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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 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''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: : Hence, the weak formulation of the Poisson equation, given ''f'' ∈ ''L''2(Ω), is to find ''u''''f'' such that : 抄文引用元・出典: フリー百科事典『 ウィキペディア（Wikipedia）』 ■ウィキペディアで「Babuška–Lax–Milgram theorem」の詳細全文を読む スポンサード リンク
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