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.
