Babuška–Lax–Milgram theorem

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

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

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

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

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.

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

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,

${\displaystyle {\begin{cases}-\Delta u(x)=f(x),&x\in \Omega$ ;\\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 L^p space V = L²(Ω); the bilinear form B associated to −Δ is the L²(Ω) inner product of the derivatives:

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

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

${\displaystyle \int _{\Omega }\nabla u_{f}(x)\cdot \nabla v(x)\,\mathrm {d} x=\int _{\Omega }f(x)v(x)\,\mathrm {d} x{\mbox{ for all }}v\in H_{0}^{1}(\Omega ).}$

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,

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

and, for all 0 ≠ v ∈ V,

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

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

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

Moreover, the solution depends continuously on the given data:

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

• Lions–Lax–Milgram theorem

• Babuška, Ivo (1970–1971). "Error-bounds for finite element method". Numerische Mathematik. 16 (4): 322–333. doi:10.1007/BF02165003. hdl:10338.dmlcz/103498. ISSN 0029-599X. MR 0288971. S2CID 122191183. Zbl 0214.42001.
• 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, MR 0067317, Zbl 0058.08703 – via De Gruyter

• Roşca, Ioan (2001) [1994], "Babuška–Lax–Milgram theorem", Encyclopedia of Mathematics, EMS Press