Poincaré inequality

In mathematics, the Poincaré inequality is a result in the theory of Sobolev spaces, named after the 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 methods of the calculus of variations. A very closely related result is the Friedrichs' inequality.

Let p, so that 1 ≤ p < ∞ and Ω a subset with at least one bound. Then there exists a constant C, depending only on Ω and p, so that, for every function u of the W₀^1,p(Ω) Sobolev space,

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),

where

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.

In the context of metric measure spaces (for example, sub-Riemannian manifolds), such spaces support a (q,p)-Poincare inequality for some ${\displaystyle 1\leq q,p<\infty }$ if there are constants C and ${\displaystyle \lambda \geq 1}$ so that for each ball B in the space,

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