Conditions of nontrivial solvability of the homogeneous Dirichlet problem for equations of any even order in the case of multiple characteristics without slope angles
- 29 Downloads
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.
KeywordsCharacteristic Equation Dirichlet Problem Slope Angle Nontrivial Solution Trivial Solution
Unable to display preview. Download preview PDF.
- 2.V. P. Burskii, Methods for the Investigation of Boundary-Value Problems for General Differential Equations [in Russian], Naukova Dumka, Kiev (2002).Google Scholar
- 6.V. P. Burskii and E. A. Buryachenko, “Some problems of nontrivial solvability of the homogeneous Dirichlet problem for linear equations of any even order in the disk,” Mat. Zametki, 74, No. 4, 1032–1043 (2005).Google Scholar
- 8.E. A. Buryachenko, “Nontrivial solvability of the homogeneous Dirichlet problem in a disk for equations of order 2m in the case of characteristics without slope angles,” Visn. Donets. Nats. Univ., Ser. Fiz.-Mat. Nauk, 50, No. 2, 10–21 (2007).Google Scholar
- 9.I. N. Vekua, New Methods for the Solution of Elliptic Equations [in Russian], OGIZ, Moscow (1948).Google Scholar