View source for 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, 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

<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>

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" />).

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" />.

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

<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>

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>

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" />.

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:

i) <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/b/b110/b110020/b11002035.png" /> 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" />.

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" />).

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.

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</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>