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The energy in a square membrane subject to constant viscous damping
on a subset decays exponentially in time
as soon as
satisfies a geometrical condition known as the “Bardos-Lebeau-Rauch” condition. The rate
of this decay satisfies
(see Lebeau [
(1996) 73–109]). Here denotes the spectral abscissa of the
damped wave equation operator and is a number called
the geometrical quantity of and defined as follows.
A ray in is the trajectory generated by the
free motion of...
We prove that the mass endomorphism associated to the Dirac operator on a Riemannian manifold is non-zero for generic Riemannian metrics. The proof involves a study of the mass endomorphism under surgery, its behavior near metrics with harmonic spinors, and analytic perturbation arguments.
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