Green's function and Dirichlet boundary problem

In summary, there is always a Green's function for the Dirichlet boundary problem, which is a function that satisfies the conditions of div (\epsilon grad G(r,r')) =- \delta(r,r') inside volume V and G(r,r') is 0 on the boundary of V. However, this solution is only applicable in physically realizable situations. In the case of a mathematical problem with only one differential equation and boundary condition, without any underlying physics, the existence of a solution is uncertain.
  • #1
paweld
255
0
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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  • #2
G is the potential of a point charge in a volume bounded by a grounded surface.
This always exists.
 
  • #3
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)?
 

1. What is a Green's function?

A Green's function is a mathematical tool used in solving partial differential equations. It is a type of response function that describes the relationship between a source term and the resulting solution to a differential equation.

2. How is Green's function used in solving Dirichlet boundary problems?

In solving Dirichlet boundary problems, Green's function is used to find the solution of the problem by taking into account the boundary conditions. By using Green's function, the problem can be reduced to a simpler form which can be more easily solved.

3. What are the properties of Green's function?

Some properties of Green's function include symmetry, positivity, and singularity at the source point. It also satisfies the boundary conditions of the problem and can be used to find the solution at any point in the domain.

4. What is the significance of Dirichlet boundary conditions in Green's function?

Dirichlet boundary conditions specify the values of the solution at the boundary of the domain. These conditions are important in Green's function because they determine the behavior of the solution at the boundary and help in simplifying the problem.

5. Can Green's function be used to solve other types of boundary value problems?

Yes, Green's function can also be used to solve other types of boundary value problems such as Neumann boundary problems and mixed boundary problems. It can also be applied to different types of differential equations such as ordinary differential equations and integral equations.

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