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 $ u \in U $ such that

$$ \tag{a1 } b ( u,v ) = l ( v ) , \forall v \in V, $$

where $ U $ and $ V $ are real normed linear spaces (cf. Norm; Linear space), $ b $ denotes a functional on $ U \times V $ and $ l $ is an element in $ V ^ \prime $( the dual of $ V $).

The essential question here is what conditions can be imposed on $ b ( \cdot, \cdot ) $ and on the normed spaces $ U $ and $ V $ so that a unique solution to (a1) exists and depends continuously on the data $ l $.

If $ U \equiv V $ is a Hilbert space, P.D. Lax and A.N. Milgram [a1] have proved that for a bilinear continuous functional $ b ( \cdot, \cdot ) $ strong coerciveness (i.e., there is a $ \gamma $ such that for all $ u \in U $, $ | {b ( u,u ) } | \geq \gamma \| u \| ^ {2} $) 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 $ U $ and $ V $ be two real Hilbert spaces and let $ b : {U \times V } \rightarrow \mathbf R $ be a continuous bilinear functional. If it is also a weakly coercive (i.e., there exists a $ c > 0 $ such that

$$ \sup _ {\left \| v \right \| \leq 1 } \left | {b ( u,v ) } \right | \geq \left \| u \right \| , \forall u \in U, $$

and

$$ \sup _ {u \in U } \left | {b ( u,v ) } \right | > 0, \forall v \in V \setminus \{ 0 \} \textrm{ ) } , $$

then for all $ f \in V $ there exists a unique solution $ u _ {f} \in U $ such that $ b ( u _ {f} ,v ) = ( f,v ) $ for all $ v \in V $ and, moreover, $ \| {u _ {f} } \| \leq { {\| f \| } / c } $.

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

i) $ b ( \cdot, \cdot ) $ is a bilinear continuous weakly coercive functional;

ii) there exists a linear, continuous and surjective operator $ S : {V ^ \prime } \rightarrow U $ such that $ b ( Sl,v ) = \langle {l,v } \rangle $ for all $ l \in V ^ \prime $ and $ v \in V $.

This result can be used to give simple examples of bilinear weakly coercive functionals that are not strongly coercive. Indeed, let $ b : {\mathbf R ^ {n} \times \mathbf R ^ {n} } \rightarrow \mathbf R $ be the bilinear functional generated by a square non-singular matrix $ B \in {\mathcal M} _ {n} ( \mathbf R ) $( i.e., $ b ( u,v ) = ( Bu,v ) $). Then $ b ( \cdot, \cdot ) $ is weakly coercive, because for all $ l \in \mathbf R ^ {n} $ there exists a unique solution, $ u = B ^ {- 1 } l $, for (a1); however, it is strongly coercive if and only if $ B $ is either strictly positive (i.e., $ ( Bu,u ) > 0 $ for all $ u \neq 0 $) or strictly negative (i.e., $ ( Bu,u ) < 0 $ for all $ u \neq 0 $).

Using this fact one can prove that if $ b : {U \times U } \rightarrow \mathbf R $ is symmetric (i.e., $ b ( u,v ) = b ( v,u ) $) and strictly defined (i.e., $ b ( u,u ) \neq 0 $ for all $ u \neq 0 $), then it is either a strictly positive functional (i.e., $ b ( u,u ) > 0 $ for all $ u \neq 0 $) or a strictly negative functional (i.e., $ b ( u,u ) < 0 $ for all $ u \neq 0 $); moreover $ | {b ( u,v ) } | ^ {2} \leq | {b ( u,u ) } | \cdot | {b ( v,v ) } | $ for all $ u,v \in U $. The following result can also be found in [a3]: If $ b : {U \times U } \rightarrow \mathbf R $ 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 $ b ( \cdot, \cdot ) $ 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