Coercive estimates, separability and coercive solvability of a nonlinear elliptic differential operator in a weight space

O. Kh. Karimov ; Z. Zh. Hakimova

Čebyševskij sbornik, Tome 24 (2023) no. 2, pp. 197-213

Résumé

The work is devoted to establishing coercive estimates and proofs of separability theorems for a nonlinear elliptic differential operator of non-divergence form in a weighted space. On the basis of the obtained coercive estimates, the coercive solvability of a nonlinear elliptic differential second-order operator in the space $L_{2,\rho}(R^n)$ is investigated. The problem of "separability of differential expressions" was first studied by mathematicians V.N.Everitt and M. Girtz. They studied in detail the separability of the Sturm-Liouville operator. Further development of this theory belongs to K.H.Boymatov, M. Otelbaev and their students. Most of the published works on this theory relate to linear operators. There are only some papers that consider nonlinear differential operators, which are weak nonlinear perturbations of linear operators. The case when the operator under study is nonlinear, i.e. it cannot be represented as a weak perturbation of a linear operator, is considered only in some separate papers. The results obtained here also relate to this little-studied case. In this work, the coercive properties of a non-divergence nonlinear elliptic differential operator are studied $$ L[u]=-\sum_{i,j=1}^na_{ij}(x)\frac{\partial^2 u}{\partial x_i\partial x_j}+V(x,u)u(x), $$ in the weight space $L_{2,\rho}(R^n)$ and on the basis of coercive estimates, its separability in this space is proved. Based on the separability of the considered elliptic operator of nondivergent form, we study the coercive solvability of a nonlinear elliptic differential equation in a weighted Hilbert space $L_{2,\rho}(R^n)$.

Détail

Keywords: Elliptic operator, non-divergent type of operator, coercive estimates, nonlinearity, separability, solvability, Hilbert space,weight space.