Nonselfadjoint elliptic problems with the polynomial property in domains possessing cylindrical outlets to infinity

Solvability of mixed boundary value problems in domains with cylindrical outlets to infinity is investigated for a class of nonselfadjoint differential operator-matrices. The structure of these matrices is such that the corresponding asymmetric quadratic forms possess the polynomial property, i.e., they degenerate only on finite-dimensional lineals of vector polynomials. With the help of elementary algebraic operations, this property provides to indicate the attributes of boundary value problems, namely, to calculate the operator index, to describe the kernel and co-kernel of the operator, to find out the asymptotic behavior of solutions at infinity, etc.