View source for Poincaré inequality

{{Short description|Mathematical inequality in Sobolev space theory}}
In [[mathematics]], the '''Poincaré inequality'''<ref>{{Cite journal|last=Poincaré|first=H.|date=1890|title=Sur les Equations aux Dérivées Partielles de la Physique Mathématique|url=http://www.jstor.org/stable/2369620|journal=American Journal of Mathematics|volume=12|issue=3|at=Equation (11) page 253|doi=10.2307/2369620|jstor=2369620 |issn=0002-9327}}</ref> 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 [[Friedrichs' inequality]].

==Statement of the inequality==

===The classical Poincaré inequality===

Let ''p'', so that 1&nbsp;≤&nbsp;''p''&nbsp;<&nbsp;∞ and Ω a subset bounded at least in one direction. Then there exists a constant ''C'', depending only on Ω and ''p'', so that, for every function ''u'' of the [[Sobolev space]] ''W''<sub>0</sub><sup>1,''p''</sup>(Ω) of zero-[[trace operator|trace]] (a.k.a. zero on the boundary) functions,

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

===Poincaré–Wirtinger inequality===

Assume that 1&nbsp;≤&nbsp;''p''&nbsp;≤&nbsp;∞ and that Ω is a [[bounded set|bounded]] [[connected set|connected]] [[open set|open subset]] of the ''n''-[[dimension]]al [[Euclidean space]] <math>\mathbb{R}^{n}</math> 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 {{math|''W''<sup>1,''p''</sup>(Ω)}},
<math display="block">\| u - u_{\Omega} \|_{L^{p} (\Omega)} \leq C \| \nabla u \|_{L^{p} (\Omega)},</math>
where
<math display="block">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 {{math|(''p'',''p'')}}-Poincaré inequality; for more general domains Ω, the above is more familiarly known as a [[Sobolev inequality]].

The necessity to subtract the average value can be seen by considering constant functions for which the derivative is zero while, without subtracting the average, we can have the integral of the function as large as we wish. There are other conditions instead of subtracting the average that we can require in order to deal with this issue with constant functions, for example, requiring trace zero, or subtracting the average over some proper subset of the domain. The constant C in the Poincare inequality may be different from condition to condition. Also note that the issue is not just the constant functions, because it is the same as saying that adding a constant value to a function can increase its integral while the integral of its derivative remains the same. So, simply excluding the constant functions will not solve the issue.

===Generalizations===

In the context of [[metric measure space]]s, the definition of a Poincaré inequality is slightly different. One definition is: a metric measure space supports a (q,p)-Poincare inequality for some <math>1\le q,p<\infty</math> if there are constants ''C'' and {{math|λ ≥ 1}} so that for each ball B in the space,
<math display="block">\mu(B)^{-\frac{1}{q}} \left \|u-u_B \right \|_{L^q(B)}\le C \operatorname{rad}(B) \mu(B)^{-\frac{1}{p}} \| \nabla u\|_{L^p(\lambda B)}.</math>
Here we have an enlarged ball in the right hand side. 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.<ref>{{cite journal | 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 | volume=181 | pages=1–61| doi-access=free }}</ref>

Whether a space supports a Poincaré inequality has turned out to have deep connections to the geometry and analysis of the space.  For example, Cheeger has shown that a [[doubling space]] satisfying a Poincaré inequality admits a notion of differentiation.<ref>{{cite journal |last1=Cheeger |first1=J. |title=Differentiability of Lipschitz functions on metric measure spaces |journal=Geometric and Functional Analysis |date=1 August 1999 |volume=9 |issue=3 |pages=428–517 |doi=10.1007/s000390050094|s2cid=120149485 }}</ref>  Such spaces include [[sub-Riemannian manifold]]s and [[Laakso space]]s.

