N. A. Zhura’s research while affiliated with P.N. Lebedev Physical Institute of the Russian Academy of Sciences and other places

We study a boundary value problem of the Riemann-Hilbert type for strictly hyperbolic first-order systems with constant coefficients and without lower-order terms in bounded domains of a special shape on the plane. Sufficient conditions for the unique solvability of this problem in weighted function classes are obtained.

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.

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.

В работе рассматривается строго гиперболическая система первого порядка с постоянными коэффициентами, состоящая из трех уравнений, в ограниченной кусочно-гладкой области. Предполагается, что граница этой области составлена из шести гладких нехарактеристических дуг. В этой области ставится краевая задача по заданным попеременно на этих дугах одного или двух линейных соотношений искомого решения. Показано, что при некоторых дополнительных условиях на коэффициенты этих соотношений, границу области и характер поведения решения вблизи характеристик, проходящих через угловые точки области, эта задача однозначно разрешима. Библиография: 16 наименований.

We obtain sufficient conditions for the fundamental Faddeev–Marchenko theorem to be true. In addition, we derive a representation of the solution of the inverse Sturm–Liouville problem on the entire line on the basis of the solution of a boundary value problem for the Jost functions and the corresponding singular integral equation.

We examine a piecewise analytic function that is defined in sectors of a disk whose real and imaginary parts obey contact conditions on adjacent boundary parts. Under the assumption of the power behavior, the sharp asymptotics of this function is established at the center of the disk.