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A boundary-value problem for a first-order hyperbolic system in a two-dimensional domain
Abstract
We consider a strictly hyperbolic first-order system of three equations with constant coefficients in a bounded piecewise-smooth domain. The boundary of the domain is assumed to consist of six smooth noncharacteristic arcs. A boundary-value problem in this domain is posed by alternately prescribing one or two linear combinations of the components of the solution on these arcs. We show that this problem has a unique solution under certain additional conditions on the coefficients of these combinations, the boundary of the domain and the behaviour of the solution near the characteristics passing through the corner points of the domain. © 2017 Russian Academy of Sciences (DoM), London Mathematical Society, Turpion Ltd.
... In this domain, it is natural to state the problem where n 1 and n 2 linear combinations of solutions to the system are given on alternating arcs respectively, where n 1 + n 2 = 4. Thus, the following three cases can occur: n 1 = 0 (the Cauchy problem), n 1 = 1, and n 1 = 2. The first two cases can be treated in the same way as in [4]. The remaining case n 1 = n 2 = 2 corresponds to a Dirichlet type problem and is studied in this paper. ...
... for the vector u = (u 1 , u 2 , u 3 , u 4 ), where the real constant (4 × 4)-matrix A admits only pairwise distinct real eigenvalues ν 1 , ν 2 , . . . , ν 4 . The characteristics of this system are segments or lines parallel to the lines l i : x 1 + ν i x 2 = 0, i = 1, 2, . . . ...
... In other words, the small diagonals of the octagon G are characteristics. The further construction is similar to that done in [4]. ...
We consider a first order strictly hyperbolic system of four equations with constant coefficients in a bounded domain with piecewise boundary consisting of eight smooth noncharacteristic arcs. In this domain, we consider boundary value problems with two linear relations between components of the solution and show show that these problems are uniquely solvable under certain assumptions.
... In the case of finite domains, there is no systematic theory of boundary value problems for hyperbolic type equations with many characteristics. The particular case of three characteristics was considered in [3]. However, even in this case, a correct statement of boundary valued problems requires certain conditions on the domain D where the solution is looked for. ...
... It is obvious that δ k is involutive, i.e., δ k [δ k (t)] = t, t ∈ Γ, fixed points are the singular points τ 0 k , τ 1 k , and Γ 0 k goes to Γ 1 k under the transformation δ k . In the case of three characteristics, the statement of a boundary value problem proposed in [3] suggests that the contour Γ is divided into two curves Γ ± such that each involution δ k is a homeomorphism from Γ + onto Γ − . Then the boundary value problem consists of two linear combinations of components of the solution u on Γ + and one combination on Γ − . ...
We consider a first order strictly hyperbolic system of n equations with constant coefficients in a bounded domain. It is assumed that the domain is strictly convex relative to characteristics, so that the projection along each characteristic is an involution having two fixed singular points. The natural statement of boundary value problems for such systems requires that singular points go to singular points under such transformations. We present a necessary and sufficient condition for the existence of such domains, called characteristically closed.
A mixed boundary value problem with arbitrary continuous, not necessarily satisfying boundary conditions, functions in initial conditions and inhomogeneities of the equation is solved. A method is proposed for finding a generalized solution by a modification of the interpolation operators of functions constructed from solutions of Cauchy problems with second-order differential expression. Methods of finding the Fourier coefficients of auxiliary functions using the Stieltjes integral or the resolvent of the third-order Cauchy differential operator are proposed.
Решена смешанная краевая задача с произвольными непрерывными, необязательно удовлетворяющими граничным условиям, функциями в начальных условиях и неоднородности уравнения. Предложен метод нахождения обобщенного решения с помощью модификации операторов интерполирования функций, построенных с помощью решений задач Коши с дифференциальным выражением второго порядка. Найдены способы нахождения коэффициентов Фурье вспомогательных функций с помощью интеграла Стилтьеса или резольвенты дифференциального оператора Коши третьего порядка. Библиография: 39 наименований.
We consider the Dirichlet problem for the linear non-elliptic fourth order partial differential equation in the unit disk. It supposed that in the equation only fourth order terms and coefficients are constant. The solvability conditions of in-homogeneous problem and the solutions of the corresponding homogeneous problem are determined in explicit form. The solutions are obtained in the form of expansions by Chebyshev polynomials.
The goal of this note is to consider the application of the complex interpolation space of Morrey spaces. Actually, the boundedness of Riesz potentials acting on Morrey spaces, which is obtained by Adams, is refined by means of the complex interpolation.
In this paper, we study an inhomogeneous Hilbert boundary value problem with a finite index and a boundary condition on a circle for a generalized Cauchy-Riemann equation with a singular coefficient. To solve this problem, we conducted a complete study of the solvability of the Hilbert boundary value problem of the theory of analytic functions with an infinite index due to a finite number of points of a special type of vorticity. Based on these results, we have derived a formula for the general solution and studied the existence and number of solutions to the boundary value problem of the theory of generalized analytic functions.
The Dirichlet problem for sixth order improperly elliptic equation is considered. The functional class of boundary functions, where this problem is normally solvable is determined. If the roots of the characteristic equation satisfy some conditions, the number of linearly independent solutions of homogeneous problem and the number of linearly independent solvability conditions of in-homogeneous problem are obtained. Solutions of homogeneous problem and solvability conditions of in-homogeneous problem are obtained in explicit form.
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