I've been given the following conditions for some 3D function, $\phi$;
$$\nabla^2 \phi(x,y,x) = 0$$ $$\phi(x,y,0) = f(x,y)$$
My question is, would the equivalent Green's function problem be; $$\nabla^2 G = \delta(\vec x - \vec x_0)$$ $$G(x,y,0) = 0$$
Or... $$\nabla^2 G = 0$$ $$G(x,y,0) = \delta(\vec x - \vec x_0)$$
Any insight would be great!! :)
By definition, Green's function of a PDE is a function that solves the PDE (in a distributional sense) with the source term replaced by delta function. In this case, $\nabla^2 G=\delta$. (I.e., what uvs said in a comment.)
I would not say it's an equivalent problem: after all, finding Green's function is just one (big) step toward solving the PDE; also, other methods exist that do not involve Green's function.