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There is porbably a rather simple answer to this, but I was wondering why it is not a problem that our Green's function fundamnetal solution in $\mathbb{R}^2$ goes to infinity as $r\rightarrow \infty$?

In general, when we are dealing with finite domains, say $\Omega_3 \subset \mathbb{R}^3$ or $\Omega_2 \subset \mathbb{R}^2$, and given Dirichlet conditions, in both cases, we want

$\nabla ^2 G = \delta(\textbf{r}-\textbf{r'})$; $\textbf{r},\textbf{r'} \in \Omega_3$ or $\Omega_2$ and

$G=0$ on $\partial \Omega_3$ or $\partial\Omega_3$

whereas given Neumann conditons we take

$\frac{\partial G}{\partial n}= \frac{1}{A}$ for problems in $\mathbb{R}^3$ or $\frac{1}{L}$ for problems in $\mathbb{R}^2$

Now the fundamnetal solution is just 'the fundamental solution', it is not stated to be a tool for solving problems specifically given Dirichlet or Neumann conditions. And yet we can see that the fundamental solution in $\mathbb{R}^3$ satisfies both the Dirichlet-problem and Neumann-problem conditions given aove, if we extend our domain to all of $\mathbb{R}^3$, while the fundamnetal solution in $\mathbb{R}^2$ goes to infinity as we extend our domain (the Neumann conditions remain satisfied).

I was wondering if there is any significance of this in solving problems? I know that when we are dealing with finite domains it doesn't really matter, because we use image sources outside of our domain to obtain the required boundary conditions for $G$. However if we are solving for all of $\mathbb{R}^2$, or some subsets of $\mathbb{R}^2$ that extend to infinity in some direction, using the fundamnetal solution will give an overall Green's function that goes to infinty.

I can think of two solutions to this:

a) In some cases, when we take, for instance, a half plane, if it so happens that we have as many $+ln$ terms as $-ln$ terms, then the infinities 'cancel' to zero at infinty

b) maybe in some cases we don't really care that the Green's function goes to infinity, porvided our forcing term goes to 0 faster so that the solution to Poisson's equation we are trying to find goes to 0 at infintiy...

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    $\begingroup$ Can you give the setting more precisely? What is the equation and what are the boundary (or asymptotic) conditions? There are Green's functions for many different PDEs. $\endgroup$ – Joonas Ilmavirta Mar 30 '18 at 12:22

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