Green's function and Dirichlet boundary problem

[paweld]

Is it true that there always exists Green's function for Dirichlet boundary problem.
I mean a function G(r,r') which fullfils the following conditions:
$$div (\epsilon grad G(r,r')) =- \delta(r,r')$$ inside volume V and G(r,r') is 0 on
boundary of V. If V is whole space there exists obvious solution (Coulomb potential)
but I wonder if there exists solution for all V.

 

[Meir Achuz]

G is the potential of a point charge in a volume bounded by a grounded surface.
This always exists.

 

[paweld]

Yes, you are right but only when the situation is physically realizable.
What is the answer for mathematical problem describe above
(we have only one differential equation + boundary condition,
without any physics behind)?