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.
Let be a Banach space, its dual space, , and . Finding the solution of the equation
is equivalent to finding such that, for all ,
Here, is called a test vector or test function.
To bring this into the generic form of a weak formulation, find such that
by defining the bilinear form
Example 1: linear system of equations
Now, let and be a linear mapping. Then, the weak formulation of the equation
involves finding such that for all the following equation holds:
where denotes an inner product.
Since 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 on its boundary, and to specify the solution space later, one can use the -scalar product
to derive the weak formulation. Then, testing with differentiable functions yields
The left side of this equation can be made more symmetric by integration by parts using Green's identity and assuming that on :
This is what is usually called the weak formulation of Poisson's equation. Functions in the solution space 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 and with zero boundary conditions, so .
The generic form is obtained by assigning
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 be a Hilbert space and a bilinear form on , which is
- bounded: and
Then, for any , there is a unique solution to the equation
and it holds
Application to example 1
Here, application of the Lax–Milgram theorem is a stronger result than is needed.
- Boundedness: all bilinear forms on are bounded. In particular, we have
- Coercivity: this actually means that the real parts of the eigenvalues of are not smaller than . Since this implies in particular that no eigenvalue is zero, the system is solvable.
Additionally, this yields the estimate
is the minimal real part of an eigenvalue of .
Application to example 2
Here, choose with the norm
where the norm on the right is the -norm on (this provides a true norm on by the Poincaré inequality).
But, we see that and by the Cauchy–Schwarz inequality, .
Therefore, for any , there is a unique solution of Poisson's equation and we have the estimate
- 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, MR 0067317, Zbl 0058.08703