Monotone class theorem - Revision history

130.60.188.148: /* Monotone class theorem for functions */

2014-08-06T13:24:41Z

Monotone class theorem for functions

← Older revision		Revision as of 15:24, 6 August 2014
Line 1:		Line 1:
	~~In [[mathematics]],~~ the '''Poincaré inequality''' is a result in the theory of [[Sobolev space]]s, named after the [[France\|French]] [[mathematician]] [[Henri Poincaré]]. The inequality allows one to obtain bounds on a function using bounds on its derivatives and the geometry of its domain of definition. Such bounds are of great importance in the modern, [[Direct method in calculus of variations\|direct methods of the calculus of variations]]. ~~A very closely related result~~ is ~~the [[Friedrichs' inequality]].~~		Alyson Meagher is the name her parents gave her but she doesn't like when individuals use her complete title. He is an order clerk and it's something he really enjoy. Ohio is exactly where his house is and his family members enjoys it. To climb is something I truly enjoy doing.<br><br>My web site ... love psychic readings ([http://www.taehyuna.net/xe/?document_srl=78721 simply click the up coming site])

	~~==Statement of the inequality==~~
	~~===The classical Poincaré inequality===~~

	~~Assume that 1 ≤ ''p'' ≤ ∞~~ and ~~that Ω is a [[bounded set\|bounded]] [[connected set\|connected]] [[open set\|open subset]] of the~~ '~~'n''-[[dimension]]al [[Euclidean space]] '''R'''<sup>''n''</sup> with a [[Lipschitz boundary]] (i.e~~.~~, Ω~~ is a [[Lipschitz domain\|Lipschitz]] [[Domain (mathematical analysis)\|domain]]). Then there exists a constant ''C'', depending only on Ω and ''p'', such that for every function ''u'' in the Sobolev space ''W''<sup>1,''p''</sup>(Ω),

	~~:<math>\\| u - u_{\Omega} \\|_{L^{p} (\Omega)} \leq C \\| \nabla u \\|_{L^{p} (\Omega)},</math>~~

	where

	~~:<math>u_{\Omega} = \frac{1}{\|\Omega\|} \int_{\Omega} u(y) \, \mathrm{d} y</math>~~

	is ~~the average value of ''u'' over Ω, with \|Ω\| standing for the [[Lebesgue measure]] of the domain Ω. When Ω is a ball, the above inequality is~~
	~~called a (p,p)-Poincare inequality; for more general domains Ω, the above is more familiarly known as a Sobolev inequality.~~

	~~===Generalizations===~~

	~~In the context of metric measure spaces (for example, sub-Riemannian manifolds), such spaces support a (q,p)-Poincare inequality for some~~
	~~<math>1\le q,p<\infty</math> if there are constants C~~ and ~~<math>\lambda\ge 1</math> so that for each ball B in the space,~~

	~~<math>\mu(B)^{-1/q}\\|u-u_B\\|_{L^q(B)}\le C \text{rad}(B) \mu(B)^{-1/p} \\| \nabla u\\|_{L^p(\lambda B)}~~.~~</math>~~

	~~In the context of metric measure spaces, <math>\|\nabla u\|</math>~~ is ~~the minimal p-weak upper gradient of u in the sense of~~
	~~Heinonen and Koskela [J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with con- trolled geometry, Acta Math. 181 (1998), 1–61]~~


	~~There exist other generalizations of the Poincaré inequality to other Sobolev spaces~~. ~~For example, the following (taken from {{harvtxt\|Garroni\|Müller\|2005}}) is a Poincaré inequality for the Sobolev space ''H''~~<~~sup~~>~~1/2~~<~~/sup>('''T'''<sup~~>~~2</sup>), i~~.e. ~~the space of functions ''u'' in the [[Lp space\|''L''<sup>2</sup> space]] of the unit [[torus]] '''T'''<sup>2</sup> with [[Fourier transform]] ''û'' satisfying~~

	~~:<math>[ u ]_{H^{1/2} (\mathbf{T}^{2})}^{2} = \sum_{k \in \mathbf{Z}^{2}} \| k \| \big\| \hat{u} (k) \big\|^{2} < + \infty:</math>~~

