The Poisson problem is the boundary value problem of finding a function 
on a domain 
such that
 |
together with prescribed boundary conditions on 
. The common Dirichlet
problem specifies 
on the boundary, while Neumann boundary
conditions specify the outward normal derivative of 
.
The differential equation is Poisson's equation. When 
, it reduces to Laplace's
equation. Under standard regularity hypotheses, Dirichlet
boundary conditions determine a unique solution. For pure Neumann
boundary conditions 
, a solution
requires the compatibility condition
 |
and is then unique only up to an additive constant.
