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