Uniqueness theorem for Poisson's equation: Difference between revisions

Revision as of 18:50, 23 March 2019

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# [[Dirichlet boundary condition]]: <math>\varphi</math> is well defined at all of the boundary surfaces. As such <math>\varphi_1=\varphi_2</math> so at the boundary <math>\varphi = 0</math> and correspondingly the surface integral vanishes.

# [[Neumann boundary condition]]: <math>\mathbf{\nabla}\varphi</math> is well defined at all of the boundary surfaces. As such <math>\mathbf{\nabla}\varphi_1=\mathbf{\nabla}\varphi_2</math> so at the boundary <math>\mathbf{\nabla}\varphi=0</math> and correspondingly the surface integral vanishes.

# Modified [[Neumann boundary condition]] (also called [[Robin boundary condition]] -– conditions where boundaries are specified as conductors with known charges): <math>\mathbf{\nabla}\varphi</math> is also well defined by applying locally [[Gauss's Law]]. As such, the surface integral also vanishes.

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

The boundary surfaces may also include boundaries at infinity (describing unbounded domains) -– for these the uniqueness theorem holds if the surface integral vanishes, which is the case (for example) when at large distances the integrand decays faster than the surface area grows.

==See also==