Single and Double Layer Potentials on Domains with Conical Points I: Straight Cones
Single and Double Layer Potentials on Domains with Conical Points I: Straight Cones
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圆锥点域上的单层和双层势 I：直锥体
DOI：
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发表时间：
2012
期刊：
影响因子：
--
通讯作者：
V. Nistor
中科院分区：
文献类型：
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作者：
Yu Qiao;V. Nistor
设$${omega=mathbb{R}^+omega}$$是$${mathbb{R}^n，ngeq3}$$中的开直锥体，其中$${omega子集S^{n-1}}$$是单位球面的光滑子域.K和S表示与Ω和拉普拉斯算子Δ相关的双层和单层位势算子。设r是到原点的距离。我们考虑了∂Ω上一类自然的伸缩不变算子，称为梅林卷积算子，证明了$${Ka：=r^{a}Kr^{-a}}$$和$${S_b：=r^{b-FRAC{1}{2}}sr^{-b-FRAC{1}{2}}$$是${a in(-1，n-1)}$$和${b在(FRAC{1}{2}，n-FRAC{3}{2})}$$中的Mellin卷积算子.众所周知，梅林卷积算子T可逆的充要条件是它的梅林变换$${HAT T(λ)}$$对任何实数的梅林变换都是可逆的。我们建立了一个约化过程，将Ka和Sb的Mellin变换分别与ω上其他一些椭圆算子的单层位势算子和双层位势算子联系起来，利用光滑域上经典的层势算子理论可以证明这些算子是可逆的。由此证明了$${frac{1}{2}pm K}$$和S在适当的加权Sobolev空间之间是可逆的。这些算子的可逆性的一个经典结果是Ω上Dirichlet问题在加权Soblev空间中的可解性结果。
Let $${Omega = mathbb{R}^+ omega}$$ be an open straight cone in $${mathbb{R}^n, ngeq3}$$ , where $${omega subset S^{n-1}}$$ is a smooth subdomain of the unit sphere. Denote by K and S the double and single layer potential operators associated to Ω and the Laplace operator Δ. Let r be the distance to the origin. We consider a natural class of dilation invariant operators on ∂Ω, called Mellin convolution operators and show that $${K_a :=r^{a}Kr^{-a}}$$ and $${S_b := r^{b-frac{1}{2}}Sr^{-b-frac{1}{2}}}$$ are Mellin convolution operators for $${a in (-1, n-1)}$$ and $${b in (frac{1}{2}, n-frac{3}{2})}$$ . It is known that a Mellin convolution operator T is invertible if, and only if, its Mellin transform $${hat T( lambda)}$$ is invertible for any real λ. We establish a reduction procedure that relates the Mellin transforms of Ka and Sb to the single and, respectively, double layer potential operators associated to some other elliptic operators on ω, which can be shown to be invertible using the classical theory of layer potential operators on smooth domains. This reduction procedure thus allows us to prove that $${frac{1}{2}pm K}$$ and S are invertible between suitable weighted Sobolev spaces. A classical consequence of the invertibility of these operators is a solvability result in weighted Sobolev spaces for the Dirichlet problem on Ω.
Bouete de Monvel 的微积分和群胚 I
影响因子：
0.9
作者：
J. Aastrup;S. Melo;B. Monthubert;E. Schrohe
通讯作者：
E. Schrohe