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 Izv. Vyssh. Uchebn. Zaved. Mat., 2009, Number 8, Pages 57–70 (Mi ivm3056)

Solution of boundary value problems for a degenerating elliptic equation of the second kind by the method of potentials

F. G. Mukhlisov, A. M. Nigmetzyanova

Tatar State Humanitarian-Pedagogical University, Kazan, Russia

Abstract: In this paper we study basic boundary value problems for one multidimensional degenerating elliptic equation of the second kind. Using the method of potentials we prove the unique solvability of the mentioned problems. We construct the fundamental solution and obtain an integral representation for the solution to the equation. Using this representation we study properties of solutions, in particular, the principle of maximum. We state the basic boundary value problems and prove their unique solvability. We introduce potentials of single and double layers and study their properties. With the help of these potentials we reduce the boundary value problems to the integral Fredholm equations of the second kind and prove their unique solvability.

Keywords: multidimensional degenerating elliptic equation, method of potentials, interior and exterior Dirichlet and Neumann problems.

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English version:
Russian Mathematics (Izvestiya VUZ. Matematika), 2009, 53:8, 46–57

Bibliographic databases:

UDC: 517.956
Revised: 26.11.2008

Citation: F. G. Mukhlisov, A. M. Nigmetzyanova, “Solution of boundary value problems for a degenerating elliptic equation of the second kind by the method of potentials”, Izv. Vyssh. Uchebn. Zaved. Mat., 2009, no. 8, 57–70; Russian Math. (Iz. VUZ), 53:8 (2009), 46–57

Citation in format AMSBIB
\Bibitem{MukNig09} \by F.~G.~Mukhlisov, A.~M.~Nigmetzyanova \paper Solution of boundary value problems for a~degenerating elliptic equation of the second kind by the method of potentials \jour Izv. Vyssh. Uchebn. Zaved. Mat. \yr 2009 \issue 8 \pages 57--70 \mathnet{http://mi.mathnet.ru/ivm3056} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=2584259} \transl \jour Russian Math. (Iz. VUZ) \yr 2009 \vol 53 \issue 8 \pages 46--57 \crossref{https://doi.org/10.3103/S1066369X09080088} 

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Citing articles on Google Scholar: Russian citations, English citations
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This publication is cited in the following articles:
1. R. M. Askhatov, R. N. Abaidullin, “Reshenie osnovnykh kraevykh zadach dlya odnogo vyrozhdayuschegosya ellipticheskogo uravneniya metodom potentsialov”, Uchen. zap. Kazan. un-ta. Ser. Fiz.-matem. nauki, 157, no. 1, Izd-vo Kazanskogo un-ta, Kazan, 2015, 5–14
2. Askhatov R.M., Abaydullin R.N., “Applying the Potential Method to Solving Main Boundary-Value Problems For a Degenerate Elliptic Equation”, Lobachevskii J. Math., 37:3 (2016), 333–341
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