Dirichlet problem for a bounded domain - Revision history

Vipul: 2 revisions

2008-05-18T19:12:15Z

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Vipul at 19:40, 3 May 2008

2008-05-03T19:40:06Z

← Older revision		Revision as of 19:40, 3 May 2008
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	* The restriction of <math>g</math> to <math>\partial U</math>is precisely <math>f</math>		* 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]]		* 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===		===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>.		A special case of the above, where <math>n = 2</math>, and we identify <math>\R^2</math> with <math>\mathbb{C}</math>.

Vipul: New page: ==Definition== ===In Euclidean space=== Suppose $U \subset \mathbb{R}^n$ is a bounded, connected open subset, and $f: \partial U \to \R$ is a continuous function. T...

2008-05-03T19:30:50Z

New page: ==Definition== ===In Euclidean space=== Suppose <math>U \subset \mathbb{R}^n</math> is a bounded, connected open subset, and <math>f: \partial U \to \R</math> is a continuous function. T...

New page

==Definition==

===In Euclidean space===

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

* 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]]

===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>.

Dirichlet problem for a bounded domain - Revision history

Vipul: 2 revisions

Vipul at 19:40, 3 May 2008

Vipul: New page: ==Definition== ===In Euclidean space=== Suppose U \subset \mathbb{R}^n is a bounded, connected open subset, and f: \partial U \to \R is a continuous function. T...

Vipul: New page: ==Definition== ===In Euclidean space=== Suppose $U \subset \mathbb{R}^n$ is a bounded, connected open subset, and $f: \partial U \to \R$ is a continuous function. T...