Babuška–Lax–Milgram Theorem

In mathematics, the Generalized–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 Necas 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

:$\Lambda u = f.$

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

:$B(u, v) = \langle f, v \rangle \mbox v \in V.$

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.

:$, B(u, u) , \geq c \, u \, ^$

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'',

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

the space ''U'' could be taken to be the Sobolev space ''H''₀¹(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:

:$B(u, v) = \int_ \nabla u(x) \cdot \nabla v(x) \, \mathrm x.$

Hence, the weak formulation of the Poisson equation, given ''f'' ∈ ''L''²(Ω), is to find ''u''_''f'' such that

:$\int_ \nabla u_(x) \cdot \nabla v(x) \, \mathrm x = \int_ f(x) v(x) \, \mathrm x \mbox v \in H_^ (\Omega).$

Statement of the theorem

In 1962 J Necas 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'',

:$\sup_ , B(u, v) , \geq c \, u \,$

and, for all 0 ≠ ''v'' ∈ ''V'',

:$\sup_ , B(u, v) , > 0$

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

:$B(u_, v) = \langle f, v \rangle \mbox v \in V.$

Moreover, the solution depends continuously on the given data:

:$\, u_ \, \leq \frac \, f \, .$
Necas' proof extends directly to the situation where $U$ is a Banach space and $V$ a reflexive Banach space.

See also

* Lions–Lax–Milgram theorem

References

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*Nečas, Jindřich, Sur une méthode pour résoudre les équations aux dérivées partielles du type elliptique, voisine de la variationnelle, Annali della Scuola Normale Superiore di Pisa - Scienze Fisiche e Matematiche, Serie 3, Volume 16 (1962) no. 4, pp. 305-326.

External links

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{{DEFAULTSORT:Babuska-Lax-Milgram theorem
Theorems in mathematical analysis
Partial differential equations