	~~there exists a constant ''C'' such that, for every ''u'' ∈ ''H''<sup>1/2</sup>('''T'''<sup>2</sup>) with ''u'' identically zero on an open set ''E'' ⊆ '''T'''<sup>2</sup>,~~

	~~:<math>\int_{\mathbf{T}^{2}} \| u(x) \|^{2} \, \mathrm{d} x \leq C \left( 1 + \frac1{\mathrm{cap} (E \times \{ 0 \})} \right) [ u ]_{H^{1/2} (\mathbf{T}^{2})}^{2},</math>~~

	~~where cap(''E'' × {0}) denotes the [[harmonic capacity]] of ''E'' × {0} when thought of as a subset of '''R'''<sup>3</sup>.~~

	~~==The Poincaré constant==~~

	~~The optimal constant ''C'' in the Poincaré inequality is sometimes known as the '''Poincaré constant''' for the domain Ω~~. Determining the Poincaré constant is, in general, a very hard task that depends upon the value of ''p'' and the geometry of the domain Ω. Certain special cases are tractable, however. For example, if Ω is a [[bounded set\|bounded]], [[convex set\|convex]], Lipschitz domain with diameter ''d'', then the Poincaré constant is at most ''d''/2 for ''p'' = 1, ''d''/π for ''p'' = 2 ({{harvnb\|Acosta\|Durán\|2004}}; {{harvnb\|Payne\|Weinberger\|1960}}), and this is the best possible estimate on the Poincaré constant in terms of the diameter alone. For smooth functions, this can be understood as an application of the [[isoperimetric inequality]] to the function's [[level sets]]. [http://~~maze5~~.net/?~~page_id=790] In one dimension, this is [[Wirtinger's inequality for functions]].~~

	~~However, in some special cases the constant ''C'' can be determined concretely. For example, for ''p'' ~~= 2, it is well known that over the domain of unit isosceles right triangle, ''C'' = 1/π ( < ''d''/π where <math>\scriptstyle{d=\sqrt{2}}</math> ). (See, for instance,{{harvtxt\|Kikuchi\|Liu\|2007}}.)

	~~Furthermore, for a smooth, bounded domain <math>\Omega</math>, since the [[Rayleigh quotient]] for~~ the ~~[[Laplace operator]~~] in the space <math>W^{1,2}_0(\Omega)</math> is minimized by the eigenfunction corresponding to the minimal eigenvalue λ<sub>1</sub> of the (negative) Laplacian, it is a simple consequence that, for any <math>u\in W^{1,2}_0(\Omega)~~</math>,~~

	~~<math>\displaystyle \|\|u\|\|_{L^2}^2\leq \lambda_1^{-1}\|\|\nabla u\|\|_{L^2}^2</math>~~

	~~and furthermore, that the constant λ<sub>1</sub> is optimal.~~

	~~==References==~~

	* {{citation
	~~\| last1 = Acosta\|first1=Gabriel\|last2=Durán\|first2=Ricardo G.~~
	~~\| title = An optimal Poincaré inequality in ''L''<sup>1</sup> for convex domains~~
	~~\| journal = Proc. Amer. Math. Soc.~~
	~~\| volume = 132~~
	~~\| year = 2004~~
	~~\| issue = 1~~
	~~\| pages = 195–202 (electronic)~~
	~~\| doi = 10.1090/S0002-9939-03-07004-7~~
	}}
	* {{citation
	~~\| last = Evans\|first=Lawrence C.~~
	~~\| title = Partial differential equations~~
	~~\| location = Providence, RI~~
	~~\| publisher = American Mathematical Society~~
	~~\| year = 1998~~
	~~\| isbn = 0-8218-0772-2~~
	}}
	* {{citation
	~~\| first1 = Fumio~~
	~~\| last1 = Kikuchi~~
	~~\| first2= Xuefeng\|last2=Liu~~
	~~\| title = Estimation of interpolation error constants for the P0 and P1 triangular finite elements~~
	~~\| journal = Comput. Methods. Appl. Mech. Engrg.~~
	~~\| volume = 196~~
	~~\| year = 2007~~
	~~\| pages = 3750–3758~~
	~~\| doi = 10.1016/j.cma.2006.10.029~~
	~~\| issue = 37–40~~
	~~}} {{MathSciNet\|id=2340000}}~~
	* {{citation
	~~\| last1 = Garroni~~
	~~\| first1 = Adriana~~
	~~\| last2 = Müller \|first2 = Stefan~~
	~~\| title = Γ-limit of a phase-field model of dislocations~~
	~~\| journal = SIAM J. Math. Anal.~~
	~~\| volume = 36~~
	~~\| year = 2005~~
	~~\| issue = 6~~
	~~\| pages = 1943–1964 (electronic)~~
	~~\| doi = 10.1137/S003614100343768X~~
	~~}} {{MathSciNet\|id=2178227}}~~
	*{{Citation \| last1=Payne \| first1=L. E. \| last2=Weinberger \| first2=H. F. \| title=An optimal Poincaré inequality for convex domains \| year=1960 \| journal=Archive for Rational Mechanics and Analysis \| issn=0003-9527 \| pages=286–292}}
	*{{Citation \| last1=Heinonen \| first1=J. \| last2=Koskela \| first2=P. \| title=Quasiconformal maps in metric spaces with controlled geometry \| journal=Acta Mathematica \| doi= 10.1007/BF02392747 \|issn= 1871-2509 \| year=1998 \| pages=1–61}}

