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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 Lewy's example ： ウィキペディア英語版 Lewy's example In the mathematical study of partial differential equations, Lewy's example is a celebrated example, due to Hans Lewy, of a linear partial differential equation with no solutions. It shows that the analog of the Cauchy–Kovalevskaya theorem does not hold in the smooth category. The original example is not explicit, since it employs the Hahn–Banach theorem, but there since have been various explicit examples of the same nature found by Harold Jacobowitz. The Malgrange–Ehrenpreis theorem states (roughly) that linear partial differential equations with constant coefficients always have at least one solution; Lewy's example shows that this result cannot be extended to linear partial differential equations with polynomial coefficients. ==The Example== The statement is as follows :On ℝ×ℂ, there exists a smooth complex-valued function $F(t,z)$ such that the differential equation ::$\backslash frac\; =\; F(t,z)$ :admits no solution on any open set. Note that if ''$F$'' is analytic then the Cauchy–Kovalevskaya theorem implies there exists a solution. Lewy constructs this ''$F$'' using the following result: :On ℝ×ℂ, suppose that $u(t,z)$ is a function satisfying, in a neighborhood of the origin, ::$\backslash frac\; =\; \backslash varphi^\backslash prime(t)$ :for some ''C''1 function ''φ''. Then ''φ'' must be real-analytic in a (possibly smaller) neighborhood of the origin. This may be construed as a non-existence theorem by taking ''φ'' to be merely a smooth function. Lewy's example takes this latter equation and in a sense ''translates'' its non-solvability to every point of ℝ×ℂ. The method of proof uses a Baire category argument, so in a certain precise sense almost all equations of this form are unsolvable. later found that the even simpler equation :$\backslash frac+ix\backslash frac\; =\; F(x,y)$ depending on 2 real variables ''x'' and ''y'' sometimes has no solutions. This is almost the simplest possible partial differential operator with non-constant coefficients. 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「Lewy's example」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.