Namespaces
Variants
Actions

Difference between revisions of "Lax-Milgram lemma"

From Encyclopedia of Mathematics
Jump to: navigation, search
m (→‎References: + ZBL link)
m (tex done)
 
Line 1: Line 1:
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>
+
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 $u \in V$ such that
  
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" />).
+
\begin{equation}
 +
\tag{a1}
 +
b(u, v) = l(v), \quad \forall v \in V,
 +
\end{equation}
  
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" />.
+
where $V$ is a normed [[Linear space|linear space]] (cf. also [[Norm|Norm]]), $b$ denotes a [[Functional|functional]] on $V\times V$ and $l$ is an element in $V'$ (the dual of $V$).
 +
 
 +
The essential question here is what conditions can be imposed on $b(.,.)$ and the normed space $V$ so that a unique solution to (a1) exists and depends continuously on the data $l$.
  
 
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).
 
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 $b:V\times V\longrightarrow\mathbb{C}$ 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
+
Let $V$ be a reflexive [[Banach space|Banach space]] (cf. also [[Reflexive space|Reflexive space]]) and let $b:V\times V\longrightarrow\mathbb{C}$ be a sesquilinear mapping (bilinear when $b$ 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>
+
$$
 +
|b(u,v)| \le M \|u\|. \|v\|, \quad u,v \in V
 +
$$
  
 
(continuity) and
 
(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>
+
$$
 +
|b(u,u)| \ge \gamma \|u\|^2, \quad u \in V
 +
$$
  
(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
+
(strong coercivity), where $M,\gamma > 0$. Then there exists a unique bijective linear mapping $B : V \to V'$, continuous in both directions and uniquely determined by $b$, 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>
+
\begin{gather*}
 
+
b(u,v) = \overline{\langle Bu, v\rangle}, \qquad \forall u,v \in 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/l/l110/l110080/l11008021.png" /></td> </tr></table>
+
b(B^{-1} l,v) = \overline{\langle l,v \rangle}, \qquad \forall v \in V, l\in V',
 +
\end{gather*}
  
 
and for the norms one has:
 
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>
+
\begin{gather*}
 
+
\|B\|_{\mathcal{L}(V,V')} \le M, \\
<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>
+
\|B^{-1}\|_{\mathcal{L}(V',V)} \le \frac{1}{\gamma}.
 +
\end{gather*}
  
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" />:
+
This implies that $u = B^{-1}l$ is the solution of (a1). The above theorem only establishes existence of a solution to (a1), namely $u = B^{-1}l$, but does not say anything about the construction of this solution. In 1965, W.V. Petryshyn [[#References|[a2]]] proved the following result: Let $V$ be a separable reflexive Banach space (cf. also [[Separable space|Separable space]]), $(e_i)_{i \in \N}$ a [[Basis|basis]] of $V$ and $b$ a continuous sesquilinear strongly coercive mapping on $V\times V$. Then for all $l \in V'$:
  
i) for all <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l11008032.png" /> the system
+
i) for all $n \in \N$ 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>
+
$$
 +
\sum_{j=1}^n b(e_j, e_i) t_j^{(n)} = \overline{\langle l, e_j \rangle} \qquad 1 \le i \le n,
 +
$$
  
is uniquely solvable for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/l/l110/l110080/l11008034.png" />;
+
is uniquely solvable for $[t_1^{(n)}, \ldots, t_n^{(n)}]$;
  
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).
+
ii) the sequence $\{u_n\}_{n\in\N}$ determined by $u_n = \sum_{j=1}^n t_j^{(n)}$ converges in $V$ to a $u$ 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.
+
To see that the strong coerciveness property of the sesquilinear mapping $b$ 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
+
Let $b : \R^2 \times \R^2 \to \R$ 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>
+
$$
 +
b(u,v) = u_1v_1 + u_1v_2 + u_2v_1 - u_2v_2,
 +
$$
  
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" />,
+
where $u = (u_1, u_2)$, $v = (v_1, v_2)$. It is easy to see that $b$ is bilinear and continuous. It is not strongly coercive, because $b(u,u) = 0$ when $u = (1,1-\sqrt 2)$. However, for all $l = (l_1, l_2) \in \R^2$,
  
<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>
+
$$
 +
u = \frac12 (l_1+l_2, l_1-l_2)
 +
$$
  
 
is the unique solution to (a1).
 
is the unique solution to (a1).
Line 60: Line 76:
 
</table>
 
</table>
  
{{TEX|want}}
+
{{TEX|done}}

Latest revision as of 03:17, 15 February 2024

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 $u \in V$ such that

\begin{equation} \tag{a1} b(u, v) = l(v), \quad \forall v \in V, \end{equation}

where $V$ is a normed linear space (cf. also Norm), $b$ denotes a functional on $V\times V$ and $l$ is an element in $V'$ (the dual of $V$).

