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.
We introduce weak formulations by a few examples and present the main theorem for the solution, the Lax–Milgram theorem. The theorem is named after Peter Lax and Arthur Milgram, who proved it in 1954.
Let be a Banach space. We want to find the solution of the equation
where and , with being the dual of .
This is equivalent to finding such that
for all holds:
Here, we call a test vector or test function.
We bring this into the generic form of a weak formulation, namely, find such that
by defining the bilinear form
Since this is very abstract, let us follow this by some examples.
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
Our aim is to solve Poisson's equation
on a domain with on its boundary,
and we want to specify the solution space later. We will use the -scalar product
to derive our weak formulation. Then, testing with differentiable functions , we get
We can make the left side of this equation 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. We have yet to specify a space in which to find a solution, but at a minimum it must allow us to write down this equation. Therefore, we require that the functions in are 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 we set
We obtain the generic form 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 definitely a stronger result than is needed, but we still can use it and give this problem the same structure as the others have.
- 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, we get the estimate
where is the minimal real part of an eigenvalue of .
Application to example 2
Here, as we mentioned above, we 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, 33, Princeton, N. J.: Princeton University Press, pp. 167–190, doi:10.1515/9781400882182-010, MR 0067317, Zbl 0058.08703