I Green's function boundary conditions

AI Thread Summary
The discussion focuses on the application of Green's identity to derive the potential in terms of the Green's function under specific boundary conditions. It highlights the necessity for the Green's function to satisfy certain conditions for Dirichlet and Neumann boundaries to simplify the integral expression for the potential. The query raised pertains to the mathematical justification for imposing these conditions on the Green's function without introducing inconsistencies. Reference to Jackson's text is made as a source of clarification on this topic. Understanding these requirements is crucial for ensuring the validity of the potential's representation.
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what is the motivation / justification behind the applied conditions on the Green's function for Dirichlet / Neumann boundary conditions
we know that, using the Green's identity ##\iiint\limits_V (\varphi \Delta\psi -\psi \Delta\varphi)\ dV =\iint_{\partial V} (\varphi \frac {\partial \psi}{\partial n}-\psi \frac {\partial\varphi}{\partial n})\ da## and substituting ##\varphi=\phi## and ##\psi=G## here, we can write the potential as:

$$\phi_{\vec r} = \iiint\limits_V \rho_{\vec r_q} G_{\vec r, \vec r_q}\ d^3r_q\ +\ \frac 1 {4\pi}\ [\iint _{\partial V} G_{\vec r, \vec r_q} \frac \partial {\partial n} \phi_{\vec r_q} - \phi_{\vec r_q} \frac{\partial G_{\vec r, \vec r_q}} {\partial n} \ da]$$

here, for the type of given boundary conditions, ( Dirichlet: ##\phi|_{\partial V}=\text{given}## or Neumann ##\frac {\partial \phi}{\partial n}|_{\partial V}=\text{given}##) we require, that the Green's function satisfies some conditions (Dirichlet: ##G|_{\partial V}=0##, Neumann: ##\frac {\partial G}{\partial n} |_{\partial V}=- \frac {4\pi}{\text{surface area of}\ \partial V}##)

I understand that these make our life easier when we substitute the Green's function into the above integral expression for ##\phi##
However, I am confused about *why* we are allowed to make these requirements on the Green's function. How are we mathematically sure that making this requirements would not cause a problem?
 
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I have found the answer in Jackson, section 1.10 page 18
 
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