Unique solvability of the Dirichlet problem for an ultrahyperbolic equation in a ball

We obtain a criterion in terms of zeros of Jacobi polynomials for the uniqueness of the solution of the first boundary value problem for an ultrahyperbolic equation in a ball. The nonuniqueness in the Dirichlet problem proves to occur if and only if the coefficient of the equation belongs to a countable dense subset of the real line.     

On a Kipriyanov problem for a singular ultrahyperbolic equation

We obtain a differential equation for the spherical means generated by a multidimensional generalized shift of an arbitrary smooth “even” function. We study the Asgeirsson property of solutions of a singular ultrahyperbolic equation that includes singular differential operators Δ _B acting in Euclidean spaces, in general, of distinct dimensions. We represent the structure of a “radial” solution of the considered equation. A theorem similar to the Asgeirsson inverse theorem is proved.     

Uniqueness in an integral geometry problem and an inverse problem for the kinetic equation

In this paper, we discuss the uniqueness in an integral geometry problem along the straight lines in a strongly convex domain. Our problem is related with the problem of finding a Riemannian metric by the distances between all pairs of the boundary points. For the proof, the problem is reduced to an inverse source problem for a kinetic equation and then the uniqueness theorem is proved using the tools of Fourier analysis.     

On a certain problem of integral geometry

Translated from Teoriya Funktsii, Funktsional'nyi Analiz i Ikh Prilozheniya, No. 53, pp. 95–99, 1990.     

On an integral equation for the two-dimensional exterior Stokes problem

This paper is concerned with solving the two-dimensional exterior Stokes problem by a combination of double- and simple-layer potentials. This leads to a boundary integral equation, which is shown to be uniquely solvable. The present approach is a variation of a technique due to Leis, Brakhage and Werner, and Panich for treating the scalar Helmholtz equation. However, it contains extra parameter which may be chosen in an optimal manner from a computational point of view. Some numerical results in this connection are also included.     

On the Dirichlet problem for a nonlinear elastic beam equation

Using a direct variational approach with no global growth conditions on the nonlinear term, we consider the existence of solutions and their dependence on a functional parameter for the fourth order Dirichlet problem connected with the elastic beam equation. We investigate also the existence of an optimal process for such an optimal control problem in which the dynamics is described by the beam equation.     

On an integral equation of the problem for an elastic strip with a slit

A new singular integral equation is obtained that describes the elastic equilibrium of a strip with both an inner and an edge slit (crack) and has a considerable advantage over existing equations /1–9/, etc.) from the viewpoint of a numerical realization and clarification of the analytical relationship with an analogous equation for a half-plane. Numerical results are given of a computation of the stress intensity coefficients at the tips of the inner and edge cracks that refine data in the literature.     

On the condition number of boundary integral operators for the exterior Dirichlet problem for the Helmholtz equation

Summary Brakhage and Werner, Leis and Panich suggested to reduce the exterior Dirichlet boundary value problem for the Helmholtz equation to an integral equation of the second kind which is uniquely solvable for all frequencies by seeking the solution in the form of a combined double- and single-layer potential. We present an analysis of the appropriate choice of the parameter coupling the double- and single-layer potential in order to minimize the condition number of the integral operator.This research was carried out while the second author was visiting the University of Göttingen on a DAAD-stipendium     

On an integral equation approach for the exterior Robin problem for the Helmholtz equation

A boundary integral equation for the exterior Robin problem for Helmholtz's equation is analyzed in this paper. This integral operator is not compact. A proof based on a suitable regularization of this integral operator and the Fredholm alternative for the regularized compact operator was given by other authors. In this paper, we will give a direct existence and uniqueness proof for the boundary non-compact integral equation in the space settings C^1,λ(S) and C^0,λ(S), where S is a closed bounded smooth surface.     

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

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. __________ Translated from Ukrains’kyi Matematychnyi Zhurnal, Vol. 58, No. 4, pp. 435–450, April, 2006.