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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 Neumann–Poincaré operator ： ウィキペディア英語版 Neumann–Poincaré operator In mathematics, the Neumann–Poincaré operator or Poincaré–Neumann operator, named after Carl Neumann and Henri Poincaré, is a non-self-adjoint compact operator introduced by Poincaré to solve boundary value problems for the Laplacian on bounded domains in Euclidean space. Within the language of potential theory it reduces the partial differential equation to an integral equation on the boundary to which the theory of Fredholm operators can be applied. The theory is particularly simple in two dimensions—the case treated in detail in this article—where it is related to complex function theory, the conjugate Beurling transform or complex Hilbert transform and the Fredholm eigenvalues of bounded planar domains. ==Dirichlet and Neumann problems== Green's theorem for a bounded region Ω in the plane with smooth boundary ∂Ω states that :$\backslash displaystyle$ One direct way to prove this is as follows. By subtraction, it is sufficient to prove the theorem for a region bounded by a simple smooth curve. Any such is diffeomorphic to the closed unit disk. By change of variables it is enough to prove the result there. Separating the ''A'' and ''B'' terms, the right hand side can be written as a double integral starting in the ''x'' or ''y'' direction, to which the fundamental theorem of calculus can be applied. This converts the integral over the disk into the integral over its boundary. Let Ω be a region bounded by a simple closed curve. Given a smooth function ''f'' on the closure of Ω its normal derivative ∂''n''''f'' at a boundary point is the directional derivative in the direction of the outward pointing normal vector. Applying Green's theorem with ''A'' = ''v''''x'' ''u'' and ''B'' = ''v''''y'' ''u'' gives the first of Green's identities: :$\backslash displaystyle$ where the Laplacian Δ is given by :$\backslash displaystyle\; \backslash Delta\; =\; -\backslash partial^2\_x\; -\backslash partial\_y^2.$ Swapping ''u'' and ''v'' and subtracting gives the second of Green's identities: :$\backslash displaystyle$ If now ''u'' is harmonic in Ω and ''v'' = 1, then this identity implies that :$\backslash displaystyle$ so the integral of the normal derivative of a harmonic function on the boundary of a region always vanishes. A similar argument shows that the average of a harmonic function on the boundary of a disk equals its value at the centre. Translating the disk can be taken to be centred at 0. Green's identity can be applied to an annulus formed of the boundary of the disk and a small circle centred on 0 with ''v'' = ''z''2: it follows that the average is independent of the circle. It tends to the value at its value at 0 as the radius of the smaller circle decreases. This result also follows easily using Fourier series and the Poisson integral. For continuous functions ''f'' on the whole plane which are smooth in Ω and the complementary region Ω''c'', the first derivative can have a jump across the boundary of Ω. The value of the normal derivative at a boundary point can be computed from inside or outside Ω. The interior normal derivative will be denoted by ∂''n''− and the exterior normal derivative by ∂''n''+. With this terminology the four basic problems of classical potential theory are as follows: *Interior Dirichlet problem: ∆''u'' = 0 in Ω, ''u'' = ''f'' on ∂Ω *Interior Neumann problem: ∆''u'' = 0 in Ω, ∂''n''− ''u'' = ''f'' on ∂Ω *Exterior Dirichlet problem: ∆''u'' = 0 in Ω''c'', ''u'' = ''f'' on ∂Ω, ''u'' continuous at ∞ *Exterior Neumann problem: ∆''u'' = 0 in Ω''c'', ∂''n''+ ''u'' = ''f'' on ∂Ω, ''u'' continuous at ∞ For the exterior problems the inversion map ''z''−1 takes harmonic functions on Ω''c'' into harmonic functions on the image of Ω''c'' under the inversion map.〔 Up to composition with complex conjugation, this is the special case of the Kelvin transform in two dimensions. In this case, since a function is harmonic if and only if it is the real part of a holomorphic function, the statement follows from the fact that the composition of holomorphic functions is holomorphic.〕 The transform ''v'' of ''u'' is continuous in a small disc |''z''| ≤ ''r'' and harmonic everywhere in the interior except possibly 0. Let ''w'' be the harmonic function given by the Poisson integral on |''z''| ≤ ''r'' with the same boundary value ''g'' as ''v'' on |''z''| = ''r''. Applying the maximum principle to ''v'' − ''w'' + ε log |''z''| on δ ≤ |''z''| ≤ ''r'', it must be negative for δ small. Hence ''v''(''z'') ≤ ''u''(''z'') for ''z'' ≠ 0. The same argument applies with ''v'' and ''w'' swapped, so ''v'' = ''w'' is harmonic in the disk. Thus the singularity at ∞ is removable. By the maximum principle the interior and exterior Dirichlet problems have unique solutions. For the interior Neumann problem, if a solution ''u'' is harmonic in 0 and its interior normal derivative vanishes, then Green's first identity implies implies the ''u''''x'' = 0 = ''u''''y'', so that ''u'' is constant. This shows the interior Neumann problem has a unique solution up to adding constants. Applying inversion, the same holds for the external Neumann problem. For both Neumann problems, a necessary condition for a solution to exist is :$\backslash displaystyle$ For the interior Neumann problem, this follows by setting ''v'' = 1 in Green's second identity. For the exterior Neumann problem, the same can be done for the intersection of Ω''c'' and a large disk |''z''| < ''R'', giving :$\backslash displaystyle\backslash partial\_n\; u.\}$ At ∞ ''u'' is the real part of a holomorphic function ''F'' with :$\backslash displaystyle\; +\; \backslash cdots\}$ The interior normal derivative on |''z''| = ''R'' is just the radial derivative ∂''r'', so that for |''z''| = ''R'' :$\backslash displaystyle$ Hence :$\backslash displaystyle,\}$ so the integral over ∂Ω must vanish. The fundamental solution of the Laplacian is given by :$\backslash displaystyle$ ''N''(''z'') = − ''E''(''z'') is called the Newtonian potential in the plane. Using polar coordinates, it is easy to see that ''E'' is in L''p'' on any closed disk for any finite ''p'' ≥ 1. To say that ''E'' is a fundamental solution of the Laplacian means that for any smooth function φ of compact support :$\backslash displaystyle$ The standard proof uses Green's second identity on the annulus ''r'' ≤ |''z''| ≤ ''R'' where the support of φ is contained in |''z''| < ''R''. In fact, since ''E'' is harmonic away from 0, :$\backslash displaystyle\backslash int\_0^\; \backslash varphi(r\; e^)\; -\; r\; \backslash cdot\; \backslash log\; r\backslash cdot\backslash partial\_n\backslash varphi(re^)\; \backslash ,\; d\backslash theta.\}$ As ''r'' tends to zero, the first term on the right hand side tends to φ(0) and the second to 0, since ''r'' log ''r'' tends to 0 and the normal derivatives of φ are uniformly bounded. (That both sides are equal even before taking limits follows from the fact that the average of a harmonic function on the boundary of a disk equals it value at the centre, while the integral of its normal derivative vanishes.) 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「Neumann–Poincaré 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.