Poincaré inequality

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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.

[edit] Statement of the inequality

Assume that $1 \leq p \leq \infty$ , $Ω$ is precompact open subset of $\mathbb{R}^n$ with Lipschitz boundary (i.e., $Ω$ is an open, bounded Lipschitz domain). Then there exists a constant $C$ , depending only on $Ω$ and $p$ , such that, for every $u \in W^{1, p} (\Omega)$ ,