Том 224, № 5 (2017)

We consider the boundary value problem for a system of differential equations consisting of the stationary nonlinear heat equation and the integro-differential radiative transfer equation and describing stationary radiative-conductive heat transfer in a system of semitransparent bodies, taking into account the effects of reflection and refraction of radiation according to the Fresnel laws at the boundaries of bodies. We take into account the dependence of radiation intensity and optical properties of bodies on the radiation frequency. We establish the existence and uniqueness of a weak solution. We prove the comparison theorem and derive a priori estimates for weak solutions and obtain a regularity result.

We obtain estimates for nonnegative semiadditive functionals on the space of continuous 2π-periodic functions defined in terms of Steklov functions.

We consider a nontrivial example of distributional eigenfunction of the planar Fourier transform. This eigenfunction is not a tensor product of univariate eigenfunctions. As a consequence, we obtain a formula for multi-dimensional eigenfunctions in dimension 2N.

We present some results conjectured by Bildhauer, Fuchs, and Weickert who investigated analytical aspects of coupled variational models with applications to mathematical imaging. We focus on variants of linear growth, which require a treatment within the framework of relaxation theory and convex analysis. We establish existence and regularity of (dual-) solutions.

We study homogenization for the Beltrami equation \( {A}_{\varepsilon }{u}_{\varepsilon}\equiv {\partial}_{\overline{z}}{u}_{\varepsilon }+{\mu}^{\varepsilon }{\partial}_z{u}_{\varepsilon }+{\nu}^{\varepsilon}\overline{\partial_z{u}_{\varepsilon }}=f \) with measurable ε-periodic coefficients μ^ε and ν^ε, where ε is a small parameter. The coefficients of the equation satisfy the uniform ellipticity condition. The equation is considered in a bounded domain Ω of the complex plane with the Riemann–Hilbert condition on the boundary ∂Ω. For the resolvent \( {A}_{\varepsilon}^{-1} \) of this boundary value problem we obtain an approximation in the operator norm of the Sobolev space W^1,2(Ω) with approximation error of order O(\( \sqrt{\varepsilon } \)).

We consider a linear nonstationary system of first order partial differential equations that is not resolved with respect to the derivatives and identically degenerates in the domain. Without using the change of variables, we construct the structural form whose set of solutions coincides with the set of solutions to the original system. We obtain the hyperbolicity conditions and conditions for the correctness of initial and boundary conditions. We establish the existence of solutions to the initial-boundary value problem for hyperbolic systems of differential algebraic equations.