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

In the complex numbers

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