Vol 216, No 2 (2016)

We consider a p(x)-harmonic equation, where p(x) is measurable in Ω and is separated from 1 and infinity. It is shown that if p(x) is a radial function of x − x₀, in a neighborhood of a point x₀ ∈ Ω i.e., p(x) = p(|x − x₀|) and p(t) is nonincreasing on (0, d), then p(x) is Hӧlder continuous at the point x₀. Bibliography: 11 titles.

We study projection type methods for numerical solution of one class of weakly singular integral Fredholm equations of the second kind including the integral radiative transfer equation. We consider Galerkin type methods and their modifications: the iterated Galerkin method (the Sloan method), the Kantorovich method, and the iterated Kantorovich method. For the approximating space we take the space of piecewise constant functions or the space of piecewise linear functions related to an arbitrary nonuniform grid. Error estimates are obtained.

We establish necessary conditions for the stabilization to zero of solutions to the Dirichlet problem for a parabolic equation with elliptic operator of divergence form and coefficients depending on x and t.

For Schlömilch polynomials in even and odd Bessel j-functions we prove formulas similar to the Riesz interpolation formula. Bibliography: 8 titles.

We propose a one-dimensional asymptotic model of an L-shaped junction of two thin two-dimensional elastic beams subject to boundary conditions of different type at external bean ends. The beams are described by standard systems of the Kirchhoff ordinary differential equations, whereas the transmission conditions on coincident (internal) nodes essentially depend on the type of boundary conditions on the external beam ends. The transmission conditions are obtained by analyzing the boundary layer near internal nodes. The asymptotics is justified with the help of a weighted anisotropic Korn inequality.

We study the Neumann problem in a bounded domain Ω for a scalar elliptic equation with coefficients oscillating with respect to two groups of variables with periods of different smallness order. We prove the H¹- and L²-estimates of homogenization. The estimates have the operator form and can be expressed in terms of the resolvent of the original problem and its approximations in the operator (L²(Ω)→H¹(Ω))- and (L²(Ω)→L²(Ω))- norms.