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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 Perron method ： ウィキペディア英語版 Perron method In the mathematical study of harmonic functions, the Perron method, also known as the method of subharmonic functions, is a technique introduced by Oskar Perron for the solution of the Dirichlet problem for Laplace's equation. The Perron method works by finding the largest subharmonic function with boundary values below the desired values; the "Perron solution" coincides with the actual solution of the Dirichlet problem if the problem is soluble. The Dirichlet problem is to find a harmonic function in a domain, with boundary conditions given by a continuous function $\backslash varphi(x)$. The Perron solution is defined by taking the pointwise supremum over a family of functions $S\_\backslash varphi$, :$u(x)\; =\; \backslash sup\_\; v(x)$ where $S\_\backslash varphi$ is the set of all subharmonic functions such that $v(x)\; \backslash leq\; \backslash varphi(x)$ on the boundary of the domain. The Perron solution ''u(x)'' is always harmonic; however, the values it takes on the boundary may not be the same as the desired boundary values $\backslash varphi(x)$. A point ''y'' of the boundary satisfies a ''barrier'' condition if there exists a superharmonic function $w\_y(x)$, defined on the entire domain, such that $w\_y(y)=0$ and $w\_y(x)\; >\; 0$ for all $x\; \backslash ne\; y$. Points satisfying the barrier condition are called ''regular'' points of the boundary for the Laplacian. These are precisely the points at which one is guaranteed to obtain the desired boundary values: as $x\backslash rightarrow\; y,\; u(x)\; \backslash rightarrow\; \backslash varphi(y)$. The characterization of regular points on surfaces is part of potential theory. Regular points on the boundary of a domain $\backslash Omega$ are those points that satisfy the Wiener criterion: for any $\backslash lambda\; \backslash in\; (0,1)$, let $C\_j$ be the capacity of the set $B\_(x\_0)\; \backslash cap\; \backslash Omega^c$; then $x\_0$ is a regular point if and only if :$\backslash sum\_^\backslash infty\; C\_j\; /\; \backslash lambda^$ diverges. The Wiener criterion was first devised by Norbert Wiener; it was extended by Werner Püschel to uniformly elliptic divergence-form equations with smooth coefficients, and thence to uniformly elliptic divergence form equations with bounded measureable coefficients by Walter Littman, Guido Stampacchia, and Hans Weinberger. ==References== * * 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「Perron method」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.