en>Addbot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q6043371

2013-03-16T09:18:48Z

Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q6043371

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	The ~~writer~~ is ~~recognized by~~ the ~~name~~ of ~~Numbers Lint~~. ~~South Dakota~~ is where me and ~~my spouse reside and my family enjoys it~~. ~~My day job~~ is a ~~meter reader~~. ~~One~~ of the ~~very best issues~~ in the ~~globe~~ for me is to ~~do aerobics and I~~'~~ve been performing it~~ for ~~quite~~ a ~~while.~~<br><br>~~Feel free~~ to ~~surf to my website~~ :~~: home std test kit~~ ([~~http~~:/~~/blogzaa~~.~~com~~/~~blogs/post/9199 click through~~ the ~~following internet site~~])		In [[mathematics]], '''Gårding's inequality''' is a result that gives a lower bound for the [[bilinear form]] induced by a real [[Elliptic operator\|linear elliptic partial differential operator]]. The inequality is named after [[Lars Gårding]].

			==Statement of the inequality==

			Let Ω be a [[bounded set\|bounded]], [[open set\|open domain]] in ''n''-[[dimension]]al [[Euclidean space]] and let ''H''<sup>''k''</sup>(Ω) denote the [[Sobolev space]] of ''k''-times weakly differentiable functions ''u'' : Ω → '''R''' with weak derivatives in ''L''<sup>2</sup>. Assume that Ω satisfies the ''k''-extension property, i.e., that there exists a [[bounded linear operator]] ''E'' : ''H''<sup>''k''</sup>(Ω) → ''H''<sup>''k''</sup>('''R'''<sup>''n''</sup>) such that (''Eu'')\|<sub>Ω</sub> = ''u'' for all ''u'' in ''H''<sup>''k''</sup>(Ω).

			Let ''L'' be a linear partial differential operator of even order ''2k'', written in divergence form

			:<math>(L u)(x) = \sum_{0 \leq \| \alpha \|, \| \beta \| \leq k} (-1)^{\| \alpha \|} \mathrm{D}^{\alpha} \left( A_{\alpha \beta} (x) \mathrm{D}^{\beta} u(x) \right),</math>

			and suppose that ''L'' is uniformly elliptic, i.e., there exists a constant ''θ'' > 0 such that

			:<math>\sum_{\| \alpha \|, \| \beta \| = k} \xi^{\alpha} A_{\alpha \beta} (x) \xi^{\beta} > \theta \| \xi \|^{2 k} \mbox{ for all } x \in \Omega, \xi \in \mathbb{R}^{n} \setminus \{ 0 \}.</math>

			Finally, suppose that the coefficients ''A<sub>αβ</sub>'' are [[bounded function\|bounded]], [[continuous function]]s on the [[closure (topology)\|closure]] of Ω for \|''α''\| = \|''β''\| = ''k'' and that

			:<math>A_{\alpha \beta} \in L^{\infty} (\Omega) \mbox{ for all } \| \alpha \|, \| \beta \| \leq k.</math>

			Then '''Gårding's inequality''' holds: there exist constants ''C'' > 0 and ''G'' ≥ 0

			:<math>B[u, u] + G \\| u \\|_{L^{2} (\Omega)}^{2} \geq C \\| u \\|_{H^{k} (\Omega)}^{2} \mbox{ for all } u \in H_{0}^{k} (\Omega),</math>

			where

			:<math>B[v, u] = \sum_{0 \leq \| \alpha \|, \| \beta \| \leq k} \int_{\Omega} A_{\alpha \beta} (x) \mathrm{D}^{\alpha} u(x) \mathrm{D}^{\beta} v(x) \, \mathrm{d} x</math>

			is the bilinear form associated to the operator ''L''.

			==Application: the Laplace operator and the Poisson problem==

			As a simple example, consider the [[Laplace operator]] Δ. More specifically, suppose that one wishes to solve, for ''f'' ∈ ''L''<sup>2</sup>(Ω) the [[Poisson equation]]

			:<math>\begin{cases} - \Delta u(x) = f(x), & x \in \Omega; \\ u(x) = 0, & x \in \partial \Omega; \end{cases}</math>

			where Ω is a bounded [[Lipschitz domain]] in '''R'''<sup>''n''</sup>. The corresponding weak form of the problem is to find ''u'' in the Sobolev space ''H''<sub>0</sub><sup>1</sup>(Ω) such that

			:<math>B[u, v] = \langle f, v \rangle \mbox{ for all } v \in H_{0}^{1} (\Omega),</math>

			where

			:<math>B[u, v] = \int_{\Omega} \nabla u(x) \cdot \nabla v(x) \, \mathrm{d} x,</math>
			:<math>\langle f, v \rangle = \int_{\Omega} f(x) v(x) \, \mathrm{d} x.</math>

			The [[Lax–Milgram lemma]] ensures that if the bilinear form ''B'' is both continuous and elliptic with respect to the norm on ''H''<sub>0</sub><sup>1</sup>(Ω), then, for each ''f'' ∈ ''L''<sup>2</sup>(Ω), a unique solution ''u'' must exist in ''H''<sub>0</sub><sup>1</sup>(Ω). The hypotheses of Gårding's inequality are easy to verify for the Laplace operator Δ, so there exist constants ''C'' and ''G'' ≥ 0

			:<math>B[u, u] \geq C \\| u \\|_{H^{1} (\Omega)}^{2} - G \\| u \\|_{L^{2} (\Omega)}^{2} \mbox{ for all } u \in H_{0}^{1} (\Omega).</math>

			Applying the [[Poincaré inequality]] allows the two terms on the right-hand side to be combined, yielding a new constant ''K'' > 0 with

			:<math>B[u, u] \geq K \\| u \\|_{H^{1} (\Omega)}^{2} \mbox{ for all } u \in H_{0}^{1} (\Omega),</math>

			which is precisely the statement that ''B'' is elliptic. The continuity of ''B'' is even easier to see: simply apply the [[Cauchy-Schwarz inequality]] and the fact that the Sobolev norm is controlled by the ''L''<sup>2</sup> norm of the gradient.

			==References==

			* {{cite book
			\| author = Renardy, Michael and Rogers, Robert C.
			\| title = An introduction to partial differential equations
			\| series = Texts in Applied Mathematics 13
			\| edition = Second edition
			\|publisher = Springer-Verlag
			\| location = New York
			\| year = 2004
			\| isbn = 0-387-00444-0
			\| page = 356
			}} (Theorem 8.17)

			{{DEFAULTSORT:Garding's inequality}}
			[[Category:Functional analysis]]
			[[Category:Inequalities]]
			[[Category:Partial differential equations]]
			[[Category:Sobolev spaces]]

en>Jesse V.: various fixes, removed stub tag using AWB

2012-04-12T04:59:39Z

various fixes, removed stub tag using AWB

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The writer is recognized by the name of Numbers Lint. South Dakota is where me and my spouse reside and my family enjoys it. My day job is a meter reader. One of the very best issues in the globe for me is to do aerobics and I've been performing it for quite a while.<br><br>Feel free to surf to my website :: home std test kit ([http://blogzaa.com/blogs/post/9199 click through the following internet site])

Integrodifference equation - Revision history

en>Addbot: Bot: Migrating 1 interwiki links, now provided by Wikidata on d:q6043371

en>Jesse V.: various fixes, removed stub tag using AWB