Namespaces
Variants
Actions

Babuska-Lax-Milgram theorem

From Encyclopedia of Mathematics
Jump to: navigation, search
The printable version is no longer supported and may have rendering errors. Please update your browser bookmarks and please use the default browser print function instead.

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
How to Cite This Entry:
Babuska-Lax-Milgram theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Babuska-Lax-Milgram_theorem&oldid=22039
This article was adapted from an original article by I. RoÅŸca (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article