Poincaré inequality

In mathematics, the Poincaré inequality is a result in the theory of Sobolev spaces, named after the 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 methods of the calculus of variations. A very closely related result is the Friedrichs' inequality.

tatement of the inequality

The classical Poincaré inequality

Assume that 1 ≤ "p" ≤ ∞ and that Ω is bounded open subset of "n"-dimensional Euclidean space R^"n" having Lipschitz boundary (i.e., Ω is an open, bounded Lipschitz domain). Then there exists a constant "C", depending only on Ω and "p", such that, for every function "u" in the Sobolev space "W"^1,"p"(Ω),

:$|\; u\; -\; u\_\{Omega\}\; |\_\{L^\{p\}\; (Omega)\}\; leq\; C\; |\; abla\; u\; |\_\{L^\{p\}\; (Omega)\},$

where

:$u\_\{Omega\}\; =\; frac\{1\}\; int\_\{Omega\}\; u(y)\; ,\; mathrm\{d\}\; y$

is the average value of "u" over Ω, with |Ω| standing for the Lebesgue measure of the domain Ω.

Generalizations

There exist generalizations of the Poincaré inequality to other Sobolev spaces. For example, the following (taken from Garroni & Müller (2005)) is a Poincaré inequality for the Sobolev space "H"^1/2(T²), i.e. the space of functions "u" in the "L"² space of the unit torus T² with Fourier transform "û" satisfying

:

there exists a constant "C" such that, for every "u" ∈ "H"^1/2(T²) with "u" identically zero on an open set "E" ⊆ T²,

:$int\_\{mathbf\{T\}^\{2\; |\; u(x)\; |^\{2\}\; ,\; mathrm\{d\}\; x\; leq\; C\; left(\; 1\; +\; frac1\{mathrm\{cap\}\; (E\; imes\; \{\; 0\; \})\}\; ight)\; [\; u\; ]\; \_\{H^\{1/2\}\; (mathbf\{T\}^\{2\})\}^\{2\},$

where cap("E" × {0}) denotes the harmonic capacity of "E" × {0} when thought of as a subset of R³.

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, convex, Lipschitz domain with diameter "d", then the Poincaré constant is "d"/2 for "p" = 1, "d"/π for "p" = 2; harv|Acosta and Durán|2004. 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/&pi; ( < "d"/&pi; where $scriptstyle\{d=sqrt\{2$ ). (See, for instance,harvtxt|Kikuchi|Liu|2007.)

References

* citation
last1 = Acosta|first1=Gabriel|last2=Durán|first2=Ricardo G.
title = An optimal Poincaré inequality in "L"¹ for convex domains
journal = Proc. Amer. Math. Soc.
volume = 132
year = 2004
issue = 1
pages = 195&ndash;202 (electronic)
doi = 10.1090/S0002-9939-03-07004-7
* citation
first = Evans|last=Lawrence C.
title = Partial differential equations
location = Providence, RI
publisher = American Mathematical Society
year = 1998
id = ISBN 0-8218-0772-2
* citation
last1 = Garroni
first1 = Adriana
last2 = Müller |first2 = Stefan
title = &Gamma;-limit of a phase-field model of dislocations
journal = SIAM J. Math. Anal.
volume = 36
year = 2005
issue = 6
pages = 1943&ndash;1964 (electronic)
issn = 0036-1410
doi = 10.1137/S003614100343768X MathSciNet|id=2178227

* citation
last1 = Fumio
first1 = Kikuchi
last2= Xuefeng|first2=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&ndash;3758
issn = 0045-7825
doi = 10.1016/j.cma.2006.10.029 MathSciNet|id=2340000