EIGENMODES OF THE DAMPED WAVE EQUATION AND SMALL HYPERBOLIC SUBSETS

Gabriel Riviere (1) , Stéphane Nonnenmacher (2)

1 LPP - Laboratoire Paul Painlevé - UMR 8524
2 IPHT - Institut de Physique Théorique - UMR CNRS 3681

We study stationary solutions of the damped wave equation on a compact and smooth Riemannian manifold without boundary. In the high frequency limit, we prove that a sequence of β-damped stationary solutions cannot be completely concentrated in small neighborhoods of a small fixed hyperbolic subset made of β-damped trajectories of the geodesic flow. 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, Stéphane Nonnenmacher. EIGENMODES OF THE DAMPED WAVE EQUATION AND SMALL HYPERBOLIC SUBSETS. 2012. ⟨hal-00673138v3⟩