Volume 10, Issue 3

We consider a general elliptic formally self-adjoint problem in a bounded domain with homogeneous boundary conditions under the assumption that the boundary and coefficients are infinitely smooth. The operator in 𝐿 2 (Ω) corresponding to this problem has an orthonormal basis {𝑢 𝑙 } of eigenfunctions, which are infinitely smooth in . However, the system {𝑢 𝑙 } is not a basis in Sobolev spaces 𝐻 𝑡 (Ω) of high order. We note and discuss the following possibility: for an arbitrarily large 𝑡, for each function 𝑢 ∈ 𝐻 𝑡 (Ω) one can explicitly construct a function 𝑢 0 ∈ 𝐻 𝑡 (Ω) such that the Fourier series of the difference 𝑢 – 𝑢 0 in the functions 𝑢 𝑙 converges to this difference in 𝐻 𝑡 (Ω). Moreover, the function 𝑢(𝑥) is viewed as a solution of the corresponding nonhomogeneous elliptic problem and is not assumed to be known a priori; only the right-hand sides of the elliptic equation and the boundary conditions for 𝑢 are assumed to be given. These data are also sufficient for the computation of the Fourier coefficients of 𝑢 – 𝑢 0 . The function 𝑢 0 is obtained by applying some linear operator to these right-hand sides.

The linear sampling method is an algorithm for solving the inverse scattering problem for acoustic and electromagnetic waves. The method is based on showing that a linear integral equation of first kind has a solution that becomes unbounded as a parameter 𝑧 approaches the boundary of the scatterer 𝐷 from inside 𝐷. However, except for the case of the transmission problem, the case where z is in the exterior of 𝐷 is unresolved. Since for the inverse scattering problem 𝐷 is unknown, this step is crucial for the mathematical justification of the linear sampling method. In this paper we give a mathematical justification of the linear sampling method for arbitrary 𝑧 by using the theory of integral equations of first kind with singular kernels.

Spatial boundary value problems of statics of couple-stress elasticity for anisotropic homogeneous media (with contact on a part of the boundary) with an open crack are studied supposing that one medium has a smooth boundary and the other one has an open crack. Using the method of the potential theory and the theory of pseudodifferential equations on manifolds with boundary, the existence and uniqueness theorems are proved in Besov and Bessel-potential spaces. The smoothness and a complete asymptotics of solutions near the contact boundaries and near crack edge are studied. Properties of exponents of the first terms of the asymptotic expansion of solutions are established. Classes of isotropic, transversally-isotropic and anisotropic bodies are found, where oscillation vanishes.

The existence of distributional solutions is investigated for the time-dependent bending of a plate with transverse shear deformation under mixed boundary conditions. The problem is then reduced to nonstationary boundary integral equations and the existence and uniqueness of solutions to the latter are studied in appropriate Sobolev spaces.

Some classes of entire functions which are eigenfunctions of generalizations of the Laguerre derivative operator are considered. Since this property is an analog of the one characterizing the exponential function, we refer to such functions as Laguerre-type exponentials, or shortly 𝐿-exponentials. The definition of 𝐿-circular and 𝐿-hyperbolic functions easily follows. Applications in the framework of generalized evolution problems are also mentioned.

In this paper, oscillation and asymptotic properties of solutions of the Dirichlet boundary value problem for hyperbolic and parabolic equations are considered. We demonstrate that introducing an arbitrary constant delay essentially changes the above properties. For instance, the delay equation does not inherit the classical properties of the Dirichlet boundary value problem for the heat equation: the maximum principle is not valid, unbounded solutions appear while all solutions of the classical Dirichlet problem tend to zero at infinity, for “narrow enough zones” all solutions oscillate instead of being positive. We establish that the Dirichlet problem for the wave equation with delay can possess unbounded solutions. We estimate zones of positivity of solutions for hyperbolic equations.

We generalize the well-known Baker's superstability result for the d'Alembert functional equation with values in the field of complex numbers to the case of the integral equation where 𝐺 is a locally compact group, μ is a generalized Gelfand measure and σ is a continuous involution of 𝐺.

The phase space for nonlinear hyperbolic functional differential equations with unbounded delay is constructed. The set of axioms for generalized solutions of initial problems is presented. A theorem on the existence and continuous dependence upon initial data is given. The Cauchy problem is transformed into a system of integral functional equations. The existence of solutions of this system is proved by the method of successive approximations and by using theorems on integral inequalities. Examples of phase spaces are given.

We investigate the Dirichlet problem in domains with nonsmooth boundaries when a boundary function belongs to functional classes of the Zygmund class type. A complete picture of solvability is established also in the case of existence of solutions for which explicit expressions are found.

Two terms of asymptotics near crack are obtained for solutions of the Dirichlet boundary value problem for second-order elliptic equations in divergent form. The front of a crack is from 𝐶 1+𝑠 and the coefficients of the equations belong to 𝐶 𝑠 (0.5 < 𝑠 < 1).

Extending the previous work on the corresponding inverse Dirichlet problem, we present a factorization method for the solution of an inverse Neumann boundary value problem for harmonic vector fields.

The properties of logarithmic derivative of the Riemann mapping function of a quasidisk are applied to quantitative estimation of quasiconformal reflections across its boundary.

Boundary integral equations in the logarithmic potential theory are studied by the direct method under the assumption that the contour has a peak.

For a differential system where λ ∈]0, 1[ and ℎ 𝑖 :]𝑎, 𝑏[×]0, +∞[×ℝ → [0, +∞[ (𝑖 = 0, 1, 2) are continuous functions, we have established sufficient conditions for the existence of at least one solution satisfying one of the two boundary conditions and