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

[math]\displaystyle{ \Lambda u = f. }[/math]

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

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

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.

[math]\displaystyle{ | B(u, u) | \geq c \| u \|^{2} }[/math]

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,

[math]\displaystyle{ \begin{cases} - \Delta u(x) = f(x), & x \in \Omega; \\ u(x) = 0, & x \in \partial \Omega; \end{cases} }[/math]

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:

[math]\displaystyle{ B(u, v) = \int_{\Omega} \nabla u(x) \cdot \nabla v(x) \, \mathrm{d} x. }[/math]

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

[math]\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). }[/math]

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,

[math]\displaystyle{ \sup_{\| v \| = 1} | B(u, v) | \geq c \| u \| }[/math]

and, for all 0 ≠ v ∈ V,

[math]\displaystyle{ \sup_{\| u \| = 1} | B(u, v) | \gt 0 }[/math]

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

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

Moreover, the solution depends continuously on the given data:

[math]\displaystyle{ \| u_{f} \| \leq \frac{1}{c} \| f \|. }[/math]

See also

• Lions–Lax–Milgram theorem

References

• Babuška, Ivo (1970–1971). "Error-bounds for finite element method". Numerische Mathematik 16 (4): 322–333. doi:10.1007/BF02165003. ISSN 0029-599X. https://eudml.org/doc/132037. 
• 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, https://www.degruyter.com/view/books/9781400882182/9781400882182-010/9781400882182-010.xml 

External links

• Hazewinkel, Michiel, ed. (2001), "Babuška–Lax–Milgram theorem", Encyclopedia of Mathematics, Springer Science+Business Media B.V. / Kluwer Academic Publishers, ISBN 978-1-55608-010-4, https://www.encyclopediaofmath.org/index.php?title=B/b110020