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

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Let   be a Banach space, let   be the dual space of  , let  ,[clarification needed] and let  . A vector   is a solution of the equation

 

if and only if for all  ,

 

A particular choice of   is called a test vector (in general) or a test function (if   is a function space).

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

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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

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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

 

and

 

The Lax–Milgram theorem

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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

  1. bounded:   and
  2. coercive:  

Then, for any bounded  , there is a unique solution   to the equation

 

and it holds

 

Application to example 1

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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   where   is the minimal real part of an eigenvalue of  .

Application to example 2

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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

 

See also

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References

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  • 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
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