# 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

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

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.

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},

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:

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

## 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*,

and, for all 0 ≠ *v* ∈ *V*,

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

Moreover, the solution depends continuously on the given datum:

## See also

## References

- Babuška, Ivo (1970–1971). "Error-bounds for finite element method".
*Numerische Mathematik*.**16**: 322–333. doi:10.1007/BF02165003. ISSN 0029-599X. MR 0288971. Zbl 0214.42001. - Lax, Peter D.; Milgram, Arthur N. (1954), "Parabolic equations",
*Contributions to the theory of partial differential equations*, Annals of Mathematics Studies,**33**, Princeton, N. J.: Princeton University Press, pp. 167–190, MR 0067317, Zbl 0058.08703 – via De Gruyter, (subscription required (help))

## External links

- Roşca, Ioan (2001), "Babuška–Lax–Milgram theorem", in Hazewinkel, Michiel,
*Encyclopedia of Mathematics*, Springer, ISBN 978-1-55608-010-4