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.