Editing Laplace's equation (section)

==Boundary conditions==
[[File:Laplace's equation on an annulus.svg|thumb|right|Laplace's equation on an [[Annulus (mathematics)|annulus]] (inner radius {{math|1=''r'' = 2}} and outer radius {{math|1=''R'' = 4}}) with Dirichlet boundary conditions {{math|1=''u''(''r''=2) = 0}} and {{math|1=''u''(''R''=4) = 4 sin(5 ''θ'')}}|350px]]
{{See also|Boundary value problem}}
The [[Dirichlet problem]] for Laplace's equation consists of finding a solution {{math|''φ''}} on some domain {{mvar|D}} such that {{math|''φ''}} on the boundary of {{mvar|D}} is equal to some given function. Since the Laplace operator appears in the [[heat equation]], one physical interpretation of this problem is as follows: fix the temperature on the boundary of the domain according to the given specification of the boundary condition. Allow heat to flow until a stationary state is reached in which the temperature at each point on the domain does not change anymore. The temperature distribution in the interior will then be given by the solution to the corresponding Dirichlet problem.

The [[Neumann boundary condition]]s for Laplace's equation specify not the function {{math|''φ''}} itself on the boundary of {{mvar|D}} but its [[normal derivative]]. Physically, this corresponds to the construction of a potential for a vector field whose effect is known at the boundary of {{math|''D''}} alone. For the example of the heat equation it amounts to prescribing the heat flux through the boundary. In particular, at an adiabatic boundary, the normal derivative of {{math|''φ''}} is zero.

Solutions of Laplace's equation are called [[harmonic function]]s; they are all [[analytic function|analytic]] within the domain where the equation is satisfied. If any two functions are solutions to Laplace's equation (or any linear homogeneous differential equation), their sum (or any linear combination) is also a solution. This property, called the [[Superposition principle|principle of superposition]], is very useful. For example, solutions to complex problems can be constructed by summing simple solutions.