An anisotropic inverse boundary value problem

We consider the impedance tomography problem for anisotropic conductivities. Given a bounded region Ω in space, a diffeomorphism Ψ from Ω to itself which restricts to the identity on ? Ω, and a conductivity γ on Ω, it is easy to construct a new conductivity Ψ_*γ which will produce the same voltage and current measurements on ? Ω. We prove the converse in two dimensions (i.e., if γ₁ and γ₂ produce the same boundary measurements, then γ₁, = Ψ_*γ₂ for an appropriate Ψ) for conductivities which are near a constant.     

An inverse problem for the Helmholtz equation

The paper considers the problem of optimal determination of linear functionals of the source intensity under various assumptions. Some theorems on optimal estimates are proved and estimation errors are determined.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 66, pp. 10–17, 1988.     

An inverse problem for a linear diffusion equation with non-linear boundary condition

In this paper we consider the determination of an unknown radiation term in the non-linear boundary condition of a linear diffusion equation from an overspecified condition. It is shown that the solutions of this inverse problem is unique and stable.     

The Robin problem for the scalar Oseen equation

We study the Robin problem for the scalar Oseen equation in an open n‐dimensional set with compact Ljapunov boundary. We prescribe two types of Robin boundary conditions, and prove the unique solvability of these problems as well as a representation formula for the solution in form of a scalar Oseen single layer potential. Moreover, we prove the maximum principle for the solution to the Robin problem of the scalar Oseen equation. Copyright © 2013 John Wiley & Sons, Ltd.     

Nonclassical boundary value problem for the transport equation

We consider a nonclassical boundary value problem for the transport equation. The particle transport process is described by a stationary linear integro-differential equation, and the outgoing particle flux density on some part of the boundary is specified as the boundary condition. We find the particle flux density for a given outgoing flux and known coefficients of the equation. We show that, under some restrictions on the medium, there exists a unique solution of the problem, which can be represented by an infinite convergent series.     

The exterior Dirichlet and the interior Neumann boundary value problems for the scalar Oseen equation

The scalar Oseen equation represents a linearized form of the Navier Stokes equations. We present an explicit potential theory for this equation and solve the exterior Dirichlet and interior Neumann boundary value problems via a boundary integral equations method in spaces of continuous functions on a C²-boundary, extending the classical approach for the isotropic selfadjoint Laplace operator to the anisotropic non-selfadjoint scalar Oseen operator. (© 2012 Wiley-VCH Verlag GmbH & Co. KGaA, Weinheim)     

An inverse coefficient problem for the heat equation in the case of nonlocal boundary conditions

This paper investigates the inverse problem of finding a time-dependent coefficient in a heat equation with nonlocal boundary and integral overdetermination conditions. Under some regularity and consistency conditions on the input data, the existence, uniqueness and continuous dependence upon the data of the solution are shown by using the generalized Fourier method.     

An improved non-local boundary value problem method for a cauchy problem of the Laplace equation

In this paper, we propose an improved non-local boundary value problem method to solve a Cauchy problem for the Laplace equation. It is known that the Cauchy problem for the Laplace equation is severely ill-posed, i.e., the solution does not depend continuously on the given Cauchy data. Convergence estimates for the regularized solutions are obtained under a-priori bound assumptions for the exact solution. Some numerical results are given to show the effectiveness of the proposed method.     

An initial boundary value problem for a pseudoparabolic equation with a nonlinear boundary condition

An initial boundary value problem for a quasilinear equation of pseudoparabolic type with a nonlinear boundary condition of the Neumann–Dirichlet type is investigated in this work. From a physical point of view, the initial boundary value problem considered here is a mathematical model of quasistationary processes in semiconductors and magnets, which takes into account a wide variety of physical factors. Many approximate methods are suitable for finding eigenvalues and eigenfunctions in problems where the boundary conditions are linear with respect to the desired function and its derivatives. Among these methods, the Galerkin method leads to the simplest calculations. On the basis of a priori estimates, we prove a local existence theorem and uniqueness for a weak generalized solution of the initial boundary value problem for the quasilinear pseudoparabolic equation. A special place in the theory of nonlinear equations is occupied by the study of unbounded solutions, or, as they are called in another way, blow-up regimes. Nonlinear evolutionary problems admitting unbounded solutions are globally unsolvable. In the article, sufficient conditions for the blow-up of a solution in a finite time in a limited area with a nonlinear Neumann–Dirichlet boundary condition are obtained.     

Generalized third boundary value problem for the biharmonic equation

We study the existence and uniqueness of solutions of a generalized third boundary value problem for the inhomogeneous biharmonic equation in the unit ball.     

On the discrete boundary value problem for anisotropic equation

In this paper we consider the discrete anisotropic boundary value problem using critical point theory. Firstly we apply the direct method of the calculus of variations and the mountain pass technique in order to reach the existence of at least one nontrivial solution. Secondly we derive some version of a discrete three critical point theorem which we apply in order to get the existence of at least two nontrivial solutions.