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

Abstract

The nontrivial solvability of the homogeneous Dirichlet problem in a unit disk \( K\subset {{\mathbb{R}}^2} \) in positive Sobolev spaces is studied for a typeless differential equation of arbitrary even order 2m, m ≥ 2, with constant complex-valued coefficients and homogeneous symbol. The detailed proofs of the criteria of nontrivial solvability of the problem are given in various cases that form a complete picture. The example considered by A. V. Bitsadze is generalized for the equations of arbitrary even order 2m, m ≥ 2.