Green's function and Dirichlet boundary problem

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SUMMARY

Green's function exists for the Dirichlet boundary problem, defined by the equation div(ε grad G(r,r')) = -δ(r,r') within volume V, with G(r,r') = 0 on the boundary of V. This solution is valid for any bounded volume V, provided the physical conditions are realizable. The discussion confirms that while the Coulomb potential serves as an obvious solution in infinite space, the existence of Green's function is guaranteed mathematically under the specified conditions.

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  • Understanding of Green's functions in mathematical physics
  • Familiarity with Dirichlet boundary conditions
  • Knowledge of differential equations
  • Concept of divergence and gradient in vector calculus
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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:
[tex]div (\epsilon grad G(r,r')) =- \delta(r,r')[/tex] 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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