1202.5123 (Gabriel Riviere)

Eigenmodes of the damped wave equation and small hyperbolic subsets    [PDF]

Gabriel Riviere

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.

View original: http://arxiv.org/abs/1202.5123