Babuska-Lax-Milgram theorem

Many boundary value problems for ordinary and partial differential equations can be posed in the following abstract variational form (cf. also Boundary value problem, ordinary differential equations; Boundary value problem, partial differential equations): Find such that

(a1)

where and are real normed linear spaces (cf. Norm; Linear space), denotes a functional on and is an element in (the dual of ).

The essential question here is what conditions can be imposed on and on the normed spaces and so that a unique solution to (a1) exists and depends continuously on the data .

If is a Hilbert space, P.D. Lax and A.N. Milgram [a1] have proved that for a bilinear continuous functional strong coerciveness (i.e., there is a such that for all , ) is a sufficient condition for the existence and uniqueness of the solution to (a1) (the Lax–Milgram lemma). In 1971, I. Babuška [a2] gave the following significant generalization of this lemma: Let and be two real Hilbert spaces and let be a continuous bilinear functional. If it is also a weakly coercive (i.e., there exists a such that

and

then for all there exists a unique solution such that for all and, moreover, .

Sufficient and necessary conditions for a linear variational problem (a1) to have a unique solution depending continuously on the data are given in [a3], namely: Let be a Banach space, let be a reflexive Banach space (cf. Reflexive space) and let be a real functional on . The following statements are equivalent:

i) is a bilinear continuous weakly coercive functional;

ii) there exists a linear, continuous and surjective operator such that for all and .

This result can be used to give simple examples of bilinear weakly coercive functionals that are not strongly coercive. Indeed, let be the bilinear functional generated by a square non-singular matrix (i.e., ). Then is weakly coercive, because for all there exists a unique solution, , for (a1); however, it is strongly coercive if and only if is either strictly positive (i.e., for all ) or strictly negative (i.e., for all ).

Using this fact one can prove that if is symmetric (i.e., ) and strictly defined (i.e., for all ), then it is either a strictly positive functional (i.e., for all ) or a strictly negative functional (i.e., for all ); moreover for all . The following result can also be found in [a3]: If is a symmetric and continuous functional then it is strongly coercive if and only if it is weakly coercive and strictly defined. This implies that if is a symmetric and strictly defined functional, then it is strongly coercive if and only if it is weakly coercive.

Effective applications of the Babuška–Lax–Millgram theorem can be found in [a4].

References

[a1]	P.D. Lax, A.N. Milgram, "Parabolic equations" Ann. Math. Studies , 33 (1954) pp. 167–190
[a2]	I. Babuška, "Error bound for the finite element method" Numer. Math. , 16 (1971) pp. 322–333
[a3]	I. Roşca, "On the Babuška Lax Milgram theorem" An. Univ. Bucureşti , XXXVIII : 3 (1989) pp. 61–65
[a4]	I. Babuška, A.K. Aziz, "Survey lectures on the mathematical foundations of finite element method" A.K. Aziz (ed.) , The Mathematical Foundations of the FEM with Application to PDE , Acad. Press (1972) pp. 5–359