Boundary-value problems for an elliptic equation with complex coefficients and a certain moment problem

Burskii, V. P.

doi:10.1007/bf01060856

Cited by 11 publications

(9 citation statements)

References 2 publications

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“…We now formulate and prove the main result, which describes a relationship between properties of the moment problem (8) and the solvability of problem (4), (5).…”

Section: Remarkmentioning

confidence: 98%

On the dirichlet problem for an improperly elliptic equation

Burskii

Кириченко

2011

Ukr Math J

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517.95We consider the problem of solvability of an inhomogeneous Dirichlet problem for a scalar improperly elliptic differential equation with complex coefficients in a bounded domain. A model case where the unit disk is chosen as the domain and the equation does not contain lower terms is studied. We prove that the classes of Dirichlet data for which the problem has a unique solution in the Sobolev space are spaces of functions with exponentially decreasing Fourier coefficients.

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“…We now formulate and prove the main result, which describes a relationship between properties of the moment problem (8) and the solvability of problem (4), (5).…”

Section: Remarkmentioning

confidence: 98%

On the dirichlet problem for an improperly elliptic equation

Burskii

Кириченко

2011

Ukr Math J

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“…In [31,32], a condition of connection of traces of a solution of the Cauchy problem for second-order equations with constant complex coefficients in a bounded domain was obtained. We use this condition below.…”

Section: Connection With Other Boundary-value Problems and Generalizementioning

confidence: 99%

On Dirichlet problem for string equation, Poncelet problem, Pell-Abel equation, and some other related problems

Burskii

Zhedanov

2006

Ukr Math J

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UDC 517.95In a plane domain bounded by a biquadratic curve, we consider the problem of the uniqueness of a solution of the Dirichlet problem for the string equation. We show that this problem is equivalent to the classical Poncelet problem in projective geometry for two appropriate ellipses and also to the problem of the solvability of the Pell -Abel algebraic equation; some other related problems are also considered.In the present paper, we describe the relationships between properties of boundary-value problems for the string equation in a domain bounded by a biquadratic curve and some classical problems in algebra, geometry, and analysis that have recently been established by the authors. Boundary-Value Problems1.1. Historical Information. Investigations of ill-posed boundary-value problems date back to Hadamard [1], who first noted that the homogeneous Dirichlet problem for the string equation in a certain rectangle admits a nontrivial solution. The investigations carried out by Huber [2] and then by Bourgin and Duffin [3] resulted in the establishment of a condition for the violation of the uniqueness of a solution of the homogenous Dirichlet problem for the equation u tt -u xx = 0 in the rectangle { 0 ≤ t ≤ T; 0 ≤ x ≤ X }. It turned out that the uniqueness of a solution of this problem in classical spaces is violated if and only if the ratio T / X is rational. In the papers indicated, the method of Fourier expansion of a solution was used. In the works of Ptashnik and his disciples, this method was used for the investigation of properties of various boundary-value problems for partial differential equations in a rectangle and a parallelepiped [4]. In [5], John studied the problem of the violation of the uniqueness of a solution of the homogeneous Dirichlet problem for the string equation in a general bounded plane domain convex with respect to both families of characteristics in connection with a certain mapping of the boundary of the domain into itself (a so-called characteristic billiard, see below) used earlier in the mentioned works of Hadamard and Huber. The investigation of this relationship was continued by Aleksandryan and his disciples in the case of the Sobolev problem of oscillations of the surface of a liquid filling a cavity in a flying body [6 -8]. In [9], Arnol'd showed the relationship between this Dirichlet problem in an ellipse and the problem of small denominators in the context of the influence of the rate of approximation of a certain number associated with the problem by rational numbers on the smoothness of a solution. Some questions related to these problems were studied by Berezanskii [10], Zelenyak [11], Fokin [12], etc.Let us describe a characteristic billiard. Consider the Dirichlet problem for the string equation

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“…Note that condition (2.17) is the starting point for the study of the properties of boundary-value problems for specific equations [4,6,8,13,14,15,16].…”

Section: Spaces Of Associated Traces and The Boundary Operatormentioning

confidence: 99%

“…In the plane, the uniqueness of the solution of the Dirichlet problem for second-order hyperbolic equations on domains with a generic boundary has been studied (see the survey [37]). For other possible topics in the study of boundary-value problems, in particular, when condition (2.15) holds, see [4,6,7,8,12,13].…”

Section: Introductionmentioning

confidence: 99%