Energy decay for Maxwells equations with Ohms law in partially cubic domains

We prove a polynomial energy decay for the Maxwell’s equations with Ohm’s law in partially cubic domains with trapped rays. We extend the results of polynomial decay for the scalar damped wave equation in partially rectangular or cubic domain. Our approach have some similitude with the construction of re‡ected gaussian beams. Keywords .Maxwell’s equations; decay estimates; trapped ray. 1 Introduction and main result The problems dealing with Maxwell’s equations with nonzero conductivity are not only theoretical interesting but also very important in many industrial applications (see e.g. [3], [7], [8], [14]). This paper is concerned with the energy decay of Maxwell’s equations with Ohm’s law in a bounded cylinder R with trapped rays. Precisely, let > 0 and D be an open simply connected bounded set in R with C boundary @D. Consider = D ( ; ) with boundary @ = 0[ 1 where 0 = D f ; g and 1 = @D ( ; ). The domain is occupied by an electromagnetic medium of constant electric permittivity "o and constant magnetic permeability o. For the sake of simplicity, we assume from now that "o o = 1. Let E and H denote the electric and magnetic …elds respectively. De…ne the energy by E (t) = 1 2 Z "o jE (x; t)j + o jH (x; t)j 2 dx . (1.1) The Maxwell’s equations with Ohm’s law are described by 8><>>>>: "o@tE curlH + E = 0 in [0;+1) o@tH + curlE = 0 in [0;+1) div ( oH) = 0 in [0;+1) E = H = 0 on @ [0;+1) (E;H) ( ; 0) = (Eo;Ho) in . (1.2) Here, (Eo;Ho) is the initial data in the energy space L ( ) 6 and denotes the outward unit normal vector to @ . The conductivity is such that 2 L1 ( ) and 0. It is well-known that when 1 ha l-0 08 45 76 1, v er si on 1 17 J ul 2 01 3 Author manuscript, published in "Communications on Pure and Applied Analysis 12, 5 (2013) 2229-2266" DOI : 10.3934/cpaa.2013.12.2229