Том 231, № 3 (2018)

We consider the initial-boundary value problems for the nonstationary radiative transfer equation in a system of semitransparent bodies with boundary and initial data in the complete scale of Lebesgue spaces. We establish the unique solvability of the first initial-boundary value problem, the problem with “shooting conditions,” and the problem with Fresnel reflection and refraction conditions on the body boundaries.

The water wave problem is considered for a class of semi-infinite domains each having its shore shaped as a cliffed cape. In particular, the free surface of a water domain is supposed to be an infinite sector whose vertex angle is greater than π, whereas the water layer lying under the free surface is of constant depth. Under these assumptions it is shown that there are no trapped mode solutions of the problem for all values of a non-dimensional spectral parameter; in other words, no point eigenvalues are embedded in the continuous spectrum of the problem.

The goal of this paper is to develop a unified approach to radiation conditions for the entire class of null-solutions of the Helmholtz operator which are Clifford algebra-valued. The latter is an algebraic context which permits the simultaneous consideration of scalarvalued and vector-valued functions, as well as differential forms of any mixed degree. In such a setting, we provide a multitude of novel radiation conditions which naturally contain the classical Sommerfeld and Silver–Müller radiation conditions in the case of null-solutions for the scalar Helmholtz operator and the Maxwell system respectively, and which also encompass as a particular case the radiation condition introduced by McIntosh and Mitrea for perturbed Dirac operators.