# Burskii V. P.

### On the Third Boundary-Value Problem for an Improperly Elliptic Equation in a Disk

Ukr. Mat. Zh. - 2014. - 66, № 2. - pp. 279–283

We study the problem of solvability of the inhomogeneous third boundary-value problem in a bounded domain for a scalar improperly elliptic differential equation with complex coefficients and homogeneous symbol. It is shown that this problem has a unique solution in the Sobolev space over the circle for special classes of boundary data from the spaces of functions with exponentially decreasing Fourier coefficients.

### Neumann problem and one oblique-derivative problem for an improperly elliptic equation

Ukr. Mat. Zh. - 2012. - 64, № 4. - pp. 451-462

We investigate the solvability of an inhomogeneous Neumann problem and oblique-derivative problem for an improperly elliptic scalar differential equation with complex coefficients in a bounded domain. The model case where the domain is the unit disk and the equation does not have lower-order terms is studied. It is proved that the classes of boundary data for which the problems have unique solutions in a Sobolev space are the spaces of functions with exponentially decreasing Fourier coefficients.

### On the dirichlet problem for an improperly elliptic equation

Burskii V. P., Kirichenko E. V.

Ukr. Mat. Zh. - 2011. - 63, № 2. - pp. 156-164

The solvability of the inhomogeneous Dirichlet problem in a bounded domain for scalar improperly elliptic differential equation with complex coefficients is investigated. We study a model case where the unit disk is chosen as a domain and the equation does not contain lowest terms. We prove that the problem has a unique solution in the Sobolev space for special classes of Dirichlet data that are spaces of functions with exponential decrease of the Fourier coefficients.

### Conditions of regularity of a general differential boundary-value problem for improperly elliptic equations

Ukr. Mat. Zh. - 2010. - 62, № 6. - pp. 754–761

The Fredholm property and well-posedness of a general differential boundary-value problem for a general improperly elliptic equation are analyzed in a two-dimensional bounded domain with smooth boundary.

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

Ukr. Mat. Zh. - 2006. - 58, № 4. - pp. 435–450

In 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.

### On the Spectrum of an Equivariant Extension of the Laplace Operator in a Ball

Ukr. Mat. Zh. - 2000. - 52, № 11. - pp. 1473-1483

We study the relationship between the well-posedness of an equivariant problem for the Poisson equation in a ball and the spectrum of the operator generated by it.

### On equivariant extensions of a differential operator by the example of the Laplace operator in a circle

Ukr. Mat. Zh. - 1999. - 51, № 2. - pp. 158–169

We propose a method for investigation of both correctness of the equivariant problem and the spectrum of the corresponding operator.

### On boundary-value problems for a second-order differential equation with complex coefficients in a plane domain

Ukr. Mat. Zh. - 1996. - 48, № 11. - pp. 1457-1467

We study boundary-value problems for a homogeneous partial differential equation of the second order with arbitrary constant complex coefficients and a homogeneous symbol in a bounded domain with smooth boundary. Necessary and sufficient conditions for the solvability of the Cauchy problem are obtained. These conditions are written in the form of a moment problem on the boundary of the domain and applied to the investigation of boundary-value problems. This moment problem is solved in the case of a disk.

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

Ukr. Mat. Zh. - 1993. - 45, № 11. - pp. 1476–1483

Elliptic systems of two second-order equations, which can be written as a single equation with complex coefficients and a homogeneous operator, are studied. The necessary and sufficient conditions for the connection of traces of a solution are obtained for an arbitrary bounded domain with a smooth boundary. These conditions are formulated in the form of a certain moment problem on the boundary of a domain; they are applied to the study of boundary-value problems. In particular, it is shown that the Dirichlet problem and the Neumann problem are solvable only together. In the case where the domain is a disk, the indicated moment problem is solved together with the Dirichlet problem and the Neumann problem. The third boundary-value problem in a disk is also investigated.

### On the uniqueness of solutions to some boundary-value problems for differential equations in a domain with algebraic boundary

Ukr. Mat. Zh. - 1993. - 45, № 7. - pp. 898–906

Classes of differential equations with constant coefficients admitting unique solutions of Dirichlet and Cauchy boundary-value problems are considered in a bounded domain with algebraic boundary. For the Dirichlet problem in a ball, the necessary and sufficient conditions for the uniqueness of the solution are obtained in the form of a countable sequence of inequalities polynomial in the coefficients of the equation.

### A commutative diagram connected with a differential operator in a domain

Ukr. Mat. Zh. - 1991. - 43, № 12. - pp. 1703–1709