Let D be a bounded domain in R n with a smooth boundary ∂D. We indicate appropriate Sobolev spaces of negative smoothness to study the non-homogeneous Cauchy problem for an elliptic differential complex {A i } of first order operators. In particular, we describe traces on ∂D of tangential part τ i (u) and normal part ν i (u) of a (vector)-function u from the corresponding Sobolev space and give an adequate formulation of the problem. If the Laplacians of the complex satisfy the uniqueness condition in the small then we obtain necessary and sufficient solvability conditions of the problem and produce formulae for its exact and approximate solutions. For the Cauchy problem in the Lebesgue spaces L 2 (D) we construct the approximate and exact solutions to the Cauchy problem with maximal possible regularity. Moreover, using Hilbert space methods, we construct Carleman's formulae for a (vector-) function u from the Sobolev space H 1 (D) by its Cauchy data τ i (u) on a subset Γ ⊂ ∂D and the values of A i u in D modulo the null-space of the Cauchy problem. Some instructive examples for elliptic complexes of operators with constant coefficients are considered.Key words: Elliptic differential complexes, ill-posed Cauchy problem, Carleman's formula.It is well-known that the Cauchy problem for an elliptic system A is ill-posed (see, for instance, [1]). Apparently, the serious investigation of the problem was stimulated by practical needs. Namely, it naturally appears in applications: in hydrodynamics (as the Cauchy problem for holomorphic functions), in geophysics (as the Cauchy problem for the Laplace operator), in elasticity theory (as the Cauchy problem for the Lamé system) etc., see, for instance, the book [2] and its bibliography. The problem was actively studied through the XX century (see, for instance, [3] Differential complexes appeared as compatibility conditions for overdetermined operators (see, for instance, [13]). Thus, the Cauchy problem for them is of special interest. One of the first problems of this kind was the Cauchy problem for the Dolbeault complex (the compatibility complex for the multidimensional Cauchy-Riemann system), see [14]. The interest to it was great because of the famous example by H. Lewy of differential equation without solutions, constructed with the use of the tangential Cauchy-Riemann operator, see [15]. Recently new approaches to the problem were found in spaces of smooth functions (see [16], [17]).We consider the Cauchy problem in spaces of distributions with some restrictions on growth in order to correctly define its traces on boundaries of domains (see, for instance, [2], [18], [19], [20]). In this paper we develop the approach presented in [9] to study the homogeneous Cauchy problem for overdetermined elliptic partial differential operators. Instead we consider the nonhomogeneous Cauchy problem for elliptic complexes.
Total internal reflection occurs at the interface between two media with different refractive indices during propagation of light rays from a medium with a higher refractive index to a medium with a lower refractive index. If the thickness of the second medium is comparable to a specific light wavelength, then total internal reflection is violated partially or completely. Based on the frustrated total internal reflection, herein we discuss a two-dimensional photonic topological insulator in an array consisting of triangular, quadrangular, or hexagonal transparent prism resonators with a narrow gap between them. An array of prism resonators allows topologically stable edge solutions (eigenwaves) similar to those studied in ring resonators. Moreover, total internal reflection occurs at different angles of incidence of light. This makes it possible to obtain a set of fundamentally new edge solutions. The light is presumably concentrated at the surface; however, in the new solutions it penetrates relatively deep into the photonic topological insulator and excites several layers of prisms positioned beyond the surface. Remarkably, the direction of light propagation is precisely biased, and therefore new solutions exhibit lower symmetry than the resonator array symmetry.
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