ISSN 1066-369X, Russian Mathematics, 2019, Vol. 63, No. 2, pp. 35–43. c  Allerton Press, Inc., 2019. Russian Text c  E.N. Sattorov, Z.E. Ermamatova, 2019, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2019, No. 2, pp. 39–48. Recovery of Solutions to Homogeneous System of Maxwell’s Equations With Prescribed Values on a Part of the Boundary of Domain E. N. Sattorov * and Z. E. Ermamatova * Samarkand State University Univrsitetskii bulv. 15, Samarkand, 140101 Republic of Uzbekistan Received April 7, 2017; revised September 21, 2018; accepted September 26, 2018 Abstract—In a bounded space domain, we investigate analytic continuation of solution to the system of Maxwell’s equations with boundary values given on a part of the boundary, i.e., we study the Cauchy problem. Using the Carleman matrix method we construct its approximate solution. DOI: 10.3103/S1066369X19020051 Key words: Maxwell’s equations, ill–posed problem, regular solution, Carleman matrix. 1. STATEMENT OF THE PROBLEM AND THE MAIN RESULTS Let R 3 be the three-dimensional real Euclidean space, x =(x 1 ,x 2 ,x 3 ), y =(y 1 ,y 2 ,y 3 ) ∈ R 3 , x  =(x 1 ,x 2 ), y  =(y 1 ,y 2 ) ∈ R 2 , α 2 =(y 1 − x 1 ) 2 +(y 2 − x 2 ) 2 , r 2 = α 2 +(y 3 − x 3 ) 2 = |y − x| 2 , and Ω be a bounded simply-connected domain in R 3 with a piecewise smooth boundary ∂ Ω, consisting of a compact part T of the plane y 3 =0 and a smooth Lyapunov surface S , lying in the half-space y 3 > 0. Let Ω=Ω ∪ ∂ Ω and ∂ Ω= T ∪ S. We will assume that every ray emanating from any point x ∈ Ω intersects the surface S at no more than l points. We consider the problem of recovery of solutions to the system of Maxwell’s equations in the harmonic regime [1] rot E − ikH =0, rot H + ikE =0 (1.1) by their values known on a part of the boundary of domain, i.e., the Cauchy problem. Here the wave number k = ω √ εμ where ε and μ are electromagnetic constants (the electric and magnetic permittivities); E =(E 1 ,E 2 ,E 3 ) and H =(H 1 ,H 2 ,H 3 ) are the electric and magnetic field strengths, and ω is the frequency of electromagnetic oscillation. Denote by A(Ω) the space of vector-functions of the class C 1 (Ω) ∩ C ( Ω), satisfying (1.1) in Ω. Problem 1. Let we know the Cauchy data for a solution to system (1.1) on the surface S : [n(y),E(y)] = f (y), [n(y),H (y)] = g(y), y ∈ S, (1.2) where n =(n 1 ,n 2 ,n 3 ) is the unit outward-pointing normal to the surface ∂ Ω at a point y, and f =(f 1 ,f 2 ,f 3 ), g =(g 1 ,g 2 ,g 3 ) are continuous vector-functions. Given f (y) and g(y) on S , find E(x) and H (x), x ∈ Ω. Problem 2. Let f (y) and g(y) be given on S . Find conditions on f (y) and g(y), necessary and sufficient for existence of solution to system (1.1), satisfying (1.2) and of the class A(Ω). * E-mail: Sattorov-e@rambler.ru. 35