Dirichlet problem for a bounded domain: Difference between revisions

Latest revision as of 19:12, 18 May 2008

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 $\mathbb {C}$ .

@@ Line 7: / Line 7: @@
 * The restriction of <math>g</math> to <math>\partial U</math>is precisely <math>f</math>
 * The restriction of <math>g</math> to <math>U</math> 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 <math>f</math> 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 <math>n = 2</math>, and we identify <math>\R^2</math> with <math>\mathbb{C}</math>.