Weak formulations are tools for the analysis of mathematicalequations that permit the transfer of concepts oflinear algebra to solve problems in other fields such aspartial 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 insteadweak solutions only with respect to certain "test vectors" or "test functions". In astrong formulation, the solution space is constructed such that these equations or conditions are already fulfilled.

TheLax–Milgram theorem, named afterPeter Lax andArthur Milgram who proved it in 1954, provides weak formulations for certain systems onHilbert spaces.

Let${\displaystyle V}$ be aBanach space, let${\displaystyle V^{\prime }}$ be thedual space of${\displaystyle V}$, let${\displaystyle A:V\to V^{\prime }}$ be alinear map, and let${\displaystyle f\in V^{\prime }}$. A vector${\displaystyle u\in V}$ is a solution of the equation

$${\displaystyle Au=f}$$

if and only if for all${\displaystyle v\in V}$,

$${\displaystyle (Au)(v)=f(v).}$$

A particular choice of${\displaystyle v}$ is called atest vector (in general) or atest function (if${\displaystyle V}$ is a function space).

To bring this into the generic form of a weak formulation, find${\displaystyle u\in V}$ such that

$${\displaystyle a(u,v)=f(v)\mathrm{\forall }v\in V,}$$

by defining thebilinear form

$${\displaystyle a(u,v):=(Au)(v).}$$

Now, let${\displaystyle V={\mathbb{R}}^n}$ and${\displaystyle A:V\to V}$ be alinear mapping. Then, the weak formulation of the equation

$${\displaystyle Au=f}$$

involves finding${\displaystyle u\in V}$ such that for all${\displaystyle v\in V}$ the following equation holds:

$${\displaystyle ⟨Au,v⟩=⟨f,v⟩,}$$

where${\displaystyle ⟨\cdot ,\cdot ⟩}$ denotes aninner product.

Since${\displaystyle A}$ is a linear mapping, it is sufficient to test withbasis vectors, and we get

$${\displaystyle ⟨Au,e_i⟩=⟨f,e_i⟩,i=1,\dots ,n.}$$

Actually, expanding${\displaystyle u=\sum _{j=1}^n{u}_j{e}_j}$, we obtain thematrix form of the equation

$${\displaystyle \mathbf{A}\mathbf{u}=\mathbf{f},}$$

where${\displaystyle a_{ij}=⟨A{e}_j,e_i⟩}$ and${\displaystyle f_i=⟨f,e_i⟩}$.

The bilinear form associated to this weak formulation is

$${\displaystyle a(u,v)={\mathbf{v}}^T\mathbf{A}\mathbf{u}.}$$

To solvePoisson's equation

$${\displaystyle -{\mathrm{\nabla }}^{2}u=f,}$$

on a domain${\displaystyle \mathrm{\Omega }\subset {\mathbb{R}}^d}$ with${\displaystyle u=0}$ on itsboundary, and to specify the solution space${\displaystyle V}$ later, one can use the${\displaystyle L^2}$-scalar product

$${\displaystyle ⟨u,v⟩={\int }_{\mathrm{\Omega }}uvdx}$$

to derive the weak formulation. Then, testing withdifferentiable functions${\displaystyle v}$ yields

$${\displaystyle -{\int }_{\mathrm{\Omega }}({\mathrm{\nabla }}^{2}u)vdx={\int }_{\mathrm{\Omega }}fvdx.}$$

The left side of this equation can be made more symmetric byintegration by parts usingGreen's identity and assuming that${\displaystyle v=0}$ on${\displaystyle \mathrm{\partial }\mathrm{\Omega }}$:

$${\displaystyle {\int }_{\mathrm{\Omega }}\mathrm{\nabla }u\cdot \mathrm{\nabla }vdx={\int }_{\mathrm{\Omega }}fvdx.}$$

This is what is usually called the weak formulation ofPoisson's equation.Functions in the solution space${\displaystyle V}$ must be zero on the boundary, and have square-integrablederivatives. The appropriate space to satisfy these requirements is theSobolev space${\displaystyle H_0^1(\mathrm{\Omega })}$ of functions withweak derivatives in${\displaystyle L^2(\mathrm{\Omega })}$ and with zero boundary conditions, so${\displaystyle V=H_0^1(\mathrm{\Omega })}$.

The generic form is obtained by assigning

$${\displaystyle a(u,v)={\int }_{\mathrm{\Omega }}\mathrm{\nabla }u\cdot \mathrm{\nabla }vdx}$$

and

$${\displaystyle f(v)={\int }_{\mathrm{\Omega }}fvdx.}$$

This is a formulation of theLax–Milgram theorem which relies on properties of the symmetric part of thebilinear form. It is not the most general form.

Let${\displaystyle V}$ be a realHilbert space and${\displaystyle a(\cdot ,\cdot )}$ abilinear form on${\displaystyle V}$, which is

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

Then, for any bounded${\displaystyle f\in V^{\prime }}$, there is a unique solution${\displaystyle u\in V}$ to the equation

$${\displaystyle a(u,v)=f(v)\mathrm{\forall }v\in V}$$

and it holds

$${\displaystyle \Vert u\Vert \le \frac{1}{c}\Vert f{\Vert }_{V^{\prime }}.}$$

Here, application of the Lax–Milgram theorem is a stronger result than is needed.

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

Additionally, this yields the estimate$${\displaystyle \Vert u\Vert \le \frac{1}{c}\Vert f\Vert ,}$$where${\displaystyle c}$ is the minimal real part of an eigenvalue of${\displaystyle A}$.

Here, choose${\displaystyle V=H_0^1(\mathrm{\Omega })}$ with the norm$${\displaystyle \Vert v{\Vert }_V:=\Vert \mathrm{\nabla }v\Vert ,}$$

where the norm on the right is the${\displaystyle L^2}$-norm on${\displaystyle \mathrm{\Omega }}$ (this provides a true norm on${\displaystyle V}$ by thePoincaré inequality).But, we see that${\displaystyle |a(u,u)|=\Vert \mathrm{\nabla }u{\Vert }^2}$ and by theCauchy–Schwarz inequality,${\displaystyle |a(u,v)|\le \Vert \mathrm{\nabla }u\Vert \Vert \mathrm{\nabla }v\Vert }$.

Therefore, for any${\displaystyle f\in [H_0^1(\mathrm{\Omega }){]}^{\prime }}$, there is a unique solution${\displaystyle u\in V}$ ofPoisson's equation and we have the estimate

$${\displaystyle \Vert \mathrm{\nabla }u\Vert \le \Vert f{\Vert }_{[H_0^1(\mathrm{\Omega }){]}^{\prime }}.}$$

• Babuška–Lax–Milgram theorem
• Lions–Lax–Milgram theorem

• 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{{citation}}:ISBN / Date incompatibility (help)

• MathWorld page on Lax–Milgram theorem

• Partial differential equations
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• Theorems in functional analysis

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