View source for Lax-Milgram lemma

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/l/l110/l110080/l1100801.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/l/l110/l110080/l1100802.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/l/l110/l110080/l1100803.png" /> is a normed [[Linear space|linear space]] (cf. also [[Norm|Norm]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l1100804.png" /> denotes a [[Functional|functional]] on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l1100805.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l1100806.png" /> is an element in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l1100807.png" /> (the dual of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l1100808.png" />).

The essential question here is what conditions can be imposed on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l1100809.png" /> and the normed space <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l11008010.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/l/l110/l110080/l11008011.png" />.

The first result in this direction was obtained in 1954 by P.D. Lax and A.N. Milgram [[#References|[a1]]], who established sufficient conditions for the existence and uniqueness of the solution for (a1).

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l11008012.png" /> be a reflexive [[Banach space|Banach space]] (cf. also [[Reflexive space|Reflexive space]]) and let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l11008013.png" /> be a sesquilinear mapping (bilinear when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l11008014.png" /> is real; cf. also [[Sesquilinear form|Sesquilinear form]]) 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/l/l110/l110080/l11008015.png" /></td> </tr></table>

(continuity) 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/l/l110/l110080/l11008016.png" /></td> </tr></table>

(strong coercivity), where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l11008017.png" />. Then there exists a unique bijective linear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l11008018.png" />, continuous in both directions and uniquely determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l11008019.png" />, with

<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/l/l110/l110080/l11008020.png" /></td> </tr></table>

<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/l/l110/l110080/l11008021.png" /></td> </tr></table>

and for the norms one has:

<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/l/l110/l110080/l11008022.png" /></td> </tr></table>

<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/l/l110/l110080/l11008023.png" /></td> </tr></table>

This implies that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l11008024.png" /> is the solution of (a1). The above theorem only establishes existence of a solution to (a1), namely <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l11008025.png" />, but does not say anything about the construction of this solution. In 1965, W.V. Petryshyn [[#References|[a2]]] proved the following result: Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l11008026.png" /> be a separable reflexive Banach space (cf. also [[Separable space|Separable space]]), <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l11008027.png" /> a [[Basis|basis]] of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l11008028.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l11008029.png" /> a continuous sesquilinear strongly coercive mapping on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l11008030.png" />. Then for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l11008031.png" />:

i) for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l11008032.png" /> the system

<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/l/l110/l110080/l11008033.png" /></td> </tr></table>

is uniquely solvable for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l11008034.png" />;

ii) the sequence <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l11008035.png" /> determined by <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l11008036.png" /> converges in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l11008037.png" /> to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l11008038.png" /> that is the solution of (a1).

To see that the strong coerciveness property of the sesquilinear mapping <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l11008039.png" /> is not necessary for the existence of the solution to (a1), consider the following very simple example.

Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l11008040.png" /> be defined by

<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/l/l110/l110080/l11008041.png" /></td> </tr></table>

where <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l11008042.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l11008043.png" />. It is easy to see that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l11008044.png" /> is bilinear and continuous. It is not strongly coercive, because <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l11008045.png" /> when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l11008046.png" />. However, for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l11008047.png" />,

<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/l/l110/l110080/l11008048.png" /></td> </tr></table>

is the unique solution to (a1).

In 1971, I. Babuška [[#References|[a3]]] gave a significant generalization of the Lax–Milgram theorem using weak coerciveness (cf. [[Babuška–Lax–Milgram theorem|Babuška–Lax–Milgram theorem]]).

An extensive literature exists on applications of the Lax–Milgram lemma to various classes of boundary-value problems (see, e.g., [[#References|[a4]]], [[#References|[a5]]]).

====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">  W.V. Petryshyn,   "Constructional proof of Lax–Milgram lemma and its applications to non-k-p.d. abstract and differential operator equation"  ''SIAM Numer. Anal. Ser. B'' , '''2''' :  3  (1965)  pp. 404–420</TD></TR><TR><TD valign="top">[a3]</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">[a4]</TD> <TD valign="top">  J.T. Oden,   J.N. Reddy,   "An introduction to the mathematical theory of finite elements" , Wiley  (1976)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  J. Nečas,   "Les méthodes directes dans la théorie des équations elliptiques" , Masson  (1967)</TD></TR></table>