I read a lot of books on the uniqueness theorem of Poisson equation,but all of them are confined to a bounded domain [tex]\Omega[/tex] ,i.e.(adsbygoogle = window.adsbygoogle || []).push({});

"Dirichlet boundary condition: [tex]\varphi[/tex] is well defined at all of the boundary surfaces.

Neumann boundary condition: [tex]\nabla\varphi[/tex]is well defined at all of the boundary surfaces.

Mixed boundary conditions (a combination of Dirichlet, Neumann, and modified Neumann boundary conditions): the uniqueness theorem will still hold.

"

However,in the method of mirror image,the domain is usually unbounded.For instance,consider the electric field induced by a point charge with a infinely large grounded conductor plate.In all of the textbooks,it is stated that "because of the uniqueness theorem...",but NO book has ever proved it in such a domain!!!

Some may say that we can regard the infinity as a special surface,but we CAN'T since this "surface" has a infinite area.I tried to prove it using the same way as that in a bounded domain,i.e. with the electric potential known in a bounded surface and [tex]\varphi \to 0{\rm{ }}(r \to \infty )[/tex],

I ended up with[tex]\int_S {\phi \frac{{\partial \phi }}{{\partial n}}} dS = {\int_V {\left( {\nabla \phi } \right)} ^2}dV[/tex]in which [tex]\phi[/tex] is the difference between two possible solution of the electric potential.

Let S be the surface of an infinite sphere,we have

[tex]4\pi \int_{r \to \infty } {{r^2}\phi \frac{{\partial \phi }}{{\partial r}}} dr = {\int_V {\left( {\nabla \phi } \right)} ^2}dV[/tex]

We have[tex]\phi \to 0(r \to \infty )[/tex],but it doesn't indicate [tex]{r^2}\phi \frac{{\partial \phi }}{{\partial r}} \to 0(r \to \infty )[/tex] So we can't conclude that [tex]\nabla \phi \equiv 0[/tex] so that the uniqueness theorem doesn't hold (or we cannot prove it with the same way proving uniqueness theorem in a bounded domain)

Or can anybody here prove that [tex]{r^2}\frac{{\partial \phi }}{{\partial r}}[/tex] is bounded?

**Physics Forums | Science Articles, Homework Help, Discussion**

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# How to prove the uniqueness theorem in an unbounded domain?

**Physics Forums | Science Articles, Homework Help, Discussion**