# Weak formulation

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.

Contents 1 General concept 2 Example 1: linear system of equations 3 Example 2: Poisson's equation 4 The Lax–Milgram theorem 4.1 Application to example 1 4.2 Application to example 2 5 See also 6 References 7 External links General concept Let {displaystyle V} be a Banach space. To find the solution {displaystyle uin V} of the equation {displaystyle Au=f,} where {displaystyle Acolon Vto V'} and {displaystyle fin V'} , with {displaystyle V'} being the dual space of {displaystyle V} , is equivalent to finding {displaystyle uin V} such that for all {displaystyle vin V} , {displaystyle [Au](v)=f(v).} Here, {displaystyle v} is called a test vector or test function.

To bring this into the generic form of a weak formulation, find {displaystyle uin V} such that {displaystyle a(u,v)=f(v)quad forall vin V,} by defining the bilinear form {displaystyle a(u,v):=[Au](v).} Example 1: linear system of equations Now, let {displaystyle V=mathbb {R} ^{n}} and {displaystyle A:Vto V} be a linear mapping. Then, the weak formulation of the equation {displaystyle Au=f} involves finding {displaystyle uin V} such that for all {displaystyle vin V} the following equation holds: {displaystyle langle Au,vrangle =langle f,vrangle ,} where {displaystyle langle cdot ,cdot rangle } denotes an inner product.

Since {displaystyle A} is a linear mapping, it is sufficient to test with basis vectors, and we get {displaystyle langle Au,e_{i}rangle =langle f,e_{i}rangle ,quad i=1,ldots ,n.} Actually, expanding {displaystyle u=sum _{j=1}^{n}u_{j}e_{j}} , we obtain the matrix form of the equation {displaystyle mathbf {A} mathbf {u} =mathbf {f} ,} where {displaystyle a_{ij}=langle Ae_{j},e_{i}rangle } and {displaystyle f_{i}=langle f,e_{i}rangle } .

The bilinear form associated to this weak formulation is {displaystyle a(u,v)=mathbf {v} ^{T}mathbf {A} mathbf {u} .} Example 2: Poisson's equation To solve Poisson's equation {displaystyle -nabla ^{2}u=f,} on a domain {displaystyle Omega subset mathbb {R} ^{d}} with {displaystyle u=0} on its boundary, and to specify the solution space {displaystyle V} later, one can use the {displaystyle L^{2}} -scalar product {displaystyle langle u,vrangle =int _{Omega }uv,dx} to derive the weak formulation. Then, testing with differentiable functions {displaystyle v} yields {displaystyle -int _{Omega }(nabla ^{2}u)v,dx=int _{Omega }fv,dx.} The left side of this equation can be made more symmetric by integration by parts using Green's identity and assuming that {displaystyle v=0} on {displaystyle partial Omega } : {displaystyle int _{Omega }nabla ucdot nabla v,dx=int _{Omega }fv,dx.} This is what is usually called the weak formulation of Poisson's equation. Functions in the solution space {displaystyle V} must be zero on the boundary, and have square-integrable derivatives. The appropriate space to satisfy these requirements is the Sobolev space {displaystyle H_{0}^{1}(Omega )} of functions with weak derivatives in {displaystyle L^{2}(Omega )} and with zero boundary conditions, so {displaystyle V=H_{0}^{1}(Omega )} .

The generic form is obtained by assigning {displaystyle a(u,v)=int _{Omega }nabla ucdot nabla v,dx} and {displaystyle f(v)=int _{Omega }fv,dx.} 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 {displaystyle V} be a Hilbert space and {displaystyle a(cdot ,cdot )} a bilinear form on {displaystyle V} , which is bounded: {displaystyle |a(u,v)|leq C|u||v|,;} and coercive: {displaystyle a(u,u)geq c|u|^{2},.} Then, for any {displaystyle fin V'} , there is a unique solution {displaystyle uin V} to the equation {displaystyle a(u,v)=f(v)quad forall vin V} and it holds {displaystyle |u|leq {frac {1}{c}}|f|_{V'},.} Application to example 1 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)|leq |A|,|u|,|v|} Coercivity: this actually means that the real parts of the eigenvalues 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 |u|leq {frac {1}{c}}|f|,} where {displaystyle c} is the minimal real part of an eigenvalue of {displaystyle A} .

Application to example 2 Here, choose {displaystyle V=H_{0}^{1}(Omega )} with the norm {displaystyle |v|_{V}:=|nabla v|,} where the norm on the right is the {displaystyle L^{2}} -norm on {displaystyle Omega } (this provides a true norm on {displaystyle V} by the Poincaré inequality). But, we see that {displaystyle |a(u,u)|=|nabla u|^{2}} and by the Cauchy–Schwarz inequality, {displaystyle |a(u,v)|leq |nabla u|,|nabla v|} .

Therefore, for any {displaystyle fin [H_{0}^{1}(Omega )]'} , there is a unique solution {displaystyle uin V} of Poisson's equation and we have the estimate {displaystyle |nabla u|leq |f|_{[H_{0}^{1}(Omega )]'}.} See also Babuška–Lax–Milgram theorem Lions–Lax–Milgram theorem References 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, MR 0067317, Zbl 0058.08703 External links MathWorld page on Lax–Milgram theorem Categories: Partial differential equationsNumerical differential equations

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