TABLE OF CONTENTS We now turn to variational problems with constraints. Let X and M be two Hilbert spaces, and suppose
are continuous bilinear forms. Let f ? X ? and g ? M ?. We denote both the dual pairing of X with X ? and that of M with M ? by 
, 
. We consider the following minimum problem.
Problem (M). Find the minimum over X of
subject to the constraint
Our starting point is the same as in the classical theory of Lagrange extremal problems. If ? ? M, then J and the Lagrange function
have the same values on the set of all points which satisfy the constraints. Instead of finding the minimum of J, we can seek a minimum of 
( , ?) with fixed ?. This raises the question of whether ? ? M can be selected so that the minimum of ( , ?) over the space X is assumed by an element which satisfies the given constraints. Since ( u, ?) contains only bilinear and quadratric expressions in u and ?, we are led to the following saddle point problem:
Problem (S). Find (u, ?) ? X M with
It is easy to see that every solution (u, ?) of Problem (S)...
