Том 235, № 2 (2018)

We consider the initial-boundary value problem for the nonstationary radiative transfer equation in a system of semitransparent bodies with the conditions of diffuse reflection and refraction of radiation. We establish the unique solvability of the problem with boundary and initial data in the complete scale of Lebesgue spaces. We obtain estimates for solutions.

We prove estimates in the Dini space for the second order spatial derivative of a vector parabolic single layer potential. We obtain estimates for higher order derivatives of solutions to initial-boundary-value problems for one-dimensional parabolic systems with Dini-continuous coefficients in a curvilinear domain with nonsmooth lateral boundary.

A nontrivial example of an eigenfunction in the sense of the theory of distributions for the planar Fourier transform was described by the authors in their previous work. In this paper, a method for obtaining other eigenfunctions is proposed. Positive homogeneous distributions in ℝⁿ of order −n/2 are considered, and it is shown that F(ω)|x|^−n/2, |ω| = 1, is an eigenfunction in the sense of the theory of distributions of the Fourier transform if and only if F(ω) is an eigenfunction of a certain singular integral operator on the unit sphere of ℝⁿ. Since \( {Y}_{m,n}^{(k)}\left(\omega \right){\left|\mathbf{x}\right|}^{-n/2} \), where \( {Y}_{m,n}^{(k)} \) denote the spherical functions of order m in ℝⁿ, are eigenfunctions of the Fourier transform, it follows that \( {Y}_{m,n}^{(k)} \) are eigenfunctions of the above-mentioned singular integral operator. In the planar case, all eigenfunctions of the Fourier transform of the form F(ω)|x|⁻¹ are described by means of the Fourier coefficients of F(ω).

We discuss connections between the three problems mentioned in the title.