"Stampacchia's theorem" is a name given to any number of related results in functional analysis, and while the body
of the theorem often varies depending on the literature consulted, one commonly-encountered
result attributed to Stampacchia is a sort of "representation inequality"
for continuous, coercive bilinear forms on a real Hilbert
space 
.
This particular version of the result is considerable for a number of reasons, most
notably for its implication of the so-called Lax-Milgram
theorem.
To state the above-mentioned version of the theorem, let 
be a real Hilbert space,
let 
be a continuous and coercive bilinear form on 
, and let 
be a nonempty closed
and convex subset of 
. One result of Stampacchia says that,
under these assumptions, any element 
necessarily corresponds to a unique element 
for which the inequality
 |
(1)
|
is satisfied for all elements 
where here, 
denotes the inner
product on 
.
When 
,
take 
and 
for each 
.
The bilinearity of 
and of the inner product
gives opposite inequalities, hence
 |
(2)
|
This is the Lax-Milgram theorem. For a proper convex set 
, the variational inequality
generally cannot be extended to all of 
.
As noted above, however, the content of the results attributed to Stampacchia may be different. Yet another common form of the theorem states that if a function 
lies in the Sobolev
space 
with 
for a bounded domain 
and if 
is a real-valued Lipschitz function satisfying 
, then the function
composition 
also lies in 
provided that 
and that the function 
satisfies the generalized derivative identity
 |
(3)
|
almost everywhere in 
. In the above, 
is defined by closure
of the smooth functions with compact
support, namely
 |
(4)
|
where 
is the set of all smooth functions
on 
with compact support.
