Babuška–Lax–Milgram theorem

In mathematics, the Babuška–Lax–Milgram theorem is a generalization of the famous Lax–Milgram theorem, which gives conditions under which a bilinear form can be "inverted" to show the existence and uniqueness of a weak solution to a given boundary value problem. The result was proved by J. Nečas in 1962, and is a generalization of the famous Lax-Milgram theorem by Peter Lax and Arthur Milgram.

Background In the modern, functional-analytic approach to the study of partial differential equations, one does not attempt to solve a given partial differential equation directly, but by using the structure of the vector space of possible solutions, e.g. a Sobolev space W k,p. Abstractly, consider two real normed spaces U and V with their continuous dual spaces U∗ and V∗ respectively. In many applications, U is the space of possible solutions; given some partial differential operator Λ : U → V∗ and a specified element f ∈ V∗, the objective is to find a u ∈ U such that

Λ u = f . {\displaystyle \Lambda u=f.}

However, in the weak formulation, this equation is only required to hold when "tested" against all other possible elements of V. This "testing" is accomplished by means of a bilinear function B : U × V → R which encodes the differential operator Λ; a weak solution to the problem is to find a u ∈ U such that

B ( u , v ) = ⟨ f , v ⟩ for all v ∈ V . {\displaystyle B(u,v)=\langle f,v\rangle {\mbox{ for all }}v\in V.}

The achievement of Lax and Milgram in their 1954 result was to specify sufficient conditions for this weak formulation to have a unique solution that depends continuously upon the specified datum f ∈ V∗: it suffices that U = V is a Hilbert space, that B is continuous, and that B is strongly coercive, i.e.

| B ( u , u ) | ≥ c ‖ u ‖ 2 {\displaystyle |B(u,u)|\geq c\|u\|^{2}}

for some constant c > 0 and all u ∈ U. For example, in the solution of the Poisson equation on a bounded, open domain Ω ⊂ Rn,

{ − Δ u ( x ) = f ( x ) , x ∈ Ω ; u ( x ) = 0 , x ∈ ∂ Ω ; {\displaystyle {\begin{cases}-\Delta u(x)=f(x),&x\in \Omega ;\\u(x)=0,&x\in \partial \Omega ;\end{cases}}}

the space U could be taken to be the Sobolev space H01(Ω) with dual H−1(Ω); the former is a subspace of the Lp space V = L2(Ω); the bilinear form B associated to −Δ is the L2(Ω) inner product of the derivatives:

B ( u , v ) = ∫ Ω ∇ u ( x ) ⋅ ∇ v ( x ) d x . {\displaystyle B(u,v)=\int _{\Omega }\nabla u(x)\cdot \nabla v(x)\,\mathrm {d} x.}

Hence, the weak formulation of the Poisson equation, given f ∈ L2(Ω), is to find uf such that

∫ Ω ∇ u f ( x ) ⋅ ∇ v ( x ) d x = ∫ Ω f ( x ) v ( x ) d x for all v ∈ H 0 1 ( Ω ) . {\displaystyle \int _{\Omega }\nabla u_{f}(x)\cdot \nabla v(x)\,\mathrm {d} x=\int _{\Omega }f(x)v(x)\,\mathrm {d} x{\mbox{ for all }}v\in H_{0}^{1}(\Omega ).}

Statement of the theorem In 1962, J. Nečas provided the following generalization of Lax and Milgram's earlier result, which begins by dispensing with the requirement that U and V be the same space. Let U and V be two real Hilbert spaces and let B : U × V → R be a continuous bilinear functional. Suppose also that B is weakly coercive: for some constant c > 0 and all u ∈ U,

sup ‖ v ‖ = 1 | B ( u , v ) | ≥ c ‖ u ‖ {\displaystyle \sup _{\|v\|=1}|B(u,v)|\geq c\|u\|}

and, for all 0 ≠ v ∈ V,

sup ‖ u ‖ = 1 | B ( u , v ) | > 0 {\displaystyle \sup _{\|u\|=1}|B(u,v)|>0}

Then, for all f ∈ V∗, there exists a unique solution u = uf ∈ U to the weak problem

B ( u f , v ) = ⟨ f , v ⟩ for all v ∈ V . {\displaystyle B(u_{f},v)=\langle f,v\rangle {\mbox{ for all }}v\in V.}

Moreover, the solution depends continuously on the given data:

‖ u f ‖ ≤ 1 c ‖ f ‖ . {\displaystyle \|u_{f}\|\leq {\frac {1}{c}}\|f\|.}

Nečas' proof extends directly to the situation where U {\displaystyle U} is a Banach space and V {\displaystyle V} a reflexive Banach space.

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