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| 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 [[mathematician]]s [[Ivo Babuška]], [[Peter Lax]] and [[Arthur Milgram]].
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| ==Background==
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| In the modern, [[functional analysis|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''<sup>''k'',''p''</sup>. Abstractly, consider two [[real number|real]] [[normed space]]s ''U'' and ''V'' with their [[continuous dual space]]s ''U''<sup>∗</sup> and ''V''<sup>∗</sup> respectively. In many applications, ''U'' is the space of possible solutions; given some [[partial differential operator]] Λ : ''U'' → ''V''<sup>∗</sup> and a specified element ''f'' ∈ ''V''<sup>∗</sup>, the objective is to find a ''u'' ∈ ''U'' such that
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| :<math>\Lambda u = f.\ </math>
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| 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
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| :<math>B(u, v) = \langle f, v \rangle \mbox{ for all } v \in V.</math>
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| 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 [[continuous function|continuously]] upon the specified datum ''f'' ∈ ''V''<sup>∗</sup>: it suffices that ''U'' = ''V'' is a [[Hilbert space]], that ''B'' is continuous, and that ''B'' is strongly [[coercive function|coercive]], i.e.
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| :<math>| B(u, u) | \geq c \| u \|^{2}</math> | |
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| for some constant ''c'' > 0 and all ''u'' ∈ ''U''.
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| For example, in the solution of the [[Poisson equation]] on a [[bounded set|bounded]], [[open set|open]] domain Ω ⊂ '''R'''<sup>''n''</sup>,
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| :<math>\begin{cases} - \Delta u(x) = f(x), & x \in \Omega; \\ u(x) = 0, & x \in \partial \Omega; \end{cases}</math>
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| the space ''U'' could be taken to be the Sobolev space ''H''<sub>0</sub><sup>1</sup>(Ω) with dual ''H''<sup>−1</sup>(Ω); the former is a subspace of the [[Lp space|''L''<sup>''p''</sup> space]] ''V'' = ''L''<sup>2</sup>(Ω); the bilinear form ''B'' associated to −Δ is the ''L''<sup>2</sup>(Ω) [[inner product]] of the derivatives:
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| :<math>B(u, v) = \int_{\Omega} \nabla u(x) \cdot \nabla v(x) \, \mathrm{d} x.</math>
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| Hence, the weak formulation of the Poisson equation, given ''f'' ∈ ''L''<sup>2</sup>(Ω), is to find ''u''<sub>''f''</sub> such that
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| :<math>\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>
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| ==Statement of the theorem==
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| 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'',
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| :<math>\sup_{\| v \| = 1} | B(u, v) | \geq c \| u \|</math>
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| and, for 0 ≠ ''v'' ∈ ''V'', for some positive constant k,
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| :<math>\sup_{\| u \| = 1} | B(u, v) | \geq k \| v \|</math>
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| Then, for all ''f'' ∈ ''V''<sup>∗</sup>, there exists a unique solution ''u'' = ''u''<sub>''f''</sub> ∈ ''U'' to the weak problem
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| :<math>B(u_{f}, v) = \langle f, v \rangle \mbox{ for all } v \in V.</math> | |
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| Moreover, the solution depends continuously on the given datum:
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| :<math>\| u_{f} \| \leq \frac{1}{c} \| f \|.</math> | |
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| ==See also==
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| * [[Lions–Lax–Milgram theorem]]
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| ==References==
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| * {{cite journal
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| | last = Babuška
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| | first = Ivo
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| | authorlink = Ivo Babuška
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| | title = Error-bounds for finite element method
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| | journal = [[Numerische Mathematik]]
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| | volume = 16
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| | year = 1970/1971
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| | pages = 322–333
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| | issn = 0029-599X
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| | doi = 10.1007/BF02165003
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| | mr = 0288971
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| }}
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| * {{cite book
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| | last = Lax
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| | first = Peter D.
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| | authorlink = Peter Lax
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| | coauthors = Milgram, Arthur N.
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| | chapter = Parabolic equations
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| | title = Contributions to the theory of partial differential equations
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| | series = Annals of Mathematics Studies, no. 33
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| | pages = 167–190
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| | publisher = Princeton University Press
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| | location = Princeton, N. J.
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| | year = 1954
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| | mr = 0067317
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| }}
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| ==External links==
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| * {{springer
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| | title = Babuška–Lax–Milgram theorem
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| | id = B/b110020
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| | last = Roşca
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| | first = Ioan
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| }}
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| {{DEFAULTSORT:Babuska-Lax-Milgram theorem}}
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| [[Category:Theorems in analysis]]
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| [[Category:Partial differential equations]]
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