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

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.

The principal transmission condition

<abstract><p>The paper treats pseudodifferential operators $ P = \operatorname{Op}(p(\xi)) $ with homogeneous complex symbol $ p(\xi) $ of order $ 2a &gt; 0 $, generalizing the fractional Laplacian $ (-\Delta)^a $ but lacking its symmetries, and taken to act on the halfspace ${\mathbb R}^n_+$. The operators are seen to satisfy a principal $ \mu $-transmission condition relative to ${\mathbb R}^n_+$, but generally not the full $ \mu $-transmission condition satisfied by $ (-\Delta)^a $ and related operators (with $ \mu = a $). However, $ P $ acts well on the so-called $ \mu $-transmission spaces over ${\mathbb R}^n_+$ (defined in earlier works), and when $ P $ moreover is strongly elliptic, these spaces are the solution spaces for the homogeneous Dirichlet problem for $ P $, leading to regularity results with a factor $ x_n^\mu $ (in a limited range of Sobolev spaces). The information is then shown to be sufficient to establish an integration by parts formula over ${\mathbb R}^n_+$ for $ P $ acting on such functions. The formulation in Sobolev spaces, and the results on strongly elliptic operators going beyond certain operators with real kernels, are new. Furthermore, large solutions with nonzero Dirichlet traces are described, and a halfways Green's formula is established, as new results for these operators. Since the principal $ \mu $-transmission condition has weaker requirements than the full $ \mu $-transmission condition assumed in earlier papers, new arguments were needed, relying on work of Vishik and Eskin instead of the Boutet de Monvel theory. The results cover the case of nonsymmetric operators with real kernel that were only partially treated in a preceding paper.</p></abstract>

Eigencurves of the p(·)-Biharmonic operator with a Hardy-type term

This paper is devoted to the study of the homogeneous Dirichlet problem for a singular nonlinear equation which involves the p(·)-biharmonic operator and a Hardy-type term that depend on the solution and with a parameter λ. By using a variational approach and min-max argument based on Ljusternik-Schnirelmann theory on C1-manifolds [13], we prove that the considered problem admits at least one nondecreasing sequence of positive eigencurves with a characterization of the principal curve μ1(λ) and also show that, the smallest curve μ1(λ) is positive for all 0 ≤ λ < CH, with CH is the optimal constant of Hardy type inequality.

THE DIRICHLET PROBLEM FOR AN ORDINARY DIFFERENTIAL EQUATION OF THE SECOND ORDER WITH THE OPERATOR OF DISTRIBUTED DIFFERENTIATION

В работе исследуется линейное обыкновенное дифференциальное уравнение второго порядка с оператором непрерывно распределенного дифференцирования, и для него изучается двухточечная краевая задача методом функции Грина. Вводится в рассмотрение специальная функция, в терминах которой строится функция Грина задачи Дирехле и доказываются основные свойства. Определены достаточные условия на ядро оператора непрерывно распределенного дифференцирования, гарантирующие выполнения условия разрешимости задачи Дирихле. В случае, когда однородная задача Дирихле для рассматриваемого однородного уравнения имеет нетривиальное решение получено неравенство типа Ляпунова для ядра оператора непрерывно распределенного дифференцирования. In this paper, we study a linear ordinary differential equation of the second order with operator of continuously distributed differentiation, and for him we study the two-point boundary value problem by the Greens function method. A special function is introduced, in terms of which the Green function of the Direchle problem is constructed and the main properties are proved. Sufficient conditions on the kernel of the operator of continuously distributed differentiation are determined that guarantee the fulfillment of the solvability condition for the Dirichlet problem. In the case when the homogeneous Dirichlet problem for the homogeneous equation under consideration has a nontrivial solution, an analog of the Lyapunov inequality is obtained for the kernel of a continuously distributed ifferentiation operator.

LYAPUNOV INEQUALITY FOR AN EQUATION WITH FRACTIONAL DERIVATIVES WITH DIFFERENT ORIGINS

В работе рассмотрено обыкновенное дифференциальное уравнение дробного порядка, содержащее композицию дробных производных с различными началами, являющееся модельным уравнением движения во фрактальной среде. Для рассматриваемого уравнения найдено необходимое условие существования нетривиального решения однородной задачи Дирихле. Условие имеет форму интегральной оценки для потенциала и является аналогом неравенства Ляпунова We consider an ordinary differential equation of fractional order with the composition of left and rightsided fractional derivatives, which is a model equation of motion in fractal media. We find a necessary condition for existence of nontrivial solution of homogeneous Dirichlet problem for the equation under consideration. The condition has the form of integral estimate for the potential and is an analog of Lyapunov inequality.

Green-Rvachev's quasi-function method for constructing two-sided approximations to positive solution of nonlinear boundary value problems

A homogeneous Dirichlet problem for a semilinear elliptic equations with the Laplace operator and Helmholtz operator is investigated. To construct the two-sided approximations to a positive solution of this boundary value problem the transition to an equivalent nonlinear integral equation (with the help of the Green-Rvachev's quasi-function) with its subsequent analysis by methods of the theory of semi-ordered spaces is used. The work and efficiency of the developed method are demonstrated by a computational experiment for a test problem with exponential nonlinearity.