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 W₀^1,p(Ω) of zero-trace (a.k.a. zero on the boundary) functions,

${\displaystyle \|u\|_{L^{p}(\Omega )}\leq C\|\nabla u\|_{L^{p}(\Omega )}.}$

Assume that 1 ≤ p ≤ ∞ and that Ω is a bounded connected open subset of the n-dimensional Euclidean space ℝⁿ with a Lipschitz boundary (i.e., Ω is a 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),

$${\displaystyle \|u-u_{\Omega }\|_{L^{p}(\Omega )}\leq C\|\nabla u\|_{L^{p}(\Omega )},}$$

where

$${\displaystyle u_{\Omega }={\frac {1}{|\Omega |}}\int _{\Omega }u(y)\,\mathrm {d} y}$$

is the average value of u over Ω, with |Ω| standing for the Lebesgue measure of the domain Ω. When Ω is a ball, the above inequality is called a (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.

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 ${\displaystyle 1\leq q,p<\infty }$ if there are constants C and λ ≥ 1 so that for each ball B in the space,

$${\displaystyle \mu (B)^{-{\frac {1}{q}}}\left\|u-u_{B}\right\|_{L^{q}(B)}\leq C\operatorname {rad} (B)\mu (B)^{-{\frac {1}{p}}}\|\nabla u\|_{L^{p}(\lambda B)}.}$$

Here we have an enlarged ball in the right hand side. In the context of metric measure spaces, ${\displaystyle \|\nabla u\|}$ is the minimal p-weak upper gradient of u in the sense of Heinonen and Koskela.[2]

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.[3] 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 L² space of the unit torus T² with Fourier transform û satisfying

$${\displaystyle [u]_{H^{1/2}(\mathbf {T} ^{2})}^{2}=\sum _{k\in \mathbf {Z} ^{2}}|k|\left|{\hat {u}}(k)\right|^{2}<+\infty .}$$

In this context, the Poincaré inequality says: there exists a constant C such that, for every u ∈ H^1/2(T²) with u identically zero on an open set E ⊆ T²,

$${\displaystyle \int _{\mathbf {T} ^{2}}|u(x)|^{2}\,\mathrm {d} x\leq C\left(1+{\frac {1}{\operatorname {cap} (E\times \{0\})}}\right)[u]_{H^{1/2}(\mathbf {T} ^{2})}^{2},}$$

where cap(E × {0}) denotes the harmonic capacity of E × {0} when thought of as a subset of ℝ³.[4]

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