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 Vestnik KRAUNC. Fiz.-Mat. Nauki, 2020, Volume 32, Number 3, Pages 114–126 (Mi vkam423)

MATHEMATICS

Holmgren problem for elliptic equation with singular coefficients

T. G. Ergashevab*, A. Hasanovbc

a Tashkent Institute of Irrigation and Agricultural Mechanization Engineers
b V. I. Romanovskiy Institute of Mathematcs of the Academy of Sciences of Uzbekistan
c Ghent University

Abstract: In the present work, we investigate the Holmgren problem for an multidimensional elliptic equation with several singular coefficients. We use a fundamental solution of the equation, containing Lauricella's hypergeometric function in many variables. Then using an «abc» method, the uniqueness for the solution of the Holmgren problem is proved. Applying a method of Green's function, we are able to find the solution of the problem in an explicit form. Moreover, decomposition and summation formulae, formulae of differentiation and some adjacent relations for Lauricella's hypergeometric functions in many variables were used in order to find the explicit solution for the formulated problem.

Keywords: Holmgren problem, multidimensional elliptic equations with several singular coefficients, decomposition formula, summation formula, Lauricella hypergeometric function in many variables, Green's function.
* Author to whom correspondence should be addressed

DOI: https://doi.org/10.26117/2079-6641-2020-32-3-114-126

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UDC: 519.644
MSC: Primary 35A08; Secondary 35J25, 35J70, 35J75
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Citation: T. G. Ergashev, A. Hasanov, “Holmgren problem for elliptic equation with singular coefficients”, Vestnik KRAUNC. Fiz.-Mat. Nauki, 32:3 (2020), 114–126

Citation in format AMSBIB
\Bibitem{ErgHas20} \by T.~G.~Ergashev, A.~Hasanov \paper Holmgren problem for elliptic equation with singular coefficients \jour Vestnik KRAUNC. Fiz.-Mat. Nauki \yr 2020 \vol 32 \issue 3 \pages 114--126 \mathnet{http://mi.mathnet.ru/vkam423} \crossref{https://doi.org/10.26117/2079-6641-2020-32-3-114-126}