Poincaré inequality
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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, the Poincaré inequality is a result in the theory of
Sobolev space In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense t ...
s, named after the French
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Henri Poincaré Jules Henri Poincaré ( S: stress final syllable ; 29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science. He is often described as a polymath, and in mathematics as "The ...
. 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 Friedrichs' inequality.


Statement of the inequality


The classical Poincaré inequality

Let ''p'', so that 1 ≤ ''p'' < ∞ 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 In mathematics, a Sobolev space is a vector space of functions equipped with a norm that is a combination of ''Lp''-norms of the function together with its derivatives up to a given order. The derivatives are understood in a suitable weak sense t ...
''W''01,''p''(Ω) of zero-trace (a.k.a. zero on the boundary) functions, :\, u \, _ \leq C \, \nabla u \, _.


Poincaré–Wirtinger inequality

Assume that 1 ≤ ''p'' ≤ ∞ and that Ω is a bounded
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of the ''n''-
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al
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's Elements, Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics ther ...
''n'' with a Lipschitz boundary (i.e., Ω is a Lipschitz
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined **Domain of definition of a partial function **Natural domain of a partial function **Domain of holomorphy of a function * Do ...
). Then there exists a constant ''C'', depending only on Ω and ''p'', such that for every function ''u'' in the Sobolev space , \, u - u_ \, _ \leq C \, \nabla u \, _, where u_ = \frac \int_ u(y) \, \mathrm y is the average value of ''u'' over Ω, with , Ω, standing for the
Lebesgue measure In measure theory, a branch of mathematics, the Lebesgue measure, named after French mathematician Henri Lebesgue, is the standard way of assigning a measure to subsets of ''n''-dimensional Euclidean space. For ''n'' = 1, 2, or 3, it coincides wit ...
of the domain Ω. When Ω is a ball, the above inequality is called a -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 spaces, the definition of a Poincaré inequality is slightly different. One definition is: a metric measure space supports a (q,p)-Poincare inequality for some 1\le q,p<\infty if there are constants ''C'' and so that for each ball B in the space, \mu(B)^ \left \, u-u_B \right \, _\le C \operatorname(B) \mu(B)^ \, \nabla u\, _. Here we have an enlarged ball in the right hand side. In the context of metric measure spaces, \, \nabla u\, is the minimal p-weak upper gradient of u in the sense of Heinonen and Koskela. 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 In mathematics, a metric space with metric is said to be doubling if there is some doubling constant such that for any and , it is possible to cover the ball with the union of at most balls of radius . The base-2 logarithm of is called the d ...
satisfying a Poincaré inequality admits a notion of differentiation. Such spaces include sub-Riemannian manifolds and Laakso spaces. There exist other generalizations of the Poincaré inequality to other Sobolev spaces. For example, consider the Sobolev space ''H''1/2(T2), i.e. the space of functions ''u'' in the ''L''2 space of the unit
torus In geometry, a torus (plural tori, colloquially donut or doughnut) is a surface of revolution generated by revolving a circle in three-dimensional space about an axis that is coplanar with the circle. If the axis of revolution does not tou ...
T2 with
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''û'' satisfying ^2 = \sum_ , k , \left , \hat (k) \right , ^2 < + \infty. In this context, the Poincaré inequality says: there exists a constant ''C'' such that, for every with ''u'' identically zero on an open set , \int_ , u(x) , ^2 \, \mathrm x \leq C \left( 1 + \frac1 \right) u ^2, where denotes the harmonic capacity of when thought of as a subset of . 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,
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, Lipschitz domain with
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''d'', then the Poincaré constant is at most ''d''/2 for , d/\pi for , 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 In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In n-dimensional space \R^n the inequality lower bounds the surface area or perimeter \operatorname(S) of a set S\subset\R^n ...
to the function's
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. In one dimension, this is
Wirtinger's inequality for functions : ''For other inequalities named after Wirtinger, see Wirtinger's inequality.'' In the mathematical field of analysis, the Wirtinger inequality is an important inequality for functions of a single variable, named after Wilhelm Wirtinger. It was us ...
. 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 d=\sqrt). Furthermore, for a smooth, bounded domain , since the
Rayleigh quotient In mathematics, the Rayleigh quotient () for a given complex Hermitian matrix ''M'' and nonzero vector ''x'' is defined as: R(M,x) = . For real matrices and vectors, the condition of being Hermitian reduces to that of being symmetric, and the con ...
for the
Laplace operator In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the ...
in the space W^_0(\Omega) is minimized by the eigenfunction corresponding to the minimal eigenvalue of the (negative) Laplacian, it is a simple consequence that, for any u\in W^_0(\Omega), \, u\, _^2\leq \lambda_1^ \left \, \nabla u\right \, _^2 and furthermore, that the constant λ1 is optimal.


See also

* Friedrichs' inequality *
Korn's inequality In mathematical analysis, Korn's inequality is an inequality concerning the gradient of a vector field that generalizes the following classical theorem: if the gradient of a vector field is skew-symmetric matrix, skew-symmetric at every point, then ...
*
Spectral gap In mathematics, the spectral gap is the difference between the moduli of the two largest eigenvalues of a matrix or operator; alternately, it is sometimes taken as the smallest non-zero eigenvalue. Various theorems relate this difference to othe ...


References

* * Leoni, Giovanni (2009),
A First Course in Sobolev Spaces
', Graduate Studies in Mathematics, American Mathematical Society, pp. xvi+607 , ,
MAA
{{DEFAULTSORT:Poincare inequality Theorems in analysis Inequalities Sobolev spaces
Inequality Inequality may refer to: Economics * Attention inequality, unequal distribution of attention across users, groups of people, issues in etc. in attention economy * Economic inequality, difference in economic well-being between population groups * ...