	~~{{DEFAULTSORT:Poincare inequality}}~~
	~~[[Category:Mathematical analysis]]~~
	~~[[Category:Inequalities]]~~
	~~[[Category:Sobolev spaces]]~~

en>ChrisGualtieri: General Fixes using AWB

2013-10-23T03:10:39Z

General Fixes using AWB

← Older revision		Revision as of 05:10, 23 October 2013
Line 1:		Line 1:
	~~Jayson Berryhill~~ is ~~how I~~'~~m known as~~ and ~~my wife doesn~~'~~t like it at all~~. ~~To play lacross~~ is some ~~thing he would by no means give up~~. ~~Invoicing~~ is ~~my occupation~~. ~~Her family lives~~ in ~~Ohio~~.<br><br>~~My website~~ :: [http://~~www~~.~~taehyuna~~.~~net~~/xe/~~?document_srl~~=~~78721 psychic readers~~]		In [[mathematics]], the '''Poincaré inequality''' is a result in the theory of [[Sobolev space]]s, named after the [[France\|French]] [[mathematician]] [[Henri Poincaré]]. The inequality allows one to obtain bounds on a function using bounds on its derivatives and the geometry of its domain of definition. Such bounds are of great importance in the modern, [[Direct method in calculus of variations\|direct methods of the calculus of variations]]. A very closely related result is the [[Friedrichs' inequality]].

			==Statement of the inequality==
			===The classical Poincaré inequality===

			Assume that 1 ≤ ''p'' ≤ ∞ and that Ω is a [[bounded set\|bounded]] [[connected set\|connected]] [[open set\|open subset]] of the ''n''-[[dimension]]al [[Euclidean space]] '''R'''<sup>''n''</sup> with a [[Lipschitz boundary]] (i.e., Ω is a [[Lipschitz domain\|Lipschitz]] [[Domain (mathematical analysis)\|domain]]). Then there exists a constant ''C'', depending only on Ω and ''p'', such that for every function ''u'' in the Sobolev space ''W''<sup>1,''p''</sup>(Ω),

			:<math>\\| u - u_{\Omega} \\|_{L^{p} (\Omega)} \leq C \\| \nabla u \\|_{L^{p} (\Omega)},</math>

			where

			:<math>u_{\Omega} = \frac{1}{\|\Omega\|} \int_{\Omega} u(y) \, \mathrm{d} y</math>

			is the average value of ''u'' over Ω, with \|Ω\| standing for the [[Lebesgue measure]] of the domain Ω. When Ω is a ball, the above inequality is
			called a (p,p)-Poincare inequality; for more general domains Ω, the above is more familiarly known as a Sobolev inequality.

