Green's function and Dirichlet boundary problem

AI Thread Summary
Green's function for the Dirichlet boundary problem exists under certain conditions, specifically when the domain is physically realizable. The function G(r,r') must satisfy the equation div(ε grad G(r,r')) = -δ(r,r') within volume V, with G(r,r') equal to zero on the boundary. While solutions like the Coulomb potential are evident for whole space, the existence of solutions for all bounded volumes depends on the specific properties of the domain. The mathematical problem can be approached using differential equations and boundary conditions without needing physical context. Ultimately, the existence of Green's function is affirmed for appropriately defined domains.
paweld
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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.
 
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G is the potential of a point charge in a volume bounded by a grounded surface.
This always exists.
 
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)?
 
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