Babuška–Lax–Milgram theorem
# 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 ⊂ $R^n,$

    ${\begin{cases}-\Delta u(x)=f(x),&x\in \Omega ;\\u(x)=0,&x\in \partial \Omega ;\end{cases}}$

the space U could be taken to be the Sobolev space $H_0^1(Ω)$ with dual $H^{−1}(Ω);$ the former is a subspace of the $L^p$ space V = $L^2(Ω);$ the bilinear form B associated to −Δ is the $L^2(Ω)$ 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 ∈ $L^2(Ω),$ 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,

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

and, for all 0 ≠ v ∈ V,

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

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

    $B(u_{f},v)=\langle f,v\rangle {\mbox{ for all }}v\in V.$

Moreover, the solution depends continuously on the given data:

    $\|u_{f}\|\leq {\frac {1}{c}}\|f\|.$