			===Generalizations===

			In the context of metric measure spaces (for example, sub-Riemannian manifolds), such spaces support a (q,p)-Poincare inequality for some
			<math>1\le q,p<\infty</math> if there are constants C and <math>\lambda\ge 1</math> so that for each ball B in the space,

			<math>\mu(B)^{-1/q}\\|u-u_B\\|_{L^q(B)}\le C \text{rad}(B) \mu(B)^{-1/p} \\| \nabla u\\|_{L^p(\lambda B)}.</math>

			In the context of metric measure spaces, <math>\|\nabla u\|</math> is the minimal p-weak upper gradient of u in the sense of
			Heinonen and Koskela [J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with con- trolled geometry, Acta Math. 181 (1998), 1–61]


			There exist other generalizations of the Poincaré inequality to other Sobolev spaces. For example, the following (taken from {{harvtxt\|Garroni\|Müller\|2005}}) is a Poincaré inequality for the Sobolev space ''H''<sup>1/2</sup>('''T'''<sup>2</sup>), i.e. the space of functions ''u'' in the [[Lp space\|''L''<sup>2</sup> space]] of the unit [[torus]] '''T'''<sup>2</sup> with [[Fourier transform]] ''û'' satisfying

			:<math>[ u ]_{H^{1/2} (\mathbf{T}^{2})}^{2} = \sum_{k \in \mathbf{Z}^{2}} \| k \| \big\| \hat{u} (k) \big\|^{2} < + \infty:</math>

			there exists a constant ''C'' such that, for every ''u'' ∈ ''H''<sup>1/2</sup>('''T'''<sup>2</sup>) with ''u'' identically zero on an open set ''E'' ⊆ '''T'''<sup>2</sup>,

			:<math>\int_{\mathbf{T}^{2}} \| u(x) \|^{2} \, \mathrm{d} x \leq C \left( 1 + \frac1{\mathrm{cap} (E \times \{ 0 \})} \right) [ u ]_{H^{1/2} (\mathbf{T}^{2})}^{2},</math>

			where cap(''E'' × {0}) denotes the [[harmonic capacity]] of ''E'' × {0} when thought of as a subset of '''R'''<sup>3</sup>.

			==The Poincaré constant==

			The optimal constant ''C'' in the Poincaré inequality is sometimes known as the '''Poincaré constant''' for the domain Ω. Determining the Poincaré constant is, in general, a very hard task that depends upon the value of ''p'' and the geometry of the domain Ω. Certain special cases are tractable, however. For example, if Ω is a [[bounded set\|bounded]], [[convex set\|convex]], Lipschitz domain with diameter ''d'', then the Poincaré constant is at most ''d''/2 for ''p'' = 1, ''d''/π for ''p'' = 2 ({{harvnb\|Acosta\|Durán\|2004}}; {{harvnb\|Payne\|Weinberger\|1960}}), and this is the best possible estimate on the Poincaré constant in terms of the diameter alone. For smooth functions, this can be understood as an application of the [[isoperimetric inequality]] to the function's [[level sets]]. [http://maze5.net/?page_id=790] In one dimension, this is [[Wirtinger's inequality for functions]].

			However, in some special cases the constant ''C'' can be determined concretely. For example, for ''p'' = 2, it is well known that over the domain of unit isosceles right triangle, ''C'' = 1/π ( < ''d''/π where <math>\scriptstyle{d=\sqrt{2}}</math> ). (See, for instance,{{harvtxt\|Kikuchi\|Liu\|2007}}.)

			Furthermore, for a smooth, bounded domain <math>\Omega</math>, since the [[Rayleigh quotient]] for the [[Laplace operator]] in the space <math>W^{1,2}_0(\Omega)</math> is minimized by the eigenfunction corresponding to the minimal eigenvalue λ<sub>1</sub> of the (negative) Laplacian, it is a simple consequence that, for any <math>u\in W^{1,2}_0(\Omega)</math>,

			<math>\displaystyle \|\|u\|\|_{L^2}^2\leq \lambda_1^{-1}\|\|\nabla u\|\|_{L^2}^2</math>

			and furthermore, that the constant λ<sub>1</sub> is optimal.

