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 is named after the mathematicians Ivo Babuška, 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

\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)=\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)|\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,

{\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 L^p space V = L²(Ω); the bilinear form B associated to −Δ is the L²(Ω) inner product of the derivatives:

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 u_f such that

\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 1971, Babuška 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,