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

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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. 
@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 = {http://geodesic.mathdoc.fr/item/IVM_2009_8_a7/}
}
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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. http://geodesic.mathdoc.fr/item/IVM_2009_8_a7/