Editing Dirichlet problem (section)

{{Short description|Problem of solving a partial differential equation subject to prescribed boundary values}}
In [[mathematics]], a '''Dirichlet problem''' asks for a [[function (mathematics)|function]] which solves a specified [[partial differential equation]] (PDE) in the interior of a given region that takes prescribed values on the boundary of the region.<ref>{{Cite web |title=Dirichlet Problem |url=https://mathworld.wolfram.com/DirichletProblem.html}}</ref>

The Dirichlet problem can be solved for many PDEs, although originally it was posed for [[Laplace's equation]].  In that case the problem can be stated as follows:

:Given a function ''f'' that has values everywhere on the boundary of a region in <math>\mathbb{R}^n</math>, is there a unique [[continuous function]] <math>u</math> twice continuously differentiable in the interior and continuous on the boundary, such that <math>u</math> is [[harmonic function|harmonic]] in the interior and <math>u=f</math> on the boundary?

This requirement is called the [[Dirichlet boundary condition]].  The main issue is to prove the existence of a solution; uniqueness can be proven using the [[maximum principle]].