Geodesic
Solution of boundary value problems for a~degenerating elliptic equation of the second kind by the method of potentials
Izvestiâ vysših učebnyh zavedenij. Matematika, no. 8 (2009), pp. 57-70.

Voir la notice de l'article provenant de la source Math-Net.Ru

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.
Mots-clés : multidimensional degenerating elliptic equation, method of potentials, interior and exterior Dirichlet and Neumann problems. 
@article{IVM_2009_8_a7,
     author = {F. G. Mukhlisov and A. M. Nigmetzyanova},
     title = {Solution of boundary value problems for a~degenerating elliptic equation of the second kind by the method of potentials},
     journal = {Izvesti\^a vys\v{s}ih u\v{c}ebnyh zavedenij. Matematika},
     pages = {57--70},
     publisher = {mathdoc},
     number = {8},
     year = {2009},
     language = {ru},
     url = {https://geodesic-test.mathdoc.fr/item/IVM_2009_8_a7/}
}
TY  - JOUR
AU  - F. G. Mukhlisov
AU  - A. M. Nigmetzyanova
TI  - Solution of boundary value problems for a~degenerating elliptic equation of the second kind by the method of potentials
JO  - Izvestiâ vysših učebnyh zavedenij. Matematika
PY  - 2009
SP  - 57
EP  - 70
IS  - 8
PB  - mathdoc
UR  - https://geodesic-test.mathdoc.fr/item/IVM_2009_8_a7/
LA  - ru
ID  - IVM_2009_8_a7
ER  - 
%0 Journal Article
%A F. G. Mukhlisov
%A A. M. Nigmetzyanova
%T Solution of boundary value problems for a~degenerating elliptic equation of the second kind by the method of potentials
%J Izvestiâ vysših učebnyh zavedenij. Matematika
%D 2009
%P 57-70
%N 8
%I mathdoc
%U https://geodesic-test.mathdoc.fr/item/IVM_2009_8_a7/
%G ru
%F IVM_2009_8_a7
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. Izvestiâ vysših učebnyh zavedenij. Matematika, no. 8 (2009), pp. 57-70. https://geodesic-test.mathdoc.fr/item/IVM_2009_8_a7/

[1] Smirnov M. M., Vyrozhdayuschiesya ellipticheskie i giperbolicheskie uravneniya, Nauka, M., 1966, 292 pp. | MR

[2] Keldysh M. V., “O nekotorykh sluchayakh vyrozhdeniya uravnenii ellipticheskogo tipa na granitse oblasti”, DAN SSSR, 77:2 (1951), 181–183

[3] Bitsadze A. V., Uravneniya smeshannogo tipa, Izd-vo AN SSSR, M., 1959, 448 pp.

[4] Sabitov K. B., “O postanovke kraevykh zadach dlya uravneniya smeshannogo tipa s vyrozhdeniem vtorogo roda na granitse beskonechnoi oblasti”, Sib. matem. zhurn., 21:4 (1980), 146–150 | MR | Zbl

[5] Sabitov K. B., “Zadacha tipa Trikomi dlya uravneniya smeshannogo tipa s silnym kharakteristicheskim vyrozhdeniem”, Differents. uravneniya, 20:1 (1984), 333–337 | MR | Zbl

[6] Nigmedzyanova A. M., “O fundamentalnom reshenii odnogo vyrozhdayuschegosya ellipticheskogo uravneniya”, Tr. 2-i Vserossiiskoi nauch. konf. Matem. modelirovanie i kraevye zadachi, Ch. 3, SamGTU, Samara, 2005, 180–182

[7] Vatson G. N., Teoriya besselevykh funktsii, Ch. 1, In. lit., M., 1949, 798 pp.

[8] Mikhlin S. G., Lineinye uravneniya v chastnykh proizvodnykh, Vyssh. shkola, M., 1977, 432 pp. | MR