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I understand the derivation of the free space Green's function, for instance, in 2-D we have $$G(\vec x; \vec x_0) = \frac{\log(|\vec x - \vec x_0|)}{2 \pi}.$$

However, I don't understand how this is applied to problems with boundaries, like, for instance, the upper half plane. Say we have $\nabla^2u = 0 $ in the upper half plane $y>0$, with some boundary conditions (either Dirichlet or Neumann), then must the Green's function satisfy these conditions too? How must our Green's function be amended in order to incorporate the Boundary/Initial conditions of the problem, any why?

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Boundary data isn't built into Green's function. Only the geometry. For dirichlet problem you want $G$ such that $\Delta G(x,y) = \delta(x-y)$ and $G(x,y)=0$ for $y\in \partial \Omega$. Evans covers this, so unless you have a specific question, you might just read it there. – StuartHa Mar 27 '13 at 23:36

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