This article is cited in 4 scientific papers (total in 4 papers)
Mathematics
On the smoothness of the solution of a nonlocal boundary value problem for the multidimensional second-order equation of the mixed type of the second kind in Sobolev space
Abstract:
In this paper we prove the unique solvability and smoothness of the solution of a nonlocal boundary-value problem for a multidimensional mixed type second-order equation of the second kind in Sobolev space Wℓ2(Q)Wℓ2(Q), (2⩽ℓ is an integer). First, we have studied the unique solvability of the problems in the space W22(Q). Solution uniqueness for a nonlocal boundary-value problem for a mixed-type equation of the second kind is proved by the methods of a priori estimates.Further, to prove the solution existence in the space W22(Q), the Fourier method is used. In other words, the problem under consideration is reduced to the study of unique solvability of a nonlocal boundary value problem for an infinite number of systems of second-order equations of mixed type of the second kind. For the unique solvability of the problems obtained, the "ε-regularization" method is used, i.e, the unique solvability of a nonlocal boundary-value problem for an infinite number of systems of composite-type equations with a small parameter was studied by the methods of functional analysis. The necessary a priori estimates were obtained for the problems under consideration. Basing on these estimates and using the theorem on weak compactness as well as the limit transition, solutions for an infinite number of systems of second-order equations of mixed type of the second kind are obtained. Then, using Steklov-Parseval equality for solving an infinite number of systems of second-order equations of mixed type of the second kind, the unique solvability of original problem was obtained. At the end of the paper, the smoothness of the problem's solution is studied
Keywords:multidimensional second-order equation of the mixed type of the second kind, Sobolev space, smoothness of the solution of the boundary problem, nonlocal boundary problem.
Citation:
S. Z. Djamalov, “On the smoothness of the solution of a nonlocal boundary value problem for the multidimensional second-order equation of the mixed type of the second kind in Sobolev space”, Zhurnal SVMO, 21:1 (2019), 24–33
\Bibitem{Dja19}
\by S.~Z.~Djamalov
\paper On the smoothness of the solution of a nonlocal boundary value problem for the multidimensional second-order equation of the mixed type of the second kind in Sobolev space
\jour Zhurnal SVMO
\yr 2019
\vol 21
\issue 1
\pages 24--33
\mathnet{http://mi.mathnet.ru/svmo724}
\crossref{https://doi.org/10.15507/2079-6900.21.201901.24-33}
Linking options:
https://www.mathnet.ru/eng/svmo724
https://www.mathnet.ru/eng/svmo/v21/i1/p24
This publication is cited in the following 4 articles:
Sirojiddin Dzhamalov, Biybinaz Sipatdinova, PROBLEMS IN THE TEXTILE AND LIGHT INDUSTRY IN THE CONTEXT OF INTEGRATION OF SCIENCE AND INDUSTRY AND WAYS TO SOLVE THEM: PTLICISIWS-2, 3045, PROBLEMS IN THE TEXTILE AND LIGHT INDUSTRY IN THE CONTEXT OF INTEGRATION OF SCIENCE AND INDUSTRY AND WAYS TO SOLVE THEM: PTLICISIWS-2, 2024, 040003
S. Z. Dzhamalov, B. K. Sipatdinova, “Ob odnoi nelokalnoi kraevoi zadache periodicheskogo tipa dlya trekhmernogo uravneniya smeshannogo tipa vtorogo roda v neogranichennom parallelepipede”, Vestnik KRAUNTs. Fiz.-mat. nauki, 42:1 (2023), 58–68
S. G. Pyatkov, “Boundary value and inverse problems for some classes of nonclassical operator-differential equations”, Siberian Math. J., 62:3 (2021), 489–502
S. Z. Djamalov, S. G. Pyatkov, “On some boundary value problems for multidimensional higher order equations of mixed type”, Siberian Math. J., 61:4 (2020), 610–625