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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 Viscosity solution ： ウィキペディア英語版 Viscosity solution In mathematics, the viscosity solution concept was introduced in the early 1980s by Pierre-Louis Lions and Michael G. Crandall as a generalization of the classical concept of what is meant by a 'solution' to a partial differential equation (PDE). It has been found that the viscosity solution is the natural solution concept to use in many applications of PDE's, including for example first order equations arising in optimal control (the Hamilton-Jacobi equation), differential games (the Isaacs equation) or front evolution problems,〔I. Dolcetta and P. Lions, eds., (1995), ''Viscosity Solutions and Applications.'' Springer, ISBN 978-3-540-62910-8.〕 as well as second-order equations such as the ones arising in stochastic optimal control or stochastic differential games. The classical concept was that a PDE :$F(x,u,Du,D^2\; u)\; =\; 0$ over a domain $x\backslash in\backslash Omega$ has a solution if we can find a function ''u''(''x'') continuous and differentiable over the entire domain such that $x$, $u$, $Du$, $D^2\; u$ satisfy the above equation at every point. If a scalar equation is degenerate elliptic (defined below), one can define a type of weak solution called ''viscosity solution''. Under the viscosity solution concept, ''u'' need not be everywhere differentiable. There may be points where either $Du$ or $D^2\; u$ does not exist and yet ''u'' satisfies the equation in an appropriate generalized sense. The definition allows only for certain kind of singularities, so that existence, uniqueness, and stability under uniform limits, hold for a large class of equations. == Definition == There are several equivalent ways to phrase the definition of viscosity solutions. See for example the section II.4 of Fleming and Soner's book〔 Wendell H. Fleming, H. M . Soner., eds., (2006), ''Controlled Markov Processes and Viscosity Solutions.'' Springer, ISBN 978-0-387-26045-7.〕 or the definition using semi-jets in the Users Guide. An equation $F(x,u,Du,D^2\; u)\; =\; 0$ in a domain $\backslash Omega$ is defined to be ''degenerate elliptic'' if for any two symmetric matrices $X$ and $Y$ such that $Y-X$ is positive definite, and any values of $x\; \backslash in\; \backslash Omega$, $u\; \backslash in\; \backslash mathbb$ and $p\; \backslash in\; \backslash mathbb^n$, we have the inequality $F(x,u,p,X)\; \backslash geq\; F(x,u,p,Y)$. For example $-\backslash Delta\; u\; =\; 0$ is degenerate elliptic. Any first order equation is degenerate elliptic. An upper semicontinuous function $u$ in $\backslash Omega$ is defined to be a subsolution of a degenerate elliptic equation in the ''viscosity sense'' if for any point $x\_0\; \backslash in\; \backslash Omega$ and any $C^2$ function $\backslash phi$ such that $\backslash phi(x\_0)\; =\; u(x\_0)$ and $\backslash phi\; \backslash geq\; u$ in a neighborhood of $x\_0$, we have $F(x\_0,\backslash phi(x\_0),D\backslash phi(x\_0),D^2\; \backslash phi(x\_0))\; \backslash leq\; 0$. A lower semicontinuous function $u$ in $\backslash Omega$ is defined to be a supersolution of a degenerate elliptic equation in the ''viscosity sense'' if for any point $x\_0\; \backslash in\; \backslash Omega$ and any $C^2$ function $\backslash phi$ such that $\backslash phi(x\_0)\; =\; u(x\_0)$ and $\backslash phi\; \backslash leq\; u$ in a neighborhood of $x\_0$, we have $F(x\_0,\backslash phi(x\_0),D\backslash phi(x\_0),D^2\; \backslash phi(x\_0))\; \backslash geq\; 0$. A continuous function ''u'' is a viscosity solution of the PDE if it is both a viscosity supersolution and a viscosity subsolution. 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「Viscosity solution」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.