76.120.168.91: edit

2013-12-20T02:56:38Z

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← Older revision		Revision as of 04:56, 20 December 2013
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	~~Irwin Butts~~ is ~~what my spouse enjoys~~ to ~~call me although I don't truly like becoming known as like~~ that. ~~Hiring~~ is ~~her day job now and she will not change~~ it ~~whenever quickly. To collect badges~~ is ~~what her family members~~ and ~~her appreciate. For~~ a ~~while she~~'~~s been~~ in ~~South Dakota~~.<br><br>~~my blog:~~ [~~http:~~//~~wixothek~~.~~com~~/~~blog/557~~ at ~~home std test~~]		In [[mathematics]], in the area of [[potential theory]], a '''Lebesgue spine''' or '''Lebesgue thorn''' is a type of [[Set (mathematics)\|set]] used for discussing solutions to the [[Dirichlet problem]] and related problems of potential theory. The Lebesgue spine was introduced in 1912 by [[Henri Lebesgue]] to demonstrate that the Dirichlet problem does not always have a solution, particularly when the boundary has a sufficiently sharp edge protruding into the interior of the region.

			== Definition ==

			A typical Lebesgue spine in <math>\R^n</math>, for <math>n\ge 3,</math> is defined as follows
			:<math>S = \{(x_1,x_2,\dots,x_n)\in\R^n : x_n>0, x_1^2+x_2^2+\cdots
			+x_{n-1}^2 \le \exp(-1/x_n^2).</math>

			The important features of this set are that it is [[Connected set\|connected]] and [[path-connected]] in the [[euclidean topology]] in <math>\R^n</math> and the origin is a [[limit point]] of the set, and yet the set is '''thin''' at the origin, as defined in the article [[Fine topology (potential theory)]].

			== Observations ==

			The set <math>S</math> is not closed in the [[euclidean topology]] since it does not contain the origin which is a [[limit point]] of <math>S</math>, but the set is closed in the [[Fine topology (potential theory)\|fine topology]] in <math>\R^n</math>.

			In comparison, it is not possible in <math>\R^2</math> to construct such a connected set which is thin at the origin.

			== References ==

			* J. L. Doob. ''Classical Potential Theory and Its Probabilistic Counterpart'', Springer-Verlag, Berlin Heidelberg New York, ISBN 3-540-41206-9.
			* L. L. Helms (1975). ''Introduction to potential theory''. R. E. Krieger ISBN 0-88275-224-3.

			{{mathanalysis-stub}}

			[[Category:Potential theory]]

en>QuiteUnusual: Typo fixing, typos fixed: , → , (2), ie, → i.e., (2), Eg, → E.g., using AWB

2012-06-30T22:16:50Z

Typo fixing, typos fixed: , → , (2), ie, → i.e., (2), Eg, → E.g., using AWB

New page

Irwin Butts is what my spouse enjoys to call me although I don't truly like becoming known as like that. Hiring is her day job now and she will not change it whenever quickly. To collect badges is what her family members and her appreciate. For a while she's been in South Dakota.<br><br>my blog: [http://wixothek.com/blog/557 at home std test]

Box spread - Revision history

76.120.168.91: edit

en>QuiteUnusual: Typo fixing, typos fixed: , → , (2), ie, → i.e., (2), Eg, → E.g., using AWB