Inverse boundary value problem for Maxwell equations

We prove a uniqueness theorem for an inverse boundary value problem for the Maxwell system with boundary data assumed known only in part of the boundary. We assume that the inaccessible part of the boundary is either part of a plane, or part of a sphere. This work generalizes the results obtained by Isakov [I] for the Schrödinger equation to Maxwell equations. Introduction. Let Ω ⊂ R be a bounded domain with C boundary, and let ε, μ, σ be C functions in Ω (ε is the permittivity, μ the permeability, and σ the conductivity). We will assume that the coefficients satisfy the positivity conditions γ = ε+ iσ/ω, ε > 0, μ > 0, σ ≥ 0 in Ω. (0.1) Let D = −i∇, let ν be the exterior unit normal to ∂Ω, and consider the time-harmonic Maxwell equations for the electric field E and magnetic field H in Ω, { D ∧H + ωγE = 0, D ∧ E − ωμH = 0, (0.2) with the boundary condition ν ∧H = a on ∂Ω. (0.3) This is the magnetic boundary value problem for the Maxwell equations. Here we use ′∧′ to denote the vector product in R, and ∇∧ F is the curl of the vector field F . When posed in correct function spaces this problem admits a unique solution (E,H) when the angular ∗Supported by the Academy of Finland under CoE–project 213476 †The first author was also supported by Ministerio de Ciencia e Innovación de España, MTM200507652-C02-01 ‡Department of Mathematics, Universidad Autónoma de Madrid, 28049 Madrid, Spain §Department of Mathematics and Statistics, P.O. Box 68, 00014 University of Helsinki, Finland ¶Department of Mathematics and Statistics, P.O. Box 68, 00014 University of Helsinki, Finland