			==References==

			* {{citation
			\| last1 = Acosta\|first1=Gabriel\|last2=Durán\|first2=Ricardo G.
			\| title = An optimal Poincaré inequality in ''L''<sup>1</sup> for convex domains
			\| journal = Proc. Amer. Math. Soc.
			\| volume = 132
			\| year = 2004
			\| issue = 1
			\| pages = 195–202 (electronic)
			\| doi = 10.1090/S0002-9939-03-07004-7
			}}
			* {{citation
			\| last = Evans\|first=Lawrence C.
			\| title = Partial differential equations
			\| location = Providence, RI
			\| publisher = American Mathematical Society
			\| year = 1998
			\| isbn = 0-8218-0772-2
			}}
			* {{citation
			\| first1 = Fumio
			\| last1 = Kikuchi
			\| first2= Xuefeng\|last2=Liu
			\| title = Estimation of interpolation error constants for the P0 and P1 triangular finite elements
			\| journal = Comput. Methods. Appl. Mech. Engrg.
			\| volume = 196
			\| year = 2007
			\| pages = 3750–3758
			\| doi = 10.1016/j.cma.2006.10.029
			\| issue = 37–40
			}} {{MathSciNet\|id=2340000}}
			* {{citation
			\| last1 = Garroni
			\| first1 = Adriana
			\| last2 = Müller \|first2 = Stefan
			\| title = Γ-limit of a phase-field model of dislocations
			\| journal = SIAM J. Math. Anal.
			\| volume = 36
			\| year = 2005
			\| issue = 6
			\| pages = 1943–1964 (electronic)
			\| doi = 10.1137/S003614100343768X
			}} {{MathSciNet\|id=2178227}}
			*{{Citation \| last1=Payne \| first1=L. E. \| last2=Weinberger \| first2=H. F. \| title=An optimal Poincaré inequality for convex domains \| year=1960 \| journal=Archive for Rational Mechanics and Analysis \| issn=0003-9527 \| pages=286–292}}
			*{{Citation \| last1=Heinonen \| first1=J. \| last2=Koskela \| first2=P. \| title=Quasiconformal maps in metric spaces with controlled geometry \| journal=Acta Mathematica \| doi= 10.1007/BF02392747 \|issn= 1871-2509 \| year=1998 \| pages=1–61}}

			{{DEFAULTSORT:Poincare inequality}}
			[[Category:Mathematical analysis]]
			[[Category:Inequalities]]
			[[Category:Sobolev spaces]]

en>Hypatia271828: I changed the link associated to the word "closed" at the very beggining, from the "Closed" Wikipedia disambiguation page to the "Closure (Mathematics)" Wikipedia article, since it's that concept the one relevant to Monotone Classes.

2012-08-30T02:57:12Z

I changed the link associated to the word "closed" at the very beggining, from the "Closed" Wikipedia disambiguation page to the "Closure (Mathematics)" Wikipedia article, since it's that concept the one relevant to Monotone Classes.

New page

Jayson Berryhill is how I'm known as and my wife doesn't like it at all. To play lacross is some thing he would by no means give up. Invoicing is my occupation. Her family lives in Ohio.<br><br>My website :: [http://www.taehyuna.net/xe/?document_srl=78721 psychic readers]

← Older revision		Revision as of 15:24, 6 August 2014
Line 1:		Line 1:
	~~In [[mathematics]],~~ the '''Poincaré inequality''' is a result in the theory of [[Sobolev space]]s, named after the [[France\|French]] [[mathematician]] [[Henri Poincaré]]. The inequality allows one to obtain bounds on a function using bounds on its derivatives and the geometry of its domain of definition. Such bounds are of great importance in the modern, [[Direct method in calculus of variations\|direct methods of the calculus of variations]]. ~~A very closely related result~~ is ~~the [[Friedrichs' inequality]].~~		Alyson Meagher is the name her parents gave her but she doesn't like when individuals use her complete title. He is an order clerk and it's something he really enjoy. Ohio is exactly where his house is and his family members enjoys it. To climb is something I truly enjoy doing.<br><br>My web site ... love psychic readings ([http://www.taehyuna.net/xe/?document_srl=78721 simply click the up coming site])

