Differentiation in the parameter of the strong extensionsfor variable linear unbounded operators with variable domains

F. E. Lomovtsev

Trudy Instituta matematiki, Tome 25 (2017) no. 2, pp. 29-49

Résumé

It is introduced the concepts of the strong continuity with respect to the parameter variable of strong extensions of variable linear unbounded operators with variable domains and weak derivatives of higher order parameter of these strong extensions, and from the strong expansions of their inverse operators. It is introduced and is also used in the more general concepts of weak derivatives with respect variable, both on themselves of variable linear unbounded operators with variable domains and inverse operators to them and from their strong extensions. These concepts for strong extensions of operators in line with the concepts introduced earlier by the author for themselves operators. We obtain two basic formulas for the first weak derivative to the parameter variable from the strong extensions of the operators defined by sesquilinear forms and by explicit form of the operator. Its are need for the investigation of the Hadamard's correct solvability of linear boundary value problems for operator-differential equations with variable domains of unbounded operator coefficients and linear mixed problems for non-stationary (time-dependent) partial differential equations with non-stationary boundary conditions. The second formula for first weak derivative to the parameter from the strong extensions of variable linear unbounded operators with variable domains generalize the well-known formula of the first strong derivative to the parameter of the variable linear unbounded operators with a constant domain. Weak derivatives to parameter variable of higher orders from the strong extensions are constructed sequentially recursively. It is proved that if the weak derivatives of higher order with respect to the parameter variable from strong extensions of operators exist, they are powerful extensions of weak derivatives the orders with respect to the parameter from the operators themselves. The results are illustrated by calculating the weak derivatives of higher orders on time from strong extensions of the non-stationary third boundary value problem for second order differential operator.