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翻訳と辞書　辞書検索 [ 開発暫定版 ] スポンサード リンク Holmgren's uniqueness theorem ： ウィキペディア英語版 Holmgren's uniqueness theorem In the theory of partial differential equations, Holmgren's uniqueness theorem, or simply Holmgren's theorem, named after the Swedish mathematician Erik Albert Holmgren (1873–1943), is a uniqueness result for linear partial differential equations with real analytic coefficients.〔Eric Holmgren, "Über Systeme von linearen partiellen Differentialgleichungen", Öfversigt af Kongl. Vetenskaps-Academien Förhandlinger, 58 (1901), 91–103.〕 ==Simple form of Holmgren's theorem== We will use the multi-index notation: Let $\backslash alpha=\backslash \backslash in\; \backslash N\_0^n,$, with $\backslash N\_0$ standing for the nonnegative integers; denote $|\backslash alpha|=\backslash alpha\_1+\backslash cdots+\backslash alpha\_n$ and : $\backslash partial\_x^\backslash alpha\; =\; \backslash left(\backslash frac\backslash right)^\; \backslash cdots\; \backslash left(\backslash frac\backslash right)^\backslash ,$. Holmgren's theorem in its simpler form could be stated as follows: :Assume that ''P'' = ∑|''α''| ≤''m'' ''A''''α''(x)∂ is an elliptic partial differential operator with real-analytic coefficients. If ''Pu'' is real-analytic in a connected open neighborhood ''Ω'' ⊂ R''n'', then ''u'' is also real-analytic. This statement, with "analytic" replaced by "smooth", is Hermann Weyl's classical lemma on elliptic regularity: :If ''P'' is an elliptic differential operator and ''Pu'' is smooth in ''Ω'', then ''u'' is also smooth in ''Ω''. This statement can be proved using Sobolev spaces. 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「Holmgren's uniqueness theorem」の詳細全文を読む スポンサード リンク 翻訳と辞書 : 翻訳のためのインターネットリソース Copyright(C) kotoba.ne.jp 1997-2016. All Rights Reserved.