scholarly journals ON THE UNIQUENESS VIOLATION OF THE DIRICHLET PROBLEM SOLUTION FOR SECOND-ORDER SYSTEMS

2021 ◽  
Vol 3 (1) ◽  
pp. 83-94
Author(s):  
V. Kyrychenko ◽  
◽  
Ye. Lesina ◽  
◽  

The study of the issues of the correct posedness of boundary value problems for differential equations and systems occupies an important place in modern research. When considering correctness, the question of unique solvability of this problem is of paramount importance. In particular, the problem of violation of the uniqueness of the solution of boundary value problems for general differential equations in bounded domains with algebraic boundary is of interest. The property of nontrivial solvability of the homogeneous Dirichlet problem for incorrectly elliptic equations of the second order was first pointed out by A. V. Bitsadze, having constructed an example of an equation with constant complex obtained a condition for the violation of the uniqueness of the solution to the Dirichlet problem in the unit disc for a hyperbolic equation in the case when the slope angles of the characteristics differ in sign. V. P. Burskii, considering the homogeneous Dirichlet problem in the unit disc for second-order equations with constant complex coefficients and a homogeneous non-degenerate symbol, obtained a criterion for nontrivial solvability in the form of π-irrationality of the angle between the characteristics. In this paper, we investigate the question of violation of the uniqueness of the solution of the homogeneous Dirichlet problem for a system of typeless second-order partial differential equations in a model domain – a circle. The original system is written in the form of an equation with commuting matrix coefficients. The permutability condition allows one to obtain a necessary and sufficient condition for the nontrivial solvability of the problem under consideration in the form of equality to zero of the determinant, the elements of which are expressed in terms of the coefficients of the equation. This form of writing the criterion allows one to construct examples of systems for which the kernel of the Dirichlet problem is nontrivial and infinite-dimensional. The study was based on the integral condition for the connection of associated boundary L-traces, as well as a functional scheme, the application of which reduces the expansion of a matrix function in a Fourier series to a standard expansion of each of its elements. A theorem of nontrivial solvability of the homogeneous Dirichlet problem is proved.

2010 ◽  
Vol 2010 ◽  
pp. 1-15
Author(s):  
Piao-Piao Shi ◽  
Wen-Xia Wang

We investigate the infinite boundary value problems for second-order impulsive differential equations with supremum by establishing a new comparison result and using the lower and upper solution method, and obtain the existence results for their maximal and minimal solutions.


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