Modified harmonic Robin function

Representation of the Green's function of the classical Neumann problem for the Poisson equation in the unit ball of arbitrary dimension is given. In constructing this function, we use the representation of the fundamental solution of the Laplace equation in the form of a series. It is shown that the Green's function can be represented in terms of elementary functions and its explicit form can be written out. An explicit form of the Neumann kernel at n = 4 and n = 5.

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“…We choose the coefficient b 0 = 1/ (n − 2) for uniformity of further calculations. Substituting the found coefficients into (19), we get …”

Section: The Main Results Of the Papermentioning

confidence: 98%

“…The Green's function of the Dirichlet problem for a polyharmonic equation in the multidimensional ball was constructed in the explicit form in [13][14][15][16][17]. We also note that works [18][19][20] are devoted to the construction of the Green's function of the Robin problem in the explicit form in a circle.…”

Section: Introductionmentioning

confidence: 99%

Representation of Green’s function of the Neumann problem for a multi-dimensional ball

Sadybekov

Torebek

Turmetov

2015

Complex Variables and Elliptic Equations

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“…Обе эти функции являются частными случаями гармонической функцией Робена (см. [23]). Кроме того, функции Робена также обладают следующими свойствами:…”

Section: лемма 4 (представление коши шварца помпейюunclassified

“…(см. [23,38]). Доказательства приведенных соотношений для функций Робена основаны на свойствах задачи Шварца для аналитических функций, поскольку вещественная гармоническая функция h 1 (•, ζ) является вещественной частью аналитической в D функции для всех ζ ∈ ∂D.…”

Section: лемма 4 (представление коши шварца помпейюunclassified

Комплексные Дифференциальные Уравнения В Частных Производных

Аксой

Aksoy

Begehr

et al. 2020

Итоги науки и техники. Серия «Современная математика и ее прило

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Изучаются краевые задачи Шварца и итерированные краевые задачи Дирихле для полианалитических операторов в некоторых плоских областях, имеющих гармоническую функцию Грина. Рассмотрены гибридные полигармонические функции Грина, позволяющие исследовать краевые задач для полигармонического оператора. Эта сравнительно новая тема далека от завершения; чем выше порядок полигармонического оператора, тем богаче теория связанных гибридных функций Грина: они конструируются путем непрерывной свертки гармонических функций Грина, Неймана, Робена, а также полигармонических функций Грина - Альманси.

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“…An explicit form of the Green function in a sector for the biharmonic and triharmonic equations is given in [5,6]. The Green's function of the Neumann problem for the Poisson equation in the half-space R n + is constructed in [7], and the Green's function for the Robin problem in the disk is studied in [8]. Explicit forms of Green's function G m (x, ξ) of the Dirichlet problem for the polyharmonic equation [9] in the unit ball S = {x ∈ R n : |x| < 1} are constructed in various ways in [10][11][12][13], and so forth.…”

Section: Introductionmentioning

confidence: 99%

Dirichlet and Neumann Boundary Value Problems for the Polyharmonic Equation in the Unit Ball

Karachik

2021

Mathematics

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In the previous author’s works, a representation of the solution of the Dirichlet boundary value problem for the biharmonic equation in terms of Green’s function is found, and then it is shown that this representation for a ball can be written in the form of the well-known Almansi formula with explicitly defined harmonic components. In this paper, this idea is extended to the Dirichlet boundary value problem for the polyharmonic equation, but without invoking the Green’s function. It turned out to find an explicit representation of the harmonic components of the m-harmonic function, which is a solution to the Dirichlet boundary value problem, in terms of m solutions to the Dirichlet boundary value problems for the Laplace equation in the unit ball. Then, using this representation, an explicit formula for the harmonic components of the solution to the Neumann boundary value problem for the polyharmonic equation in the unit ball is obtained. Examples are given that illustrate all stages of constructing solutions to the problems under consideration.

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Modified harmonic Robin function

Cited by 36 publications

References 10 publications

Representation of Green’s function of the Neumann problem for a multi-dimensional ball

Representation of Green’s function of the Neumann problem for a multi-dimensional ball

Комплексные Дифференциальные Уравнения В Частных Производных

Dirichlet and Neumann Boundary Value Problems for the Polyharmonic Equation in the Unit Ball

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