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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 Sobolev spaces for planar domains ： ウィキペディア英語版 Sobolev spaces for planar domains In mathematics, Sobolev spaces for planar domains are one of the principal techniques used in the theory of partial differential equations for solving the Dirichlet and Neumann boundary value problems for the Laplacian in a bounded domain in the plane with smooth boundary. The methods use the theory of bounded operators on Hilbert space. They can be used to deduce regularity properties of solutions and to solve the corresponding eigenvalue problems. ==Sobolev spaces with boundary conditions== Let be a bounded domain with smooth boundary. Since is contained in a large square in , it can be regarded as a domain in by identifying opposite sides of the square. The theory of Sobolev spaces on can be found in , an account which is followed in several later textbooks such as and . For an integer, the (restricted) Sobolev space is defined as the closure of in the standard Sobolev space . * . *Vanishing properties on boundary: For the elements of are referred to as " functions on which vanish with their first derivatives on ." In fact if agrees with a function in , then is in . Let be such that in the Sobolev norm, and set . Thus in . Hence for and , ::$\backslash iint\_\backslash Omega\; \backslash left\; (g\; (Dh)\; +\; (Dg)\; h\; \backslash right\; )\; \backslash ,\; dx\; \backslash ,\; dy\; =\; \backslash lim\_\; \backslash iint\_\backslash Omega\; \backslash left\; (g\; (Dh\_n)\; +\; (Dg)\; h\_n\; \backslash right\; )\; \backslash ,\; dx\; \backslash ,\; dy\; =0.$ :By Green's theorem this implies ::$\backslash int\_\; gk\; =0,\; \backslash qquad\; k\; =\; h\; \backslash cos\; \backslash left\; (\backslash mathbf\; \backslash cdot\; (a,\; b)\; \backslash right\; ),$ :where the unit normal to the boundary. Since such form a dense subspace of , it follows that on . *Support properties: Let be the complement of and define restricted Sobolev spaces analogously for . Both sets of spaces have a natural pairing with . The Sobolev space for is the annihilator in the Sobolev space for of and that for is the annihilator of . In fact this is proved by locally applying a small translation to move the domain inside itself and then smoothing by a smooth convolution operator. :Suppose in annihilates . By compactness, there are finitely many opens covering such that the closure of is disjoint from and each is an open disc about a boundary point such that in small translations in the direction of the normal vector carry into . Add an open with closure in to produce a cover of and let be a partition of unity subordinate to this cover. If translation by is denoted by , then ::$\backslash left\; \backslash \; g\; \backslash right\; \backslash \}\; \backslash longrightarrow\; g,$ :as decreases to and still lie in the annihilator, indeed they are in the annihilator for a larger domain than , the complement of which lies in . Convolving by smooth functions of small support produces smooth approximations in the annihilator of a slightly smaller domain still with complement in . These are necessarily smooth functions of compact support in . *Further vanishing properties on the boundary: The characterization in terms of annihilators shows that lies in if (and only if) it and its derivatives of order less than vanish on . In fact can be extended to by setting it to be on . The extension defines as element in using the formula ::$\backslash |h\backslash |\_^2=\backslash sum\_^k\; \backslash left\; \backslash |\backslash partial\_x^j\backslash partial\_y^\; h\; \backslash right\backslash |^2,$ :and satisfies for ''g'' in . *Duality: For , define to be the orthogonal complement of in . Let be the orthogonal projection onto , so that is the orthogonal projection onto . When , this just gives . If and , then ::$(f,g)=\; (f,\; P\_kg).$ :This implies that under the pairing between and , and are each other's duals. *Approximation by smooth functions: The image of is dense in for . This is obvious for since the sum + is dense in . Density for follows because the image of is dense in and annihilates . *Canonical isometries: The operator gives an isometry of into and of onto . In fact the first statement follows because it is true on . That is an isometry on follows using the density of in : for we have: ::$\backslash begin$ \left \|P_k (I+\Delta)^k f \right \|_ &= \sup_ \left|\left ((I+\Delta)^kf,g \right )_ \right | \\ &= \sup_ |(f,g)| \\ &=\|f\|_. \end :Since the adjoint map between the duals can by identified with this map, it follows that is a unitary map. 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「Sobolev spaces for planar domains」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.