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Why does the Green's function $G(r,r_0)$ of the Laplace's equation $\nabla^2 u=0$, the domain being the half plane, is equal $0$ on the boundary? How can I interpret the Laplace's equation physically? Is there a way to intuitively interpret the Green's function in general?

Based on @Matt's comment, I am guessing that it is not generally true that the Green's function is 0 on the boundary?

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    I hate to keep writing this in comments, but can you post more information. Is your domain just the disk? Where is Laplace's equation coming up for you? I ask that because there are entire books on physical interpretations of Laplace's equation, so it is surprising that wherever you are seeing this doesn't have any.2011-12-21
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    @Matt: Thanks. Based on your comment then I am guessing that it is not generally true that the Green's function is 0 on the boundary? I am thinking of the domain being the half plane. But it would be nice if some other cases could be considered.2011-12-21
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    If the Greens function is equal to $0$ at the boundary (regardless of the domain), it can be used to solve the Dirichlet problem (prescribed boundary values). You often find the name 'Green's function of the first kind' for this. Another problem which is of interest is prescribing the normal derivative of the solution at the boundary. For this, the Greens function should have zero normal derivative at the boundary (...Green's function of the second kind). See any general text book about PDE in general or about elliptic PDE. The concept as such is not restricted to the Laplace equation.2011-12-21
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    @Thomas: Thanks. Would you mind elaborating why for a Dirichlet problem we should have G=0 on the boundary?2011-12-21
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    One more clarification. What is your definition of Green's function? I just looked up what it was when I saw it and the definition is a function $G(x,y)$ such that $\Delta_y G(x,y)=\delta_x$ for points inside the domain and $G(x,y)=0$ on the boundary. I only asked about the domain because common domains like the disk and upper half plane have straightforward to derive Green's functions that you can then just check directly they are zero.2011-12-21
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    @Matt: Thanks. I am only aware that the Green's function satisfies $LG(r;r_0)=\delta (r-r_0)$ Why is it necessary to have to have G=0 on the boundary?2011-12-21

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