	~~==Statement of the inequality==~~
	~~===The classical Poincaré inequality===~~

	~~Assume that 1 ≤ ''p'' ≤ ∞~~ and ~~that Ω is a [[bounded set\|bounded]] [[connected set\|connected]] [[open set\|open subset]] of the~~ '~~'n''-[[dimension]]al [[Euclidean space]] '''R'''<sup>''n''</sup> with a [[Lipschitz boundary]] (i.e~~.~~, Ω~~ is a [[Lipschitz domain\|Lipschitz]] [[Domain (mathematical analysis)\|domain]]). Then there exists a constant ''C'', depending only on Ω and ''p'', such that for every function ''u'' in the Sobolev space ''W''<sup>1,''p''</sup>(Ω),

	~~:<math>\\| u - u_{\Omega} \\|_{L^{p} (\Omega)} \leq C \\| \nabla u \\|_{L^{p} (\Omega)},</math>~~

	where

	~~:<math>u_{\Omega} = \frac{1}{\|\Omega\|} \int_{\Omega} u(y) \, \mathrm{d} y</math>~~

	is ~~the average value of ''u'' over Ω, with \|Ω\| standing for the [[Lebesgue measure]] of the domain Ω. When Ω is a ball, the above inequality is~~
	~~called a (p,p)-Poincare inequality; for more general domains Ω, the above is more familiarly known as a Sobolev inequality.~~

	~~===Generalizations===~~

	~~In the context of metric measure spaces (for example, sub-Riemannian manifolds), such spaces support a (q,p)-Poincare inequality for some~~
	~~<math>1\le q,p<\infty</math> if there are constants C~~ and ~~<math>\lambda\ge 1</math> so that for each ball B in the space,~~

	~~<math>\mu(B)^{-1/q}\\|u-u_B\\|_{L^q(B)}\le C \text{rad}(B) \mu(B)^{-1/p} \\| \nabla u\\|_{L^p(\lambda B)}~~.~~</math>~~

	~~In the context of metric measure spaces, <math>\|\nabla u\|</math>~~ is ~~the minimal p-weak upper gradient of u in the sense of~~
	~~Heinonen and Koskela [J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with con- trolled geometry, Acta Math. 181 (1998), 1–61]~~


	~~There exist other generalizations of the Poincaré inequality to other Sobolev spaces~~. ~~For example, the following (taken from {{harvtxt\|Garroni\|Müller\|2005}}) is a Poincaré inequality for the Sobolev space ''H''~~<~~sup~~>~~1/2~~<~~/sup>('''T'''<sup~~>~~2</sup>), i~~.e. ~~the space of functions ''u'' in the [[Lp space\|''L''<sup>2</sup> space]] of the unit [[torus]] '''T'''<sup>2</sup> with [[Fourier transform]] ''û'' satisfying~~

	~~:<math>[ u ]_{H^{1/2} (\mathbf{T}^{2})}^{2} = \sum_{k \in \mathbf{Z}^{2}} \| k \| \big\| \hat{u} (k) \big\|^{2} < + \infty:</math>~~

	~~there exists a constant ''C'' such that, for every ''u'' ∈ ''H''<sup>1/2</sup>('''T'''<sup>2</sup>) with ''u'' identically zero on an open set ''E'' ⊆ '''T'''<sup>2</sup>,~~

	~~:<math>\int_{\mathbf{T}^{2}} \| u(x) \|^{2} \, \mathrm{d} x \leq C \left( 1 + \frac1{\mathrm{cap} (E \times \{ 0 \})} \right) [ u ]_{H^{1/2} (\mathbf{T}^{2})}^{2},</math>~~

	~~where cap(''E'' × {0}) denotes the [[harmonic capacity]] of ''E'' × {0} when thought of as a subset of '''R'''<sup>3</sup>.~~

	~~==The Poincaré constant==~~

	~~The optimal constant ''C'' in the Poincaré inequality is sometimes known as the '''Poincaré constant''' for the domain Ω~~. Determining the Poincaré constant is, in general, a very hard task that depends upon the value of ''p'' and the geometry of the domain Ω. Certain special cases are tractable, however. For example, if Ω is a [[bounded set\|bounded]], [[convex set\|convex]], Lipschitz domain with diameter ''d'', then the Poincaré constant is at most ''d''/2 for ''p'' = 1, ''d''/π for ''p'' = 2 ({{harvnb\|Acosta\|Durán\|2004}}; {{harvnb\|Payne\|Weinberger\|1960}}), and this is the best possible estimate on the Poincaré constant in terms of the diameter alone. For smooth functions, this can be understood as an application of the [[isoperimetric inequality]] to the function's [[level sets]]. [http://~~maze5~~.net/?~~page_id=790] In one dimension, this is [[Wirtinger's inequality for functions]].~~

	~~However, in some special cases the constant ''C'' can be determined concretely. For example, for ''p'' ~~= 2, it is well known that over the domain of unit isosceles right triangle, ''C'' = 1/π ( < ''d''/π where <math>\scriptstyle{d=\sqrt{2}}</math> ). (See, for instance,{{harvtxt\|Kikuchi\|Liu\|2007}}.)

