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| 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]].
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| ==Statement of the inequality==
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| ===The classical Poincaré inequality===
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| 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),
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| :<math>\| u - u_{\Omega} \|_{L^{p} (\Omega)} \leq C \| \nabla u \|_{L^{p} (\Omega)},</math>
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| where | |
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| :<math>u_{\Omega} = \frac{1}{|\Omega|} \int_{\Omega} u(y) \, \mathrm{d} y</math>
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| 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.
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| ===Generalizations===
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| In the context of metric measure spaces (for example, sub-Riemannian manifolds), such spaces support a (q,p)-Poincare inequality for some
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| <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,
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| <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>
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| In the context of metric measure spaces, <math>|\nabla u|</math> is the minimal p-weak upper gradient of u in the sense of
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| Heinonen and Koskela [J. Heinonen and P. Koskela, Quasiconformal maps in metric spaces with con- trolled geometry, Acta Math. 181 (1998), 1–61]
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| 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
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| :<math>[ u ]_{H^{1/2} (\mathbf{T}^{2})}^{2} = \sum_{k \in \mathbf{Z}^{2}} | k | \big| \hat{u} (k) \big|^{2} < + \infty:</math>
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| 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>,
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| :<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>
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| where cap(''E'' × {0}) denotes the [[harmonic capacity]] of ''E'' × {0} when thought of as a subset of '''R'''<sup>3</sup>.
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| ==The Poincaré constant==
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| 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]].
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| 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}}.)
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| 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>,
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| <math>\displaystyle ||u||_{L^2}^2\leq \lambda_1^{-1}||\nabla u||_{L^2}^2</math>
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| and furthermore, that the constant λ<sub>1</sub> is optimal.
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| ==References==
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| * {{citation
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| | last1 = Acosta|first1=Gabriel|last2=Durán|first2=Ricardo G.
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| | title = An optimal Poincaré inequality in ''L''<sup>1</sup> for convex domains
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| | journal = Proc. Amer. Math. Soc.
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| | volume = 132
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| | year = 2004
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| | issue = 1
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| | pages = 195–202 (electronic)
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| | doi = 10.1090/S0002-9939-03-07004-7
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| }}
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| * {{citation
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| | last = Evans|first=Lawrence C.
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| | title = Partial differential equations
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| | location = Providence, RI
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| | publisher = American Mathematical Society
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| | year = 1998
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| | isbn = 0-8218-0772-2
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| }}
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| * {{citation
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| | first1 = Fumio
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| | last1 = Kikuchi
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| | first2= Xuefeng|last2=Liu
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| | title = Estimation of interpolation error constants for the P0 and P1 triangular finite elements
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| | journal = Comput. Methods. Appl. Mech. Engrg.
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| | volume = 196
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| | year = 2007
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| | pages = 3750–3758
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| | doi = 10.1016/j.cma.2006.10.029
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| | issue = 37–40
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| }} {{MathSciNet|id=2340000}}
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| * {{citation
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| | last1 = Garroni
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| | first1 = Adriana
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| | last2 = Müller |first2 = Stefan
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| | title = Γ-limit of a phase-field model of dislocations
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| | journal = SIAM J. Math. Anal.
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| | volume = 36
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| | year = 2005
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| | issue = 6
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| | pages = 1943–1964 (electronic)
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| | doi = 10.1137/S003614100343768X
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| }} {{MathSciNet|id=2178227}}
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| *{{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}}
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| *{{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}}
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| {{DEFAULTSORT:Poincare inequality}}
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| [[Category:Mathematical analysis]]
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| [[Category:Inequalities]]
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| [[Category:Sobolev spaces]]
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