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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 Elliptic operator ： ウィキペディア英語版 Elliptic operator In the theory of partial differential equations, elliptic operators are differential operators that generalize the Laplace operator. They are defined by the condition that the coefficients of the highest-order derivatives be positive, which implies the key property that the principal symbol is invertible, or equivalently that there are no real characteristic directions. Elliptic operators are typical of potential theory, and they appear frequently in electrostatics and continuum mechanics. Elliptic regularity implies that their solutions tend to be smooth functions (if the coefficients in the operator are smooth). Steady-state solutions to hyperbolic and parabolic equations generally solve elliptic equations. ==Definitions== A linear differential operator ''L'' of order ''m'' on a domain $\backslash Omega$ in R''d'' given by :$Lu\; =\; \backslash sum\_\; a\_\backslash alpha(x)\backslash partial^\backslash alpha\; u\backslash ,$ (where $\backslash alpha\; =\; (\backslash alpha\_1,\; ...,\; \backslash alpha\_n)$ is a multi-index, and $\backslash partial^\backslash alpha\; u\; =\; \backslash partial^\; \backslash cdots\; \backslash partial^\; u$) is called ''elliptic'' if for every ''x'' in $\backslash Omega$ and every non-zero $\backslash xi$ in R''d'', :$\backslash sum\_\; a\_\backslash alpha(x)\backslash xi^\backslash alpha\; \backslash neq\; 0,\backslash ,$ where $\backslash xi^\backslash alpha\; =\; \backslash xi\_1^\; \backslash cdots\; \backslash xi\_n^$. In many applications, this condition is not strong enough, and instead a ''uniform ellipticity condition'' may be imposed for operators of degree ''m = 2k'': :$(-1)^k\backslash sum\_\; a\_\backslash alpha(x)\; \backslash xi^\backslash alpha\; >\; C\; |\backslash xi|^,\backslash ,$ where ''C'' is a positive constant. Note that ellipticity only depends on the highest-order terms.〔Note that this is sometimes called ''strict ellipticity'', with ''uniform ellipticity'' being used to mean that an upper bound exists on the symbol of the operator as well. It is important to check the definitions the author is using, as conventions may differ. See, e.g., Evans, Chapter 6, for a use of the first definition, and Gilbarg and Trudinger, Chapter 6, for a use of the second.〕 A nonlinear operator :$L(u)\; =\; F(x,\; u,\; (\backslash partial^\backslash alpha\; u)\_)\backslash ,$ is elliptic if its first-order Taylor expansion with respect to ''u'' and its derivatives about any point is a linear elliptic operator. ;Example 1 :The negative of the Laplacian in R''d'' given by ::$-\; \backslash Delta\; u\; =\; -\backslash sum\_^d\; \backslash partial\_i^2u\backslash ,$ :is a uniformly elliptic operator. The Laplace operator occurs frequently in electrostatics. If ρ is the charge density within some region Ω, the potential Φ must satisfy the equation ::$-\; \backslash Delta\; \backslash Phi\; =\; 4\backslash pi\backslash rho.\backslash ,$ ;Example 2 :Given a matrix-valued function ''A(x)'' which is symmetric and positive definite for every ''x'', having components ''a''''ij'', the operator ::$Lu\; =\; -\backslash partial\_i(a^(x)\backslash partial\_ju)\; +\; b^j(x)\backslash partial\_ju\; +\; cu\backslash ,$ :is elliptic. This is the most general form of a second-order divergence form linear elliptic differential operator. The Laplace operator is obtained by taking ''A = I''. These operators also occur in electrostatics in polarized media. ;Example 3 :For ''p'' a non-negative number, the p-Laplacian is a nonlinear elliptic operator defined by ::$L(u)\; =\; -\backslash sum\_^d\backslash partial\_i\; (|\backslash nabla\; u|^\backslash partial\_i\; u).\backslash ,$ :A similar nonlinear operator occurs in glacier mechanics. The Cauchy stress tensor of ice, according to Glen's flow law, is given by ::$\backslash tau\_\; =\; B\backslash left(\backslash sum\_^3(\backslash partial\_lu\_k)^2\backslash right)^\}\backslash cdot\backslash frac(\backslash partial\_ju\_i\; +\; \backslash partial\_iu\_j)\backslash ,$ :for some constant ''B''. The velocity of an ice sheet in steady state will then solve the nonlinear elliptic system ::$\backslash sum\_^3\backslash partial\_j\backslash tau\_\; +\; \backslash rho\; g\_i\; -\; \backslash partial\_ip\; =\; Q,\backslash ,$ :where ρ is the ice density, ''g'' is the gravitational acceleration vector, ''p'' is the pressure and ''Q'' is a forcing term. 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「Elliptic operator」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.