An Equivalence Between the Dirichlet and the Neumann Problem for the Laplace Operator

Beznea Lucian; Pascu Mihai N.; Pascu Nicolae R.

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An Equivalence Between the Dirichlet and the Neumann Problem for the Laplace Operator

机译：Laplace算子Dirichlet和Neumann问题之间的等价关系

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We give a representation of the solution of the Neumann problem for the Laplace operator on the n-dimensional unit ball in terms of the solution of an associated Dirichlet problem. The representation is extended to other operators besides the Laplacian, to smooth simply connected planar domains, and to the infinite-dimensional Laplacian on the unit ball of an abstract Wiener space, providing in particular an explicit solution for the Neumann problem in this case. As an application, we derive an explicit formula for the Dirichlet-to-Neumann operator, which may be of independent interest.

机译：我们用相关的Dirichlet问题的解来表示n维单位球上Laplace算子的Neumann问题的解。除了拉普拉斯算子，该表示法还扩展到其他算子，以平滑简单连接的平面域，并扩展到抽象维纳空间的单位球上的无穷维拉普拉斯算子，在这种情况下，尤其为诺伊曼问题提供了一个明确的解决方案。作为应用，我们为Dirichlet-to-Neumann运算符导出了一个明确的公式，该公式可能具有独立的意义。

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来源
《Potential analysis: An international journal devoted to the interactions between potential theory, probability theory, geometry and functional analysis》 |2016年第4期|共18页
作者
Beznea Lucian; Pascu Mihai N.; Pascu Nicolae R.;
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原文格式 PDF
正文语种 eng
中图分类数学;
关键词
Dirichlet problem; Neumann problem; Laplace operator; Infinite-dimensional Laplace operator; Dirichlet-to-Neumann operator;

机译：Dirichlet问题;Neumann问题;Laplace算子;无限维Laplace算子;Dirichlet-to-Neumann算子;

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专利

1. An Equivalence Between the Dirichlet and the Neumann Problem for the Laplace Operator [J] . Beznea Lucian, Pascu Mihai N., Pascu Nicolae R. Potential analysis: An international journal devoted to the interactions between potential theory, probability theory, geometry and functional analysis . 2016,第4期

机译：Laplace算子Dirichlet和Neumann问题之间的等价关系
2. Some multipoint boundary value problems of Neumann–Dirichlet type involving a multipoint p-Laplace like operator [J] . M. Garcia-Huidobro, Chaitan P. Gupta, R. Manasevich Journal of Mathematical Analysis and Applications . 2007,第1期

机译：涉及多点p-Laplace像算子的Neumann–Dirichlet型的一些多点边值问题
3. On an inequality between Dirichlet and Neumann eigenvalues for the Laplace operator [J] . N. Filonov St. Petersburg mathematical journal . 2005,第2期

机译：关于Laplace算子的Dirichlet和Neumann特征值之间的不等式
4. A numerical algorithm for solving Laplace and Poisson type partial differential equations on a uniform rectangular mesh with Dirichlet and Neumann boundary conditions [C] . Vasos Pavlika WSEAS International Conferences . 2008

机译：用Dirichlet和Neumann边界条件求解LAPLACH和泊松型偏微分方程的数值算法
5. On analyticity of variations of the Dirichlet-Neumann operators, and computational concerns. [D] . Fazioli, Christopher Carlo. 2009

机译：关于Dirichlet-Neumann算子的变分的解析性和计算问题。
6. Laplace operators of infinite-dimensional Lie algebras and theta functions [O] . Victor G. Kac 1984

机译：无限维李代数和θ函数的拉普拉斯算子
7. Some multipoint boundary value problems of Neumann–Dirichlet type involving a multipoint p-Laplace like operator [O] . García-Huidobro M., Gupta Chaitan P., Manásevich R. 2007

机译：涉及多点p-Laplace算子的Neumann–Dirichlet型的一些多点边值问题
8. Laplace's equation and the Dirichlet-Neumann map in multiply connected domains. [R] . A. Greenbaum L. Greengard G. B. McFadden 1991

机译：拉普拉斯方程和多重连通域中的Dirichlet-Neumann映射。

An Equivalence Between the Dirichlet and the Neumann Problem for the Laplace Operator

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