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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 Dirichlet form ： ウィキペディア英語版 Dirichlet form In the branch of mathematics known as potential theory, a Dirichlet form is a generalization of the Laplacian that can be defined on any measure space, without the need for mentioning partial derivatives. This allows mathematicians to study the Laplace equation and heat equation on spaces that are not manifolds: for example, fractals. To accomplish this generalization, one focuses not on the Laplacian itself but on the quantity :$\backslash mathcal(u)\; =\; \backslash int\_\; |\backslash nabla\; u|^2\backslash ;dx$ that is minimized when the Laplacian vanishes. Technically, a Dirichlet form is a Markovian closed symmetric form on an ''L''2-space.〔Fukushima, M, Oshima, Y., & Takeda, M. (1994). ''Dirichlet forms and symmetric Markov processes.'' Walter de Gruyter & Co , ISBN 3-11-011626-X〕 Such objects are studied in abstract potential theory, based on the classical Dirichlet's principle. The theory of Dirichlet forms originated in the work of on Dirichlet spaces. A Dirichlet form on a measure space $(X,\; \backslash mu)$ is a bilinear function :$\backslash mathcal:\; D\backslash times\; D\; \backslash to\; \backslash mathbb$ such that 1) The domain $D$ is a dense subset of $L^2(X,\; \backslash mu)$ 2) $\backslash mathcal$ is symmetric, that is $\backslash mathcal(u,v)=\backslash mathcal(v,u)$ for any $u,v\; \backslash in\; D$. 3) $\backslash mathcal(u,u)\; \backslash geq\; 0$ for any $u\; \backslash in\; D$. 4) The set $D$ equipped with the inner product defined by $(u,v)\_\; +\; \backslash mathcal(u,v)$ is a real Hilbert space. 5) For any $u\; \backslash in\; D$ we have that $u\_$ * = \min (\max(u, 0) , 1) \in D and $\backslash mathcal(u\_$ *,u_ *)\leq \mathcal(u,u) In other words, a Dirichlet form is nothing but a positive symmetric bilinear form defined on a dense subset of $L^2(X,\; \backslash mu)$ such that 4) and 5) hold. Alternatively, the quadratic form $u\; \backslash to\; \backslash mathcal(u,u)$ itself is known as the Dirichlet form and it is still denoted by $\backslash mathcal$, so $\backslash mathcal(u):=\backslash mathcal(u,u)$. The best known Dirichlet form is the Dirichlet energy of functions on $\backslash mathbb^n$ :$\backslash mathcal(u)\; =\; \backslash int\_\; |\backslash nabla\; u|^2\backslash ;dx$ which gives rise to the space $H^1(\backslash mathbb^n)$. Another example of a Dirichlet form is given by :$\backslash mathcal(u)\; =\; \backslash iint\_^n\}\; (u(y)-u(x))^2\; k(x,y)\backslash ,\; \backslash mathrmx\; \backslash mathrm\; y$ where $k:\backslash mathbb\; R^n\backslash times\; \backslash mathbb\; R^n\backslash to\; \backslash mathbb\; R$ is some non-negative symmetric integral kernel. If the kernel $k$ satisfies the bound $k(x,y)\; \backslash leq\; \backslash Lambda\; |x-y|^$, then the quadratic form is bounded in $\backslash dot\; H^$. If moreover, $\backslash lambda\; |x-y|^\; \backslash leq\; k(x,y)$, then the form is comparable to the norm in $\backslash dot\; H^$ squared and in that case the set $D\; \backslash subset\; L^2(\backslash mathbb^n)$ defined above is given by $H^(\backslash mathbb^n)$. Thus Dirichlet forms are natural generalizations of the Dirichlet integrals :$\backslash mathcal(u)\; =\; \backslash int\; (A\backslash nabla\; u,\backslash nabla\; u)\backslash ;\; \backslash mathrm\; x,$ where $A(x)$ is a positive symmetric matrix. The Euler-Lagrange equation of a Dirichlet form is a non-local analogue of an elliptic equations in divergence form. Equations of this type are studied using variational methods and they are expected to satisfy similar properties.〔〔〔 == 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「Dirichlet form」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.