	~~Furthermore, for a smooth, bounded domain <math>\Omega</math>, since the [[Rayleigh quotient]] for~~ the ~~[[Laplace operator]~~] in the space <math>W^{1,2}_0(\Omega)</math> is minimized by the eigenfunction corresponding to the minimal eigenvalue λ<sub>1</sub> of the (negative) Laplacian, it is a simple consequence that, for any <math>u\in W^{1,2}_0(\Omega)~~</math>,~~

	~~<math>\displaystyle \|\|u\|\|_{L^2}^2\leq \lambda_1^{-1}\|\|\nabla u\|\|_{L^2}^2</math>~~

	~~and furthermore, that the constant λ<sub>1</sub> is optimal.~~

	~~==References==~~

	* {{citation
	~~\| last1 = Acosta\|first1=Gabriel\|last2=Durán\|first2=Ricardo G.~~
	~~\| title = An optimal Poincaré inequality in ''L''<sup>1</sup> for convex domains~~
	~~\| journal = Proc. Amer. Math. Soc.~~
	~~\| volume = 132~~
	~~\| year = 2004~~
	~~\| issue = 1~~
	~~\| pages = 195–202 (electronic)~~
	~~\| doi = 10.1090/S0002-9939-03-07004-7~~
	}}
	* {{citation
	~~\| last = Evans\|first=Lawrence C.~~
	~~\| title = Partial differential equations~~
	~~\| location = Providence, RI~~
	~~\| publisher = American Mathematical Society~~
	~~\| year = 1998~~
	~~\| isbn = 0-8218-0772-2~~
	}}
	* {{citation
	~~\| first1 = Fumio~~
	~~\| last1 = Kikuchi~~
	~~\| first2= Xuefeng\|last2=Liu~~
	~~\| title = Estimation of interpolation error constants for the P0 and P1 triangular finite elements~~
	~~\| journal = Comput. Methods. Appl. Mech. Engrg.~~
	~~\| volume = 196~~
	~~\| year = 2007~~
	~~\| pages = 3750–3758~~
	~~\| doi = 10.1016/j.cma.2006.10.029~~
	~~\| issue = 37–40~~
	~~}} {{MathSciNet\|id=2340000}}~~
	* {{citation
	~~\| last1 = Garroni~~
	~~\| first1 = Adriana~~
	~~\| last2 = Müller \|first2 = Stefan~~
	~~\| title = Γ-limit of a phase-field model of dislocations~~
	~~\| journal = SIAM J. Math. Anal.~~
	~~\| volume = 36~~
	~~\| year = 2005~~
	~~\| issue = 6~~
	~~\| pages = 1943–1964 (electronic)~~
	~~\| doi = 10.1137/S003614100343768X~~
	~~}} {{MathSciNet\|id=2178227}}~~
	*{{Citation \| last1=Payne \| first1=L. E. \| last2=Weinberger \| first2=H. F. \| title=An optimal Poincaré inequality for convex domains \| year=1960 \| journal=Archive for Rational Mechanics and Analysis \| issn=0003-9527 \| pages=286–292}}
	*{{Citation \| last1=Heinonen \| first1=J. \| last2=Koskela \| first2=P. \| title=Quasiconformal maps in metric spaces with controlled geometry \| journal=Acta Mathematica \| doi= 10.1007/BF02392747 \|issn= 1871-2509 \| year=1998 \| pages=1–61}}

	~~{{DEFAULTSORT:Poincare inequality}}~~
	~~[[Category:Mathematical analysis]]~~
	~~[[Category:Inequalities]]~~
	~~[[Category:Sobolev spaces]]~~