Eigenmodes of the damped wave equation and small hyperbolic subsets

Gabriel Riviere (1)

1 LPP - Laboratoire Paul Painlevé - UMR 8524

We study high frequency stationary solutions of the damped wave equation on a compact and smooth Riemannian manifold without boundary. For a fixed damping parameter, we describe concentration properties of eigenmodes in neighborhoods of a fixed small hyperbolic subset with the same damping parameter. Precisely, we prove that, in the high frequency limit, a sequence of such modes cannot be completely localized in a shrinking tubes around such subsets. The article also includes an appendix (by S. Nonnenmacher and the author) where we establish the existence of an inverse logarithmic strip without eigenvalues below the real axis, under a pressure condition on the set of undamped trajectories.

Mots clés: nonselfadjoint operators; semiclassical analysis; hyperbolic dynamical systems; eigenmodes; topological pressure

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

Gabriel Riviere. Eigenmodes of the damped wave equation and small hyperbolic subsets. 2012. ⟨hal-00673138v2⟩