: $$ G(x,y)_{2d}= \int_{-\infty}^{\infty} G(x,y,z)_{3d} dz $$ If it really the plain Laplacian Green function you are after, you will need an infra-red cutoff as strictly the standard 2d Green function $$ G(x,y)_{2d}= - \frac 1 {2\pi}\ln \mu |x-y|, $$ as given by $$ G(x,y)_{2d}=- \int \frac{d^2k}{(2\pi)^2} \frac{e^{ik(x-y)}}{k^2} $$ is divergent at samll $k$. This is the origin of the undetermined constant $\mu$ in the standard expression.

For your sphere geometry the two probelms are rather unrelated, so there is no obvious answer.

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    The $3$-D Green (or Green's) function is given by $G(\vec r|\vec r')=\frac{1}{4\pi |\vec r-\vec r'|}$. And $\int_{-\infty}^\infty G(\vec r|\vec r')\,dz'$ diverges. – 2017-02-16