There exist other generalizations of the Poincaré inequality to other Sobolev spaces.  For example, consider 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 display="block">[u]_{H^{1/2} (\mathbf{T}^{2})}^2 = \sum_{k \in \mathbf{Z}^2} | k | \left | \hat{u} (k) \right |^2 < + \infty.</math>
In this context, the Poincaré inequality says: there exists a constant ''C'' such that, for every {{math|''u'' ∈ ''H''<sup>1/2</sup>('''T'''<sup>2</sup>)}} with ''u'' identically zero on an open set {{math|''E'' ⊆ '''T'''<sup>2</sup>}},
<math display="block">\int_{\mathbf{T}^2} | u(x) |^2 \, \mathrm{d} x \leq C \left( 1 + \frac1{\operatorname{cap} (E \times \{ 0 \})} \right) [ u ]_{H^{1/2} (\mathbf{T}^2)}^2,</math>
where {{math|cap(''E'' × {0})}} denotes the [[harmonic capacity]] of {{math|''E'' × {0}}} when thought of as a subset of <math>\mathbb{R}^{3}</math>.<ref>{{cite journal
| last1 = Garroni | first1 = Adriana | author1-link = Adriana Garroni
| 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 = 1943–1964 (electronic)
| doi = 10.1137/S003614100343768X
}} {{MathSciNet|id=2178227}}</ref>

Yet another generalization involves weighted Poincaré inequalities where the Lebesgue measure is replaced by a weighted version.

==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 {{math|1=''p'' = 1}}, <math>d/\pi</math> for {{math|1=''p'' = 2}},<ref>{{cite journal
| last1 = Acosta|first1=Gabriel|last2=Durán|first2=Ricardo G.
| title = An optimal Poincaré inequality in ''L''<sup>1</sup> for convex domains
| journal = [[Proceedings of the American Mathematical Society]]
| volume = 132
| year = 2004
| issue = 1
| pages = 195–202 (electronic)
| doi = 10.1090/S0002-9939-03-07004-7
| doi-access = free
}}</ref><ref>{{cite journal | 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 | volume=5 | issue=1 | issn=0003-9527 | pages=286–292 | doi=10.1007/BF00252910 | bibcode=1960ArRMA...5..286P | s2cid=121881343 }}</ref> 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]].<ref>{{cite web |last1=Alger |first1=Nick |title=L1 Poincare Inequality |url=http://maze5.net/?page_id=790 |archive-url=https://web.archive.org/web/20120303090859/http://maze5.net/?page_id=790 |archive-date=March 3, 2012}}</ref>  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''&nbsp;=&nbsp;2, it is well known that over the domain of unit isosceles right triangle, ''C''&nbsp;=&nbsp;1/π (&nbsp;<&nbsp;''d''/π where <math>d=\sqrt{2}</math>).<ref>{{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
| bibcode = 2007CMAME.196.3750K 
}} {{MathSciNet|id=2340000}}</ref>

Furthermore, for a smooth, bounded domain {{math|Ω}}, 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 {{math|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 display="block"> \|u\|_{L^2}^2\leq \lambda_1^{-1} \left \|\nabla u\right \|_{L^2}^2</math>
and furthermore, that the constant λ<sub>1</sub> is optimal.

==Poincaré inequality on metric-measure spaces==

Since the 90s there have been several fruitful ways to make sense of Sobolev functions on general metric measure spaces (metric spaces equipped with a measure that is often compatible with the metric in certain senses). For example, the approach based on "upper gradients" leads to Newtonian-Sobolev space of functions. Thus, it makes sense to say that a space "supports a Poincare inequality".

It turns out that whether a space supports any Poincare inequality and if so, the critical exponent for which it does, is tied closely to the geometry of the space. For example, a space that supports a Poincare inequality must be path connected. Indeed, between any pair of points there must exist a rectifiable path with length comparable to the distance of the points. Much deeper connections have been found, e.g. through the notion of modulus of path families. A good and rather recent reference is the monograph "Sobolev Spaces on Metric Measure Spaces, an approach based on upper gradients" written by Heinonen et al.

== Sobolev Slobodeckij Spaces and Poincaré Inequality ==

Given <math>0 < s < 1</math> and <math>p \in [1, \infty)</math>, the Sobolev Slobodeckij space <math>W^{s,p}(\Omega)</math> is defined as the set of all functions <math>u</math> such that <math>u \in L^p(\Omega)</math> and the seminorm <math>[u]_{s,p}</math> is finite. The seminorm <math>[u]_{s,p}</math> is defined by:
: <math>[u]_{s,p} = \left( \int_\Omega \int_\Omega \frac{|u(x) - u(y)|^p}{|x - y|^{n + sp}} \, dx \, dy \right)^{1/p}</math>

The Poincaré Inequality in this context can be generalized as follows:
: <math>\|u - u_\Omega\|_{L^p(\Omega)} \leq C [u]_{s,p}</math>
where <math>u_\Omega</math> is the average of <math>u</math> over <math>\Omega</math> and <math>C</math> is a constant dependent on <math>s, p</math>, and <math>\Omega</math>. This inequality holds for every bounded <math>\Omega</math>.

