Dirichlet problem

In mathematics, a Dirichlet problem asks for a 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.^[1]

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 ${\displaystyle \mathbb {R} ^{n}}$, is there a unique continuous function ${\displaystyle u}$ twice continuously differentiable in the interior and continuous on the boundary, such that ${\displaystyle u}$ is harmonic in the interior and ${\displaystyle u=f}$ 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.