Boundary problems for nonclassical systems of equations of the second order

B. B. Oshorov

Sibirskij žurnal čistoj i prikladnoj matematiki, Tome 16 (2016) no. 2, pp. 68-78

Résumé

Boundary problems for systems of equations of the second order where leading part is the second degree of some operator of the first order are investigated. Methods of research are development of methods from works of the author where for the Bitsadze’s generalized systems of equations of elliptic type the nonclassical problem of Poincare is investigated. The nonclassical system of equations of the first order on the plane which has no certain type on classification Petrovsky is considered. For this system resolvability of a boundary problem in plane domain is proved. This problem is Riemann–Gilbert problem with discontinuous boundary conditions. The leading part of system of the second order is the second degree of the considered operator of the first order. Resolvability of the boundary problem for this system when boundary conditions are received by addition of new conditions to statements of the problem for system of the first order is proved. In three-dimensional space for Moisila–Teodoresku system the nonclassical boundary Riemann–Gilbert problem with discontinuous boundary conditions is solvable. The corresponding operator of the second order is Laplace operator for a four-dimensional vector function. It gives the possibility to give statements of both classical, and nonclassical boundary problems.

Détail

Keywords: system of partial differential equations, boundary problem, Bitsadze system, nonclassical boundary problem, Riemann–Gilbert problem, Poincare problem, nonclassical system, Moisila–Teodoresko system, generalized solution, prior estimate.