Let ${\displaystyle V}$  be a Banach space, let ${\displaystyle V'}$  be the dual space of ${\displaystyle V}$ , let ${\displaystyle A\colon V\to V'}$ ,^{[clarification needed]} and let ${\displaystyle f\in V'}$ . A vector ${\displaystyle u\in V}$  is a solution of the equation

if and only if for all ${\displaystyle v\in V}$ ,

Here, ${\displaystyle v}$  is called a test vector (in general) or a test function (if ${\displaystyle V}$  is a function space).

To bring this into the generic form of a weak formulation, find ${\displaystyle u\in V}$  such that

by defining the bilinear form

Now, let ${\displaystyle V=\mathbb {R} ^{n}}$  and ${\displaystyle A:V\to V}$  be a linear mapping. Then, the weak formulation of the equation

involves finding ${\displaystyle u\in V}$  such that for all ${\displaystyle v\in V}$  the following equation holds:

where ${\displaystyle \langle \cdot ,\cdot \rangle }$  denotes an inner product.

Since ${\displaystyle A}$  is a linear mapping, it is sufficient to test with basis vectors, and we get

Actually, expanding ${\displaystyle u=\sum _{j=1}^{n}u_{j}e_{j}}$ , we obtain the matrix form of the equation

where ${\displaystyle a_{ij}=\langle Ae_{j},e_{i}\rangle }$  and ${\displaystyle f_{i}=\langle f,e_{i}\rangle }$ .

The bilinear form associated to this weak formulation is

To solve Poisson's equation

on a domain ${\displaystyle \Omega \subset \mathbb {R} ^{d}}$  with ${\displaystyle u=0}$  on its boundary, and to specify the solution space ${\displaystyle V}$  later, one can use the ${\displaystyle L^{2}}$ -scalar product

to derive the weak formulation. Then, testing with differentiable functions ${\displaystyle v}$  yields

The left side of this equation can be made more symmetric by integration by parts using Green's identity and assuming that ${\displaystyle v=0}$  on ${\displaystyle \partial \Omega }$ :

This is what is usually called the weak formulation of Poisson's equation. Functions in the solution space ${\displaystyle V}$  must be zero on the boundary, and have square-integrable derivatives. The appropriate space to satisfy these requirements is the Sobolev space ${\displaystyle H_{0}^{1}(\Omega )}$  of functions with weak derivatives in ${\displaystyle L^{2}(\Omega )}$  and with zero boundary conditions, so ${\displaystyle V=H_{0}^{1}(\Omega )}$ .

The generic form is obtained by assigning

and

This is a formulation of the Lax–Milgram theorem which relies on properties of the symmetric part of the bilinear form. It is not the most general form.

Let ${\displaystyle V}$  be a Hilbert space and ${\displaystyle a(\cdot ,\cdot )}$  a bilinear form on ${\displaystyle V}$ , which is

1. bounded: ${\displaystyle |a(u,v)|\leq C\|u\|\|v\|\,;}$  and
2. coercive: ${\displaystyle a(u,u)\geq c\|u\|^{2}\,.}$ 

Then, for any ${\displaystyle f\in V'}$ , there is a unique solution ${\displaystyle u\in V}$  to the equation

and it holds

Application to example 1 edit

Here, application of the Lax–Milgram theorem is a stronger result than is needed.

• Boundedness: all bilinear forms on ${\displaystyle \mathbb {R} ^{n}}$  are bounded. In particular, we have
  $${\displaystyle |a(u,v)|\leq \|A\|\,\|u\|\,\|v\|}$$

   
• Coercivity: this actually means that the real parts of the eigenvalues of ${\displaystyle A}$  are not smaller than ${\displaystyle c}$ . Since this implies in particular that no eigenvalue is zero, the system is solvable.

Additionally, this yields the estimate

Application to example 2 edit

Here, choose ${\displaystyle V=H_{0}^{1}(\Omega )}$  with the norm

where the norm on the right is the ${\displaystyle L^{2}}$ -norm on ${\displaystyle \Omega }$  (this provides a true norm on ${\displaystyle V}$  by the Poincaré inequality). But, we see that ${\displaystyle |a(u,u)|=\|\nabla u\|^{2}}$  and by the Cauchy–Schwarz inequality, ${\displaystyle |a(u,v)|\leq \|\nabla u\|\,\|\nabla v\|}$ .

Therefore, for any ${\displaystyle f\in [H_{0}^{1}(\Omega )]'}$ , there is a unique solution ${\displaystyle u\in V}$  of Poisson's equation and we have the estimate

• Babuška–Lax–Milgram theorem
• Lions–Lax–Milgram theorem

• Lax, Peter D.; Milgram, Arthur N. (1954), "Parabolic equations", Contributions to the theory of partial differential equations, Annals of Mathematics Studies, vol. 33, Princeton, N. J.: Princeton University Press, pp. 167–190, doi:10.1515/9781400882182-010, ISBN 9781400882182, MR 0067317, Zbl 0058.08703

• MathWorld page on Lax–Milgram theorem