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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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b1100201.png" /> such that
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<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b1100202.png" /></td> <td valign="top" style="width:5%;text-align:right;">(a1)</td></tr></table>
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{{TEX|auto}}
 +
{{TEX|done}}
  
where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b1100203.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b1100204.png" /> are real normed linear spaces (cf. [[Norm|Norm]]; [[Linear space|Linear space]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b1100205.png" /> denotes a [[Functional|functional]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b1100206.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b1100207.png" /> is an element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b1100208.png" /> (the dual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b1100209.png" />).
+
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
  
The essential question here is what conditions can be imposed on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002010.png" /> and on the normed spaces <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002011.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002012.png" /> so that a unique solution to (a1) exists and depends continuously on the data <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002013.png" />.
+
$$ \tag{a1 }
 +
b ( u,v ) = l ( v ) ,  \forall v \in V,
 +
$$
  
If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002014.png" /> is a [[Hilbert space|Hilbert space]], P.D. Lax and A.N. Milgram [[#References|[a1]]] have proved that for a bilinear continuous functional <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002015.png" /> strong coerciveness (i.e., there is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002016.png" /> such that for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002017.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002018.png" />) 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002020.png" /> be two real Hilbert spaces and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002021.png" /> be a continuous [[Bilinear functional|bilinear functional]]. If it is also a weakly coercive (i.e., there exists a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002022.png" /> such that
+
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 $).
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002023.png" /></td> </tr></table>
+
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
 
and
  
<table class="eq" style="width:100%;"> <tr><td valign="top" style="width:94%;text-align:center;"><img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002024.png" /></td> </tr></table>
+
$$
 +
\sup  _ {u \in U } \left | {b ( u,v ) } \right | > 0,  \forall v \in V \setminus  \{ 0 \} \textrm{ )  } ,
 +
$$
  
then for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002025.png" /> there exists a unique solution <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002026.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002027.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002028.png" /> and, moreover, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002029.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002030.png" /> are given in [[#References|[a3]]], namely: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002031.png" /> be a [[Banach space|Banach space]], let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002032.png" /> be a reflexive Banach space (cf. [[Reflexive space|Reflexive space]]) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002033.png" /> be a real functional on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002034.png" />. The following statements are equivalent:
+
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) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002035.png" /> is a bilinear continuous weakly coercive functional;
+
i) b ( \cdot, \cdot ) $
 +
is a bilinear continuous weakly coercive functional;
  
ii) there exists a linear, continuous and surjective operator <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002036.png" /> such that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002037.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002038.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002039.png" />.
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002040.png" /> be the bilinear functional generated by a square non-singular matrix <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002041.png" /> (i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002042.png" />). Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002043.png" /> is weakly coercive, because for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002044.png" /> there exists a unique solution, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002045.png" />, for (a1); however, it is strongly coercive if and only if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002046.png" /> is either strictly positive (i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002047.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002048.png" />) or strictly negative (i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002049.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002050.png" />).
+
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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002051.png" /> is symmetric (i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002052.png" />) and strictly defined (i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002053.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002054.png" />), then it is either a strictly positive functional (i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002055.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002056.png" />) or a strictly negative functional (i.e., <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002057.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002058.png" />); moreover <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002059.png" /> for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002060.png" />. The following result can also be found in [[#References|[a3]]]: If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002061.png" /> 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 <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002062.png" /> is a symmetric and strictly defined functional, then it is strongly coercive if and only if it is weakly coercive.
+
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]]].
 
Effective applications of the Babuška–Lax–Millgram theorem can be found in [[#References|[a4]]].

Revision as of 10:26, 27 April 2020


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
How to Cite This Entry:
Babuska-Lax-Milgram theorem. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Babuska-Lax-Milgram_theorem&oldid=45580
This article was adapted from an original article by I. RoÅŸca (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article