Weak formulations are an important tool 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, an equation is 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".

We introduce weak formulations by a few examples and present the main theorem for the solution, the Lax–Milgram theorem.

General concept

Let $V$ be a Banach space. We want to find the solution $u\; \backslash in\; V$ of the equation

where $A:V\backslash to\; V\text{'}$ and $f\backslash in\; V\text{'}$, the dual of $V$.
Calculus of variations tells us that this is equivalent to finding $u\backslash in\; V$ such that
for all $v\backslash in\; V$ holds:

Here, we call $v$ a test vector or test function.

We bring this into the generic form of a weak formulation, namely, find $u\backslash in\; V$ 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 $V\; =\; \backslash mathbb\; R^n$ and $A:V\backslash to\; V$ a linear mapping. Then, the weak formulation of the equation

involves finding $u\backslash in\; V$ such that for all $v\backslash in\; V$ the following equation holds:

where $(\backslash cdot,\backslash cdot)$ denotes an inner product.

Since $A$ is a linear mapping, it is sufficient to test with basis vectors, we get

Actually, expanding $u=\backslash sum\_\{j=1\}^n\; u\_je\_j$, we obtain the matrix form of the equation

where $a\_\{ij\}\; =\; (Ae\_j,\; e\_i)$ and $f\_i\; =\; (f,e\_i)$.

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 $\backslash Omega\backslash subset\; \backslash mathbb\; R^d$ with $u=0$ on its boundary,
and we want to specify the solution space $V$ later. We will use the $L^2$-scalar product

to derive our weak formulation. Then, testing with differentiable functions $v$, we get

We can make the left side of this equation more symmetric by integration by parts using Green's identity:

This is what is usually called the weak formulation of Poisson's equation; what's missing is the space $V$. Well, this a bit tricky and way beyond the scope of this article. The space must allow us to write down this equation. Therefore, we should require that the derivatives of functions in this space are square integrable. Now, there is actually the Sobolev space $H^1\_0(\backslash Omega)$ of functions with weak derivatives in $L^2(\backslash Omega)$ and with zero boundary conditions, which fulfills this purpose.

We obtain the generic form 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 $a(\backslash cdot,\backslash cdot)$ a bilinear form on $V$, which is

1. bounded: $|a(u,v)|\; \backslash le\; C\; \backslash |u\backslash |\; \backslash |v\backslash |$ and
   
2. coercive: $a(u,u)\; \backslash ge\; c\; \backslash |u\backslash |^2.$

Then, for any $f\backslash in\; V\text{'}$, there is a unique solution $u\backslash in\; V$ to the equation

and it holds

Application to example 1

Here, application of the Lax–Milgram theorem is definitely overkill, but we still can use it and give this problem the same structure as the others have.

• Boundedness: all bilinear forms on $\backslash mathbb\; R^n$ are bounded. In particular, we have
  
  $|a(u,v)|\; \backslash le\; \backslash |A\backslash |\backslash ,\backslash |u\backslash |\backslash ,\backslash |v\backslash |$
• Coercivity: this actually means that the real parts of the eigenvalues of $A$ are not smaller than $c$. Since this implies in particular that no eigenvalue is zero, the system is solvable.
  

Additionally, we get the estimate

where $c$ is the minimal real part of an eigenvalue of $A$.

Application to Example 2

Here, as we mentioned above, we choose $V\; =\; H^1\_0(\backslash Omega)$ with the norm

where the norm on the right is the $L^2$-norm on $\backslash Omega$ (this provides a true norm on $V$ by the Poincaré inequality).
But, we see that $a(u,u)\; =\; \backslash |\backslash nabla\; u\backslash |^2$ and by Cauchy-Schwarz inequality $a(u,v)\; \backslash le\; \backslash |\backslash nabla\; u\backslash |\backslash ,\backslash |\backslash nabla\; v\backslash |$.

Therefore, for any $f\backslash in\; [H^1\_0(\backslash Omega)]\text{'}$, there is a unique solution $u\backslash in\; V$ of Poisson's equation and we have the estimate

See also

• Babuška-Lax-Milgram theorem