Some aspects of the nontrivial solvability of homogeneous Dirichlet problems for linear equations of arbitrary even order in the disk

Conditions of nontrivial solvability of the homogeneous Dirichlet problem for equations of any even order in the case of multiple characteristics without slope angles

“…Hence, the function u x ( ) given by relation (13) satisfies the first boundary condition in (2). By using the result established in [6], we conclude that the function u x ( ) given by relation (13) also satisfies the other boundary conditions of the problem because …”

“…In [6], the role of F y j ( ) is played by a function equal (to within a constant) to the (m -1)th primitive of the Chebyshev polynomial of the first kind of degree N = n -m + 1: …”

“…In [7,8], the results obtained in [3,6] are generalized to the case of general equations of any even order 2m, m ≥ 2, with multiple characteristics and simple roots of the characteristic equation equal to ± i. The relationship between the value of multiplicity of the roots unequal to ± i and the existence of nontrivial solutions of the corresponding Dirichlet problem in a disk is also established in these papers.…”

“…The problem of existence of nontrivial solutions of the Dirichlet problem for general equations of the principal type without multiple characteristics is solved and a criterion of existence of these solutions is established in [6].…”

“…At the same time, earlier, in almost all papers (see, e.g., [2][3][4]6]), similar criteria of the unique solvability of boundary-value problems were expressed just in terms of the slope angles of the characteristics ϕ j .…”