The essential question here is what conditions can be imposed on $b(.,.)$ and the normed space $V$ so that a unique solution to (a1) exists and depends continuously on the data $l$.

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

Let $V$ be a reflexive Banach space (cf. also Reflexive space) and let $b:V\times V\longrightarrow\mathbb{C}$ be a sesquilinear mapping (bilinear when $b$ is real; cf. also Sesquilinear form) such that

$$ |b(u,v)| \le M \|u\|. \|v\|, \quad u,v \in V $$

(continuity) and

$$ |b(u,u)| \ge \gamma \|u\|^2, \quad u \in V $$

(strong coercivity), where $M,\gamma > 0$. Then there exists a unique bijective linear mapping $B : V \to V'$, continuous in both directions and uniquely determined by $b$, with

\begin{gather*} b(u,v) = \overline{\langle Bu, v\rangle}, \qquad \forall u,v \in V, \\ b(B^{-1} l,v) = \overline{\langle l,v \rangle}, \qquad \forall v \in V, l\in V', \end{gather*}

and for the norms one has:

\begin{gather*} \|B\|_{\mathcal{L}(V,V')} \le M, \\ \|B^{-1}\|_{\mathcal{L}(V',V)} \le \frac{1}{\gamma}. \end{gather*}

This implies that $u = B^{-1}l$ is the solution of (a1). The above theorem only establishes existence of a solution to (a1), namely $u = B^{-1}l$, but does not say anything about the construction of this solution. In 1965, W.V. Petryshyn [a2] proved the following result: Let $V$ be a separable reflexive Banach space (cf. also Separable space), $(e_i)_{i \in \N}$ a basis of $V$ and $b$ a continuous sesquilinear strongly coercive mapping on $V\times V$. Then for all $l \in V'$:

i) for all $n \in \N$ the system

$$ \sum_{j=1}^n b(e_j, e_i) t_j^{(n)} = \overline{\langle l, e_j \rangle} \qquad 1 \le i \le n, $$

is uniquely solvable for $[t_1^{(n)}, \ldots, t_n^{(n)}]$;

ii) the sequence $\{u_n\}_{n\in\N}$ determined by $u_n = \sum_{j=1}^n t_j^{(n)}$ converges in $V$ to a $u$ that is the solution of (a1).

To see that the strong coerciveness property of the sesquilinear mapping $b$ is not necessary for the existence of the solution to (a1), consider the following very simple example.

Let $b : \R^2 \times \R^2 \to \R$ be defined by

$$ b(u,v) = u_1v_1 + u_1v_2 + u_2v_1 - u_2v_2, $$

where $u = (u_1, u_2)$, $v = (v_1, v_2)$. It is easy to see that $b$ is bilinear and continuous. It is not strongly coercive, because $b(u,u) = 0$ when $u = (1,1-\sqrt 2)$. However, for all $l = (l_1, l_2) \in \R^2$,

$$ u = \frac12 (l_1+l_2, l_1-l_2) $$

is the unique solution to (a1).

In 1971, I. Babuška [a3] gave a significant generalization of the Lax–Milgram theorem using weak coerciveness (cf. 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., [a4], [a5]).

References

[a1] P.D. Lax, A.N. Milgram, "Parabolic equations" Ann. Math. Studies , 33 (1954) pp. 167–190 Zbl 0058.08703
[a2] 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
[a3] I. Babuška, "Error bound for the finite element method" Numer. Math. , 16 (1971) pp. 322–333
[a4] J.T. Oden, J.N. Reddy, "An introduction to the mathematical theory of finite elements" , Wiley (1976)
[a5] J. Nečas, "Les méthodes directes dans la théorie des équations elliptiques" , Masson (1967)
How to Cite This Entry:
Lax-Milgram lemma. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Lax-Milgram_lemma&oldid=55499
This article was adapted from an original article by I. RoÅŸca (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article