View source for Babuska-Lax-Milgram theorem

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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, ordinary differential equations]]; [[Boundary value problem, partial 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|Norm]]; [[Linear space|Linear space]]),  $  b $
denotes a [[Functional|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|Hilbert space]], P.D. Lax and A.N. Milgram [[#References|[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|Lax–Milgram lemma]]). In 1971, I. Babuška [[#References|[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|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 [[#References|[a3]]], namely: Let  $  U $
be a [[Banach space|Banach space]], let  $  V $
be a reflexive Banach space (cf. [[Reflexive space|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 [[#References|[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 [[#References|[a4]]].

====References====
<table>
<TR><TD valign="top">[a1]</TD> <TD valign="top">  P.D. Lax,   A.N. Milgram,   "Parabolic equations"  ''Ann. Math. Studies'' , '''33'''  (1954)  pp. 167–190 {{ZBL|0058.08703}}</TD></TR>
<TR><TD valign="top">[a2]</TD> <TD valign="top">  I. Babuška,   "Error bound for the finite element method"  ''Numer. Math.'' , '''16'''  (1971)  pp. 322–333</TD></TR>
<TR><TD valign="top">[a3]</TD> <TD valign="top">  I. Roşca,   "On the Babuška Lax Milgram theorem"  ''An. Univ. Bucureşti'' , '''XXXVIII''' :  3  (1989)  pp. 61–65</TD></TR>
<TR><TD valign="top">[a4]</TD> <TD valign="top">  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</TD></TR>
</table>