"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 an arbitrary 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 Hilbert space, let 
be a continuous and coercive bilinear
form on 
,
and let 
be a closed and convex subset of 
. One result of Stampacchia says that, under these assumptions,
any function 
necessarily corresponds to a unique function 
for which the inequality
 |
(1)
|
is satisfied for all functions 
where here, 
denotes the inner
product on 
.
Note that this form of the result is particularly nice in the sense that, by examining
an arbitrary element 
and by selecting an element 
for which the above inequality holds for all 
, the convexity of 
implies that
 |
(2)
|
for all 
,
and because of the bilinearity of both 
and of the inner product, the above inequality restated for
necessarily yields
 |
(3)
|
Combining these latter two inequalities and considering the case for 
, one then arrives at the Lax-Milgram
theorem which states that, under the assumptions stated above, any element 
necessarily corresponds to a unique
element 
satisfying
 |
(4)
|
for all 
.
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 
for a bounded domain 
and if 
is a real-valued Lipschitz function satisfying 
, then the composition 
also lies in 
provided that 
and that the function 
satisfies the generalized derivative identity
 |
(5)
|
almost everywhere in 
. In the above, 
is defined to be the collection of functions
in 
with zero trace, i.e.,
 |
(6)
|
where 
is the collection of all smooth functions on 
with compact
support.
