In mathematics, the Poincaré inequality[1] is a result in the theory of Sobolev spaces, named after the French mathematician Henri Poincaré.
Such bounds are of great importance in the modern, direct methods of the calculus of variations.
A very closely related result is Friedrichs' inequality.
zero on the boundary) functions, Assume that 1 ≤ p ≤ ∞ and that Ω is a bounded connected open subset of the n-dimensional Euclidean space
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.
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.
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
is the minimal p-weak upper gradient of u in the sense of Heinonen and Koskela.
For example, Cheeger has shown that a doubling space satisfying a Poincaré inequality admits a notion of differentiation.
There exist other generalizations of the Poincaré inequality to other Sobolev spaces.
In this context, the Poincaré inequality says: there exists a constant C such that, for every u ∈ H1/2(T2) with u identically zero on an open set E ⊆ T2,
where cap(E × {0}) denotes the harmonic capacity of E × {0} when thought of as a subset of
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.
For example, if Ω is a bounded, convex, Lipschitz domain with diameter d, then the Poincaré constant is at most d/2 for p = 1,
for p = 2,[5][6] and this is the best possible estimate on the Poincaré constant in terms of the diameter alone.
However, in some special cases the constant C can be determined concretely.
[8] Furthermore, for a smooth, bounded domain Ω, since the Rayleigh quotient for the Laplace operator in the space
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".
For example, a space that supports a Poincare inequality must be path connected.
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.
is defined by: The Poincaré Inequality in this context can be generalized as follows: where
By applying Jensen's inequality, we obtain: By exploiting the boundedness of
We can derive a growth constant for Balls in a manner similar to previous cases.
The proof proceeds similarly to the classical one, by using the scaling
Then, by using a form of chain rule for the fractional derivative, we get