On the breakdown of the uniqueness of a solution of the Dirichlet problem for typeless differential equations of arbitrary even order in a disk

Maximum principle for the weak solutions of the Cauchy problem for the fourth‐order hyperbolic equations

“…Definition 3 Minimum operator. [6]. Let us consider the differential operation (3) on functions from the space 𝐶 ∞ 0 (Ω).…”

On the dimension of the kernel of the Dirichlet problem for fourth-order equations

“…In [11], Theorem 1 was generalized to the case of equations of arbitrary even order 2m, m ≥ 2. However, the problem on the dimension of the kernel of the Dirichlet problem (1), (2) remained open; i.e., the following problems were not studied.…”

“…linearly independent polynomial solutions of the form (11). Here A n is the matrix occurring in relation (9), and the numbers μ j are related to the roots λ j of the characteristic equation by formulas (8).…”

“…A criterion for the kernel of the Dirichlet problem to be nontrivial was obtained in [11] for typeless equations of arbitrary even order 2m whose characteristic polynomial has roots ±i. Let us state this assertion for fourth-order equations, i.e., for m = 2. be satisfied for some n ∈ N, n > 3.…”

“…For higher-order equations (of order ≥ 4), one can single out classes of improperly elliptic equations for which the kernel of the Dirichlet problem has finite nonzero dimension. In [9][10][11], the uniqueness of the solution of the Dirichlet problem in the disk was studied for general fourth-order equations, including those with degenerate symbol; however, the dimension of the kernel was not studied there. The present paper deals with this problem.…”