Statement of the inequality[edit]
The classical Poincaré inequality[edit]
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 W01,p(Ω) of zero-trace (a.k.a. zero on the boundary) functions,
The Poincaré constant[edit]
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 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. For smooth functions, this can be understood as an application of the isoperimetric inequality to the function's level sets.[7] 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/π ( < d/π where ).[8]
Furthermore, for a smooth, bounded domain Ω, since the Rayleigh quotient for the Laplace operator in the space is minimized by the eigenfunction corresponding to the minimal eigenvalue λ1 of the (negative) Laplacian, it is a simple consequence that, for any ,
and furthermore, that the constant λ1 is optimal.
Poincaré inequality on metric-measure spaces[edit]
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.