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Chebyshevskii Sbornik, 2019, Volume 20, Issue 4, Pages 170–187
DOI: https://doi.org/10.22405/2226-8383-2018-20-4-170-187
(Mi cheb843)
 

Оn separability and coercive solvability of second-order nonlinear differential equations in the weight space

O. Kh. Karimov

Dzhuraev Institute of Mathematics, Academy of Sciences of Republic of Tajikistan, Dushanbe
References:
Abstract: The work focuses on obtaining coercive estimates and separability theorems of second-order nonlinear differential operators. Based on the obtained coercive estimates, the coercive solvability of the second-order nonlinear differential equations in the space $L_{2,\rho}(R^n)$ is investigated. For the first time the problem of the differential operators separability was dealt with by the English mathematicians V.N.Everitt and M.Girz. They studied in details the separability of the Sturm-Liouville operator and its degrees. Further development of this theory belongs to K.H.Boimatov, M.Otelbayev and their students. The main part of the published works on this theory applies to linear operators. There are only individual works that consider nonlinear differential operators, which are a weak nonlinear perturbations of linear operators. The case where the operator under study is strictly nonlinear, that is, it cannot be represented as a weak perturbation of the linear operator, is considered only in some individual separate works. The results obtained in this work also refer to this insufficiently studied case. The paper examined the coercive properties of a second-order nonlinear differential operator in the Hilbert space $L_{2,\rho}(R^n)$
$$ L[u]=-\sum_{i,j=1}^na_{ij}(x)\frac{\partial^2 u}{\partial x_i\partial x_j}+\sum_{j=1}^n b_{j}(x)\frac{\partial u}{\partial x_j}+V(x,u)u(x), $$
and on the basis of coercive estimates, its separability in this space has been proved. The operator under study is not a weak perturbation of the linear operator, i.e. is strictly nonlinear. Based on obtained coercive estimates and separability, solvability of nonlinear differential equation in the space $L_{2,\rho}(R^n)$ is investigated.
Keywords: Differential operator, coercive estimates, nonlinearity, separability, solvability, Hilbert space, weight space.
Received: 04.10.2019
Accepted: 20.12.2019
Document Type: Article
UDC: 517.948
Language: Russian
Citation: O. Kh. Karimov, “Оn separability and coercive solvability of second-order nonlinear differential equations in the weight space”, Chebyshevskii Sb., 20:4 (2019), 170–187
Citation in format AMSBIB
\Bibitem{Kar19}
\by O.~Kh.~Karimov
\paper Оn separability and coercive solvability of second-order nonlinear differential equations in the weight space
\jour Chebyshevskii Sb.
\yr 2019
\vol 20
\issue 4
\pages 170--187
\mathnet{http://mi.mathnet.ru/cheb843}
\crossref{https://doi.org/10.22405/2226-8383-2018-20-4-170-187}
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