On problems with displacement in boundary conditions for two partial differential equations

In the present paper the system
$$ \sum_{i=0}^2\sum_{j=0}^ka_{ij}(x,y)D_x^iD_y^ju(x,y)=0,\quad k=1,2, $$
is investigated. The first equation of this system is the generalization of Aller's equation, and the second one is the generalization of Boussinesq–Love's equation. We consider the problems of finding the regular solutions of this system in the rectangle $D=\{x,y\in(0,1)\}$ by using the given linear relationships. The each of these relationships connect a values of unknown function in the boundary and the interior points of $D$ We obtain the sufficient conditions of existence of unique solutions of the considered problems in the terms of the coefficients of the above mentioned relationships.