Boundary-value problem with mixed conditions for linear typeless partial differential equations

Abstract

In the domain obtained as the Cartesian product of a segment $0 \leq t \leq T$ by a $p$-dimensional torus in variables $x_1, ..., x_p$, $p \geq 1$, we study the problem with mixed boundary conditions in the variable $t$ for general (no restrictions are imposed on the type) linear partial differential equations of high order with constant coefficients isotropic with respect to the order of differentiation for all independent variables. We establish conditions for the unique solvability of the problem in various functional spaces and construct its solution in the form of a series with respect to systems of orthogonal functions of the variables $x_1, ..., x_p$.