Scattering problems for symmetric systems with dissipative boundary conditions

Vesselin Petkov (1)

1 IMB - Institut de Mathématiques de Bordeaux

We study symmetric systems with dissipative boundary conditions. The solutions of the mixed problems for such systems are given by a contraction semigroup $V(t)f = e^{tG_b}f, t \geq 0$. The solutions $u(t, x) = V(t)f$, where $f$ is an eigenfunction of the generator $G_b$ with eigenvalue $\lambda,\Re \lambda < 0,$ are called asymptotically disappearing (ADS). We prove that the wave operators are not complete if there exist (ADS). This is the case for Maxwell system with special boundary conditions in the exterior of the sphere. We obtain a representation of the scattering kernel and we examine the inverse back-scattering problem related to the leading term of the scattering kernel.

Mots clés: dissipative boundary conditions; asymptotically disappearing solutions; inverse scattering problems

Domaines: Equations aux dérivées partielles [math.AP]; Physique mathématique [math-ph]; Physique mathématique [math-ph]

Vesselin Petkov. Scattering problems for symmetric systems with dissipative boundary conditions. Studies in Phase Space Analysis with Applications to PDEs, Birkhäuser, pp.337-353, 2013, Progress in Nonlinear Differential Equations and Their Applicagitons, 978-1-4614-6347-4. ⟨hal-00950069⟩