On boundary uniqueness theorems for a linear elliptic equation with constant coefficients

For the solutions of an elliptic equation with constant coefficients, we prove uniqueness theorems that generalize the classical boundary uniqueness theorems for analytic functions.     

Uniqueness theorem for linear elliptic equation of the second order with constant coefficients

The interior uniqueness theorem for analytic functions was generalized by M. B. Balk to the case of polyanalytic functions of order n. He proved that if the zeros of a polyanalytic function have an accumulation point of order n, then this function is identically zero. In this paper the interior uniqueness theorem is generalized to the solution to a linear homogeneous second order differential equation of elliptic type with constant coefficients.     

Calculating the coefficients of a discrete elliptic equation from spectral data

A numerical analytical algorithm is developed for calculating the coefficients of a discrete two-dimensional elliptic equation from given eigenvalues and certain symmetry conditions imposed on the basis eigenfunctions.     

Numerical solution of a linear elliptic partial differential equation with variable coefficients: A complex variable boundary element approach

This article presents a complex variable boundary element method for the numerical solution of a second order elliptic partial differential equation with variable coefficients. To assess the validity and accuracy of the method, it is applied to solve some specific problems with known solutions. © 2011 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2011     

On the harnack inequality for the equation formally adjoint to a linear elliptic differential equation

Baku. Translated fromSibirskiî Matematicheskiî Zhurnal, Vol. 33, No. 5, pp. 100–106, September–October, 1992.     

Combining linear and fast diffusion in a nonlinear elliptic equation

In this paper we analyse an elliptic equation that combines linear and nonlinear fast diffusion with a logistic type reaction function. We prove existence and non-existence results of positive solutions using bifurcation theory and sub-supersolution method. Moreover, we apply variational methods to obtain a pair of ordered positive solutions.     

Finding discontinuities in the coefficients of the linear nonstationary transport equations

An X-ray tomography problem that is an inverse problem for the transport differential equation is set up and investigated. The absorption and single scattering of particles are taken into account. The transport equation is nonstationary (its coefficients and the unknown function depend on time), involves multiple energy levels, and its coefficients can undergo jump discontinuities with respect to the spatial variable (in other words, the medium in which the process proceeds is inhomogeneous). The sought object is the set on which the coefficients of the equation suffer a discontinuity, which corresponds to the search for the boundaries between the different substances composing the sensed medium.     

Periodic solution of a linear functional equation with variable coefficients

Translated from Issledovaniya po Prikladnoi Matematike, No. 17, pp. 46–50, 1990.     

On optimal control by coefficients in an elliptic equation

We consider a problem of optimal control by coefficients in a linear elliptic equation. We study the well-posedness of the statement of the problem and obtain a necessary optimality condition.     

Optimal control problem for a quasilinear elliptic equation with controls in coefficients

We consider an optimal control problem for a quasilinear elliptic equation with controls in coefficients. We study the well-posedness of the problem, prove that the objective functional is differentiable, and establish a necessary optimality condition.     

Splitting and the spectrum of a linear differential equation with quasiperiodic coefficients

A linear differential equation with quasiperiodic coefficients for large values of the spectral parameter is split into an exponentially dichotomous system and a two-dimensional] system with the properties of the one-dimensional Schrödinger equation. The set of the values of the parameter, for which the equation has solution of the form A(t) epx (iat) with a real number a and a quasiperiodic function A(t), is a single out.Translated from Ukrainskii Matematicheskii Zhurnal, Vol. 42, No. 10, pp. 1383–1388, October, 1990.     

Radial solutions with a vortex to an asymptotically linear elliptic equation

In this paper, we study a two-dimensional nonlinear elliptic equation:

On uniqueness sets for an elliptic equation with constant coefficients

For the solutions of the elliptic equation

Results on entire solutions for a degenerate critical elliptic equation with anisotropic coefficients

In this paper, we study the following degenerate critical elliptic equation with anisotropic coefficients-div(|x N | 2α▽u) = K(x)|x N | α·2 * (s)-s |u| 2 * (s)-2 u in R N ,where x = (x 1 , . . . , x N ) ∈ R N , N≥3, α > 1/2, 0≤ s ≤2 and 2 * (s) = 2(N-s)/(N-2). Some basic properties of the degenerate elliptic operator -div(|x N |2α▽u) are investigated and some regularity, symmetry and uniqueness results for entire solutions of this equation are obtained. We also get some variational identities for solutions ...