We consider the homogeneous Dirichlet problem in the unit disk K ⊂ R 2 for a general typeless differential equation of any even order 2m, m ≥ 2, with constant complex coefficients whose characteristic equation has multiple roots ± i. For each value of multiplicity of the roots i and – i, we either formulate criteria of the nontrivial solvability of the problem or prove that the analyzed problem possesses solely the trivial solution. A similar result generalizes the well-known Bitsadze examples to the case of typeless equations of any even order.
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Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 62, No. 5, pp. 591–603, May, 2010.
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Buryachenko, E.A. Conditions of nontrivial solvability of the homogeneous Dirichlet problem for equations of any even order in the case of multiple characteristics without slope angles. Ukr Math J 62, 676–690 (2010). https://doi.org/10.1007/s11253-010-0380-0
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DOI: https://doi.org/10.1007/s11253-010-0380-0