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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 Schauder estimates ： ウィキペディア英語版 Schauder estimates In mathematics, the Schauder estimates are a collection of results due to concerning the regularity of solutions to linear, uniformly elliptic partial differential equations. The estimates say that when the equation has appropriately smooth terms and appropriately smooth solutions, then the Hölder norm of the solution can be controlled in terms of the Hölder norms for the coefficient and source terms. Since these estimates do not assume the existence of the solution, they are called a priori estimates. There is both an ''interior'' result, giving a Hölder condition for the solution in interior domains away from the boundary, and a ''boundary'' result, giving the Hölder condition for the solution in the entire domain. The former bound depends only on the spatial dimension, the equation, and the distance to the boundary; the latter depends on the smoothness of the boundary as well. The Schauder estimates are a necessary precondition to using the method of continuity to proving the existence and regularity of solutions to the Dirichlet problem for elliptic PDEs. This result says that when the coefficients of the equation and the nature of the boundary conditions are sufficiently smooth, there is a smooth classical solution to the PDE. == Notation == The Schauder estimates are given in terms of weighted Hölder norms; the notation will follow that given in the text of . The supremum norm of a continuous function $f\; \backslash in\; C(\backslash Omega)$ is given by :$|f|\_\; =\; \backslash sup\_\; |f(x)|$ For a function which is Hölder continuous with exponent $\backslash alpha$, that is to say, $f\; \backslash in\; C^\backslash alpha\; (\backslash Omega)$ the usual Hölder seminorm is given by :$()\_\; =\; \backslash sup\_\; \backslash frac.$ The sum of the two is the full Hölder norm of ''f'' :$|f|\_\; =\; |f|\_\; +\; ()\_\; =\; \backslash sup\_\; |f(x)|\; +\; \backslash sup\_\; \backslash frac.$ For differentiable functions ''u'', it is necessary to consider the higher order norms, involving derivatives. The norm in the space of functions with ''k'' continuous derivatives, $C^k(\backslash Omega)$, is given by :$|u|\_\; =\; \backslash sum\_\; \backslash sup\_\; |D^\backslash beta\; u(x)|$ where $\backslash beta$ ranges over all multi-indices of appropriate orders. For functions with ''k''th order derivatives which are Holder continuous with exponent $\backslash alpha$, the appropriate semi-norm is given by :$()\_\; =\; \backslash sup\_\}\; \backslash frac$ which gives a full norm of :$|u|\_\; =\; |u|\_\; +\; ()\_\; =\; \backslash sum\_\; \backslash sup\_\; |D^\backslash beta\; u(x)|\; +\; \backslash sup\_\}\; \backslash frac.$ For the interior estimates, the norms are weighted by the distance to the boundary :$d\_x\; =\; d(x,\; \backslash partial\; \backslash Omega)$ raised to the same power as the derivative, and the seminorms are weighted by :$d\_\; =\; \backslash min\; (d\_x,d\_y)$ raised to the appropriate power. The resulting weighted interior norm for a function is given by :$|u|^$ *_ = |u|^ *_ + ()^ *_ = \sum_ \sup_ |d_x^ D^\beta u(x)| + \sup_} d_^ \frac It is occasionally necessary to add "extra" powers of the weight, denoted by :$|u|^\_\; =\; |u|^\_\; +\; ()^\_\; =\; \backslash sum\_\; \backslash sup\_\; |d\_x^\; D^\backslash beta\; u(x)|\; +\; \backslash sup\_\}\; d\_^\; \backslash frac.$ 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「Schauder estimates」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.