Formula of energy parametric derivative for variable linear unbounded operators with variable domains

F. E. Lomovtsev

Trudy Instituta matematiki, Tome 24 (2016) no. 1, pp. 75-94

Résumé

Formulated and proved the basics of differential calculus to the parameter variable of linear unbounded operators with variable domains. 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. It are introduced new concepts mutually conjugate, conjugate and closed sesquilinear forms, weak derivatives of entire order in the parameter, derived two formulas of weak first derivative to the parameter for operators defined by means of sesquilinear forms and of the operator form, and proposed the methods of it calculation. In the physical processes of the first and second derivatives are a speed and acceleration energy change. The results are illustrated by calculating the time weak derivatives of two non-stationary boundary value problems for differential operators of the second and third orders.