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.
is called a test vector (in general) or a test function (if
To bring this into the generic form of a weak formulation, find
by defining the bilinear form
be a linear mapping.
Then, the weak formulation of the equation
is a linear mapping, it is sufficient to test with basis vectors, and we get
, we obtain the matrix form of the equation
The bilinear form associated to this weak formulation is
To solve Poisson's equation
on its boundary, and to specify the solution space
to derive the weak formulation.
Then, testing with differentiable functions
The left side of this equation can be made more symmetric by integration by parts using Green's identity and assuming that
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
The generic form is obtained by assigning
This is a formulation of the Lax–Milgram theorem which relies on properties of the symmetric part of the bilinear form.
be a real Hilbert space and
Here, application of the Lax–Milgram theorem is a stronger result than is needed.
Additionally, this yields the estimate
is the minimal real part of an eigenvalue of
(this provides a true norm on
of Poisson's equation and we have the estimate