# 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].

How to Cite This Entry:
Babuska-Lax-Milgram theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Babuska-Lax-Milgram_theorem&oldid=53945
This article was adapted from an original article by I. RoÅŸca (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article