Volume 219, Nº 6 (2016)

We consider the boundary value problem for the radiative transfer equation with mirror reflection and refraction conditions subject to the Fresnel laws in a system of bodies with piecewise smooth boundaries. In the case of data in the complete scale of Lebesgue spaces, we establish the existence and uniqueness of a solution. We obtain estimates for the solution and study the conjugate problem.

We study stationary states of the magnetic Schrödinger–Poisson system in the repulsive (plasma physics) Coulomb case. We prove the existence and nonlinear stability for a wide class of stationary states by using the energy-Casimir method. We generalize the known global well-posedness result for the Schrödinger–Poisson system to the case where a magnetic field is involved.

We study the spectral problem with a small parameter for a quadratic operator pencil with boundary conditions of nonperiodically and rapidly varying (alternating) type. We show that the homogenized problem admits the classical Steklov type spectral condition instead of alternating one. Bibliography: 11 titles.

We obtain sufficient positivity conditions for continuously differentiable minimal coordinate splines of the second order in a general case. These conditions are used for constructing positive exponential continuously differentiable coordinate splines. We establish the positivity of hyperbolic and fractional-rational minimal coordinate splines without any restrictions on a grid.

We establish upper estimates for nonnegative semiadditive functionals defined on the space of periodic functions L₂ in terms of deviations of Steklov functions of even order in L₂.

We consider the transmission conditions at vertices of the graph modeling a periodic rectangular lattice of thin quantum waveguides described by the spectral Dirichlet problem for the Laplace operator. The type of transmission conditions is determined by the structure of the space B_bo^R of bounded solutions to the boundary layer problem in a cross-shaped waveguide with a circular core of radius R. We describe all variants of the structure of the space B_st^R of nondecaying solutions and present methods for constructing hardly probable and very probable examples. Based on the method of matched asymptotic expansion, we construct all possible transmission conditions. We discuss numerical methods for computing critical radii, construction of the space B_st^R, and classification of “trapped”/“almost standing” waves.