Eigenvalues for Maxwell’s equations with dissipative boundary conditions

Article type: Research Article

Authors: Colombini, Ferruccio^a | Petkov, Vesselin^{b; *} | Rauch, Jeffrey^c

Affiliations: [a] Dipartimento di Matematica, Università di Pisa, Italia. E-mail: colombini@dm.unipi.it | [b] Institut de Mathématiques de Bordeaux, 351, Cours de la Libération, 33405 Talence, France. E-mail: petkov@math.u-bordeaux.fr | [c] Department of Mathematics, University of Michigan, USA. E-mail: rauch@umich.edu

Correspondence: [*] Corresponding author. E-mail: petkov@math.u-bordeaux.fr.

Abstract: Let V(t)=etGb, t⩾0, be the semigroup generated by Maxwell’s equations in an exterior domain Ω⊂R3 with dissipative boundary condition Etan−γ(x)(ν∧Btan)=0, γ(x)>0, ∀x∈Γ=∂Ω. We prove that if γ(x) is nowhere equal to 1, then for every 0<ϵ≪1 and every N∈N the eigenvalues of Gb lie in the region Λϵ∪RN, where Λϵ={z∈C:|Rez|⩽Cϵ(|Imz|12+ϵ+1),Rez<0}, RN={z∈C:|Imz|⩽CN(|Rez|+1)−N,Rez<0}.

Keywords: Maxwell’s equations, asymptotically disappearing solutions, location of the eigenvalues

DOI: 10.3233/ASY-161377

Journal: Asymptotic Analysis, vol. 99, no. 1-2, pp. 105-124, 2016