Monday, March 18, 2013

1303.3743 (Ferruccio Colombini et al.)

Spectral problems for non elliptic symmetric systems with dissipative
boundary conditions

Ferruccio Colombini, Vesselin Petkov, Jeffrey Rauch
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 $\partial_t - \sum_{j=1}^r A_j \partial_{x_j}$ 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.
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