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

O. Kh. Karimov

Čebyševskij sbornik, Tome 20 (2019) no. 4, pp. 170-187

Résumé

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.

Détail

Keywords: Differential operator, coercive estimates, nonlinearity, separability, solvability, Hilbert space, weight space.