- #1
deuteron
- 55
- 12
- TL;DR Summary
- 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?
$$\phi_{\vec r} = \iiint\limits_V \rho_{\vec r_q} G_{\vec r, \vec r_q}\ d^3r_q\ +\ \frac 1 {4\pi}\ [\iint _{\partial V} G_{\vec r, \vec r_q} \frac \partial {\partial n} \phi_{\vec r_q} - \phi_{\vec r_q} \frac{\partial G_{\vec r, \vec r_q}} {\partial n} \ da]$$
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?