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I am reading Walter Strauss's book Introduction to PDE. The definition of Green's function is as follows: The Green's function $G(x)$ for the operator $-\Delta$ and the domain $D \subset \mathbb{R^3}$(here I assume $D$ is an open set and has a smooth boundary) at the point $x_0 \in D$ is a function defined for $x \in D$ such that:

(i) $G(x)$ possesses continuous second derivatives and $\Delta G=0$ in $D$,except at the point $x=x_0$.

(ii) $G(x)=0$ for $x\in \text{bdy} D$.

(iii) The function $G(x)+1/(4\pi|x-x_0|)$ is finite at $x_0$ and has continuous second derivatives everywhere and is harmonic at $x_0$.

My first question is: How to show that such Green's function exists?

My second question is: How to prove that the solution of the problem $$\Delta u=f \text{ in } D,u=0 \text{ on bdy }D$$ is given by $$u(x_0)=\int \int \int_D f(x)G(x,x_0)dx \text{ ?}$$

I know these questions can be explained in terms of distribution but since I haven't learnt distribution yet,I am looking for a more elementary solution.Thank you.

Ben
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  • For (iii) did you mean "G(x)=" rather than "G(x)+"? – Ellya Apr 14 '14 at 08:28
  • I mean there is a harmonic function H(x) on D such that H(x)=G(x)+1/(4\pi|x-x_0|) on D{x_0} – Ben Apr 14 '14 at 08:31
  • Then should it be H in the integral? – Ellya Apr 14 '14 at 08:38
  • The G(x) in the definition is for point x_0 so in fact G(x)=G(x,x_0).The function in the integral is G(x,x_0) not H. – Ben Apr 14 '14 at 08:48
  • $H$ is called a fundamental solution. You might want to take a look at Fritz John book. To prove it classically, you'll have to separate the singularity at $x_0$ from the rest of the integral (using a ball of radius $\epsilon$), and then investigate the behavior in the limit. I've used the argument here – Pragabhava Apr 14 '14 at 18:01

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