On the solvability of some boundary value problems for the inhomogeneous polyharmonic equation with boundary operators of the Hadamard type

2017 ◽  
Vol 53 (3) ◽  
pp. 333-344 ◽  
Author(s):  
B. Kh. Turmetov
Keyword(s):  
Boundary Value Problems ◽  
Boundary Value ◽  
Polyharmonic Equation ◽  
Boundary Operators

scholarly journals On Boundary Value Problems for Elliptic Equations in a Singular Domain

1969 ◽  
Vol 36 ◽  
pp. 99-115
Author(s):  
Kazunari Hayashida
Keyword(s):  
Boundary Value Problem ◽  
Boundary Value Problems ◽  
Elliptic Operator ◽  
Bounded Domain ◽  
Elliptic Equations ◽  
Boundary Value ◽  
Tangent Line ◽  
Boundary Operators ◽  
One To One ◽  
Singular Domain

1. Let Ω be a bounded domain in the plane and denotes its closure and boundary by Ω̅ and ∂Ω, respectively. We shall say that the domain Ω is regular, if every point P ∈ ∂û has an 2-dimensional neighborhood U such that dΩ ∩ U can be mapped in a one-to-one way onto a portion of the tangent line through P by a mapping T which together with its inverse is infinitely differentiable. Let L be an elliptic operator of order 2m defined in Ω̅ and let be a normal set of boundary operators of orders mf <2m. If f is a given function in Ω, the boundary value problem II(L,f,Bj) will be to find a solution u ofsatisfyingBju = 0 on ∂Ω, j = 1, …, m.

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

Mathematics ◽  
2021 ◽  
Vol 9 (16) ◽  
pp. 1907
Author(s):  
Valery Karachik
Keyword(s):  
Boundary Value Problem ◽  
Unit Ball ◽  
Boundary Value Problems ◽  
Green's Function ◽  
Dirichlet Boundary ◽  
Boundary Value ◽  
Neumann Boundary ◽  
Dirichlet Boundary Value Problem ◽  
Polyharmonic Equation ◽  
Harmonic Components

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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