Energy decay rates for solutions of Maxwell's system with a memory boundary condition

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Energy decay rates for solutions of Maxwell's system with a memory boundary condition Serge Nicaise Universite de Valenciennes et du Hainaut Cambresis LAMAV, Institut des Sciences et Techniques de Valenciennes 59313 Valenciennes Cedex 9 France Cristina Pignotti Dipartimento di Matematica Pura e Applicata Universita di L'Aquila Via Vetoio, Loc. Coppito, 67010 L'Aquila Italy Abstract We consider the stabilization of Maxwell's equations with space variable coefficients in a bounded region with a smooth boundary, subject to dissipative boundary conditions of mem- ory type on the boundary. Under suitable conditions on the domain and on the permeability and permittivity coefficients, we prove the exponential/polynomial decay of the energy. Our result is mainly based on the use of a multiplier method and the introduction of a suitable Lyapounov functional. 2000 Mathematics Subject Classification: 93D15, 93D05, 35L10. Keywords and Phrases: Maxwell's equations, variable coefficients, memory boundary conditions, stabilization 1 Introduction Let ? ? IR3 be an open bounded domain with a smooth boundary ?. In the domain ?, we consider the homogeneous Maxwell's system D? ? curl (µB) = 0 in ? ? (0,+∞) (1.1) B? + curl (?D) = 0 in ? ? (0,+∞) (1.2) div D = div B = 0 in ? ? (0,+∞) (1.3) D(0) = D0 and B(0) = B0 in ? (1.4) ?µD? (t) = k0B(t) ? ? + ∫ t

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Energy decay rates for solutions of Maxwell’s system with a memory boundary condition Serge Nicaise UniversitedeValenciennesetduHainautCambresis LAMAV, Institut des Sciences et Techniques de Valenciennes 59313 Valenciennes Cedex 9 France Cristina Pignotti Dipartimento di Matematica Pura e Applicata UniversitadiL’Aquila Via Vetoio, Loc. Coppito, 67010 L’Aquila Italy Abstract WeconsiderthestabilizationofMaxwell’sequationswithspacevariablecoecientsina bounded region with a smooth boundary, subject to dissipative boundary conditions of mem-ory type on the boundary. Under suitable conditions on the domain and on the permeability and permittivity coecien ts, we prove the exponential/polynomial decay of the energy. Our result is mainly based on the use of a multiplier method and the introduction of a suitable Lyapounov functional. 2000 Mathematics Subject Classication: 93D15, 93D05, 35L10. Keywords and Phrases: Maxwell’sequations,variablecoecients,memoryboundary conditions, stabilization 1 Introduction Let IR 3 be an open bounded domain with a smooth boundary . In the domain , we consider the homogeneous Maxwell’s system D 0 curl ( B ) = 0 in (0 , + ∞ ) (1.1) B 0 + curl ( D ) = 0 in (0 , + ∞ ) (1.2) div D = div B = 0 in (0 , + ∞ ) (1.3) D (0) = D 0 and B (0) = B 0 in (1.4) ) + Z 0 t ) ds on (0 , + ∞ ) (1.5) D ( t ) = k 0 B ( t k ( s ) B ( t s where D, B are three-dimensional vector-valued functions of t, x = ( x 1 , x 2 , x 3 ); = ( x ) and = ( x ) are scalar functions in C 2 ( ) bounded from below by a positive constant, i.e., ( x ) 0 > 0 , ( x ) 0 > 0 , ∀ x ∈ ; (1.6) 1

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