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, 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)