=== Proof of the Poincaré Inequality ===

The proof follows that of Irene Drelichman and Ricardo G. Durán.<ref>{{Cite journal
| arxiv = 1705.04227
| title = Improved Poincaré inequalities in fractional Sobolev spaces
| last1 = Drelichman | first1 = Irene
| last2 = Durán | first2 = Ricardo G.
| journal = Annales Academiæ Scientiarum Fennicæ
| volume = 43
| issue = 2
| date = 2018
| pages = 885—903
| doi = 10.5186/aasfm.2018.4352 | doi-access = free}}</ref>
Let <math>f_\Omega = \frac{1}{|\Omega|} \int_\Omega f(x) \, dx</math>. By applying Jensen's inequality, we obtain:
: <math>\|f - f_\Omega\|^p_{L^p(\Omega)} = \left\| \frac{1}{|\Omega|} \int_\Omega (f(y) - f(x)) \, dx \right\|^p_{L^p} = \int_\Omega \left| \frac{1}{|\Omega|} \int_\Omega f(y) - f(x) \, dy \right|^p \, dx</math>
: <math>\leq \frac{1}{|\Omega|} \int_\Omega \int_\Omega |f(y) - f(x)|^p \, dy \, dx</math>

By exploiting the boundedness of <math>\Omega</math> and further estimates:
: <math>\frac{1}{|\Omega|} \int_\Omega \int_\Omega |f(y) - f(x)|^p \, dy \, dx</math>
: <math>\leq \frac{\text{diam}(\Omega)^{n+sp}}{|\Omega|} \int_\Omega \int_\Omega \frac{|f(y) - f(x)|^p}{|y - x|^{n+sp}} \, dy \, dx</math>

It follows that the constant <math>C</math> is given as <math>C = \frac{\text{diam}(\Omega)^{\frac{n}{p}+s}}{|\Omega|^{\frac{1}{p}}}</math>, however, the reference<ref>{{Cite journal
| last1 = Bourgain | first1 = Jean | authorlink1=Jean Bourgain
| last2 = Brezis | first2 = Haïm | authorlink2=Haïm Brezis
| last3 = Mironescu | first3 = Petru
| title = Limiting embedding theorems for <math>W^{s,p}</math> when <math>s\uparrow 1</math> and applications
| journal = [[Journal d'Analyse Mathématique]]
| volume = 87
| pages = 77–101
| year = 2002
| doi = 10.1007/BF02868470
| language = en
}}</ref> with Theorem 1 indicates that this is not the optimal constant.

=== Poincaré on Balls ===

We can derive a growth constant for Balls in a manner similar to previous cases. The relationship is given by the following inequality:

<math>
\|u - u_\Omega\|_{L^p(B_R(y))} \leq C R^s [u]_{\dot{W}^{s,p}(B_R(x))}
</math>

==== Sketch of the Proof ====

The proof proceeds similarly to the classical one, by using the scaling <math>u_R(x) = u(Rx)</math>. Then, by using a form of chain rule for the fractional derivative, we get <math>R^s</math> as a result.

==See also==
*[[Friedrichs' inequality]]
*[[Korn's inequality]]
*[[Spectral gap]]

==References==
{{Reflist}}
{{Refbegin}}
* {{citation
| last = Evans|first=Lawrence C. 
| title = Partial differential equations 
| location = Providence, RI 
| publisher = American Mathematical Society 
| year = 1998
| isbn = 0-8218-0772-2
}}
* Leoni, Giovanni (2009), ''[http://bookstore.ams.org/gsm-105 A First Course in Sobolev Spaces]'', Graduate Studies in Mathematics, American Mathematical Society, pp. xvi+607 {{isbn|978-0-8218-4768-8}}, {{MR|2527916}}, {{Zbl|1180.46001}}, [http://www.maa.org/press/maa-reviews/a-first-course-in-sobolev-spaces MAA]
{{Refend}}

{{DEFAULTSORT:Poincare inequality}}
[[Category:Theorems in analysis]]
[[Category:Inequalities]]
[[Category:Sobolev spaces]]
[[Category:Henri Poincaré|Inequality]]