Weak formulations are important tools for the analysis of mathematical equations that permit the transfer of concepts of linear algebra to solve problems in other fields such as partial differential equations. In a weak formulation, equations or conditions are no longer required to hold absolutely (and this is not even well defined) and has instead weak solutions only with respect to certain "test vectors" or "test functions". In a strong formulation, the solution space is constructed such that these equations or conditions are already fulfilled. The Lax–Milgram theorem, named after Peter Lax and Arthur Milgram who proved it in 1954, provides weak formulations for certain systems on Hilbert spaces.

General concept

Let V be a Banach space, let V' be the dual space of V, let, and let f \in V'. A vector u \in V is a solution of the equation Au = f if and only if for all v \in V, A particular choice of v is called a test vector (in general) or a test function (if V is a function space). To bring this into the generic form of a weak formulation, find u\in V such that by defining the bilinear form

Example 1: linear system of equations

Now, let and A:V \to V be a linear mapping. Then, the weak formulation of the equation Au = f involves finding u\in V such that for all v \in V the following equation holds: where denotes an inner product. Since A is a linear mapping, it is sufficient to test with basis vectors, and we get Actually, expanding, we obtain the matrix form of the equation where and. The bilinear form associated to this weak formulation is

Example 2: Poisson's equation

To solve Poisson's equation on a domain with u=0 on its boundary, and to specify the solution space V later, one can use the L^2-scalar product to derive the weak formulation. Then, testing with differentiable functions v yields The left side of this equation can be made more symmetric by integration by parts using Green's identity and assuming that v=0 on : This is what is usually called the weak formulation of Poisson's equation. Functions in the solution space V must be zero on the boundary, and have square-integrable derivatives. The appropriate space to satisfy these requirements is the Sobolev space of functions with weak derivatives in L^2(\Omega) and with zero boundary conditions, so. The generic form is obtained by assigning and

The Lax–Milgram theorem

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 V be a Hilbert space and a bilinear form on V, which is Then, for any bounded f\in V', there is a unique solution u\in V to the equation and it holds

Application to example 1

Here, application of the Lax–Milgram theorem is a stronger result than is needed. Additionally, this yields the estimate where c is the minimal real part of an eigenvalue of A.

Application to example 2

Here, choose with the norm where the norm on the right is the L^2-norm on \Omega (this provides a true norm on V by the Poincaré inequality). But, we see that and by the Cauchy–Schwarz inequality,. Therefore, for any, there is a unique solution u\in V of Poisson's equation and we have the estimate