Dirichlet problem for a bounded domain

Definition

In Euclidean space

Suppose $U\subset \mathbb {R} ^{n}$ is a bounded, connected open subset, and $f:\partial U\to \mathbb {R}$ is a continuous function. The Dirichlet problem for $U$ asks whether there exists a continuous function $g:{\overline {U}}\to R$ such that:

• The restriction of $g$ to $\partial U$is precisely $f$
• The restriction of $g$ to $U$ is a harmonic function

Because harmonic functions satisfy a mean-valued property, a solution to the Dirichlet problem, if it exists, is unique. Moreover, the map sending a continuous function $f$ that permits a solution, to the solution for it, is a linear operator. Solving the Dirichlet problem is often equated with finding an explicit form for the linear operator; for instance, in the form of an integral operator.

In the complex numbers

A special case of the above, where $n=2$, and we identify $\mathbb {R} ^{2}$ with $\mathbb {C}$.