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 $u \in V$ 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|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).

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

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

(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 [[#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 $n \in \N$ 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 [[#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 {{ZBL|0058.08703}}</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>

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