COLOMBINI, Ferruccio; PETKOV, Vesselin; RAUCH, Jeffrey

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COLOMBINI, Ferruccio
Dipartimento di Matematica

PETKOV, Vesselin
Institut de Mathématiques de Bordeaux [IMB]

RAUCH, Jeffrey
Department of Mathematics - University of Michigan

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Journal of Functional Analysis. 2014-09-15, vol. 267, n° 6, p. 1637-1661

Elsevier

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This paper considers and extends spectral and scattering theory to dissipative symmetric systems that may have zero speeds and in particular to strictly dissipative boundary conditions for Maxwell's equations. Consider symmetric systems $\partial_t - \sum_{j=1}^n A_j \partial_{x_j}$ in $\R^n,\: n \geq 3$, $n$ odd, in a smooth connected exterior domain $\Omega :=\R^n \setminus \bar{K}$. Assume that the rank of $A(\xi) = \sum_{j= 1}^n A_j \xi_j$ is constant for $\xi \not= 0.$ For maximally dissipative boundary conditions on $\Omega :=\R^n \setminus \bar{K}$ with bounded open domain $K$ the solution of the boundary problem in $\R^{+} \times \Omega$ is described by a contraction semigroup $V(t) = e^{t G_b},\:t \geq 0.$ Assuming coercive conditions for $G_b$ and its adjoint $G_b^*$ on the complement of their kernels, we prove that the spectrum of $G_b$ in the open half plane $\Re z < 0$ is formed only by isolated eigenvalues with finite multiplicities.< Réduire

Mots clés en anglais

non elliptic symmetric system

dissipative boundary conditions

asymptotically disappearing solutions

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Spectral problems for non elliptic symmetric systems with dissipative boundary conditions

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