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.

This topic will cover two versions of the Poincaré inequality, one is for spaces and the other is for spaces.

**The classical Poincaré inequality for **** spaces**. Assume that and that is a bounded open subset of the –dimensional Euclidean space with a Lipschitz boundary (i.e., is an open, bounded Lipschitz domain). Then there exists a constant , depending only on and , such that for every function in the Sobolev space ,

,

where

is the average value of over , with standing for the Lebesgue measure of the domain .

*Proof*. We argue by contradiction. Were the stated estimate false, there would exist for each integer a function satisfying

.

We renormalize by defining

.

Then

and therefore

.

In particular the functions are bounded in .

By mean of the Rellich-Kondrachov Theorem, there exists a subsequence and a function such that

in .

Passing to a limit, one easily gets

.

On the other hand, for each and ,

.

Consequently, with a.e. Thus is constant since is connected. Since then . This contradicts to .

**The Poincaré inequality**** for **** spaces**. Assume that is a bounded open subset of the -dimensional Euclidean space with a Lipschitz boundary (i.e., is an open, bounded Lipschitz domain). Then there exists a constant , depending only on such that for every function in the Sobolev space ,

.

*Proof*. Assume can be enclosed in a cube

.

Then for any , we have

.

Thus

.

Integration over from to gives the result.

**The Poincaré inequality**** for **** spaces**. Assume that and that is a bounded open subset of the -dimensional Euclidean space with a Lipschitz boundary (i.e., is an open, bounded Lipschitz domain). Then there exists a constant , depending only on and , such that for every function in the Sobolev space ,

,

where is defined to be .

*Proof*. The proof of this version is exactly the same to the proof of case.

**Remark**. The point on the boundary of is important. Otherwise, the constant function will not satisfy the Poincaré inequality. In order to avoid this restriction, a weight has been added like the classical Poincaré inequality for case. Sometimes, the Poincaré inequality for spaces is called the Sobolev inequality.