On a method for constructing the Green function of the Dirichlet problem for the Laplace equation

Лaплaс теңдеуi үшiн Дирихле есебiнiң Грин функциясының интегралдық көрсетiлiмi туралы
Об интегрaльном предстaвлении функции Гринa зaдaчи Дирихле для урaвнения Лaплaсa
Kalmenov T.S.
2024 E.A. Buketov Karaganda University Publish house

Bulletin of the Karaganda University. Mathematics Series
2024 #114 Issue 2 105 - 113 pp.

The study of boundary value problems for elliptic equations is of both theoretical and applied interest. A thorough study of model physical and spectral problems requires an explicit and effective representation of the problem solution. Integral representations of solutions of problems of differential equations are one of the main tools of mathematical physics. Currently, the integral representation of the Green function of classical problems for the Laplace equation for an arbitrary domain is obtained only in a two-dimensional domain by the Riemann conformal mapping method. Starting from the three-dimensional case, these classical problems are solved only for spherical sectors and for the regions lying between the faces of the hyperplane. The problem of constructing integral representations of general boundary value problems and studying their spectral problems remains relevant. In this work, using the boundary condition of the Newtonian (volume) potential and the spectral property of the potential of a simple layer, the Green function of the Dirichlet problem for the Laplace equation was constructed.

Dirichlet problem , Green function , Laplace equation , simple layer potential

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The National Academy of Sciences, Kazakhstan
The Department of Differential Operators, Institute of Mathematics and Mathematical Modeling, 125 Pushkin street, Almaty, 050010, Kazakhstan

The National Academy of Sciences
The Department of Differential Operators

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