206 research outputs found
The second Yamabe invariant
Let be a compact Riemannian manifold of dimension . We
define the second Yamabe invariant as the infimum of the second eigenvalue of
the Yamabe operator over the metrics conformal to and of volume 1. We study
when it is attained. As an application, we find nodal solutions of the Yamabe
equation
The geometrical quantity in damped wave equations on a square
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 [Math. Phys. Stud. 19 (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 a mass-point in subject to elastic
reflections on the boundary. These reflections obey the law of geometrical
optics. The geometrical quantity is then defined as the upper limit
(large time asymptotics) of the average trajectory length. We give here an
algorithm to compute explicitly when is a finite union of
squares
Optimal transportation between hypersurfaces bounding some strictly convex domains
Let be two smooth compact hypersurfaces of which bound
strictly convex domains equipped with two absolutely continuous measures
and (with respect to the volume measures of and ). We consider the
optimal transportation from to for the quadratic cost. Let be some functions which achieve the
supremum in the Kantorovich formulation of the problem and which satisfy
Define
for ,
In this short paper, we exhibit a relationship between the regularity of
and the existence of a solution to the Monge problem
About the mass of certain second order elliptic operators
Let be a closed Riemannian manifold of dimension and let
, such that the operator is positive. If
is flat near some point and vanishes around , we can define the
mass of as the constant term in the expansion of the Green function of
at . In this paper, we establish many results on the mass of such
operators. In particular, if f:= \frac{n-2}{4(n-1)} \scal_g, i.e. if is
the Yamabe operator, we show the following result: assume that there exists a
closed simply connected non-spin manifold such that the mass is
non-negative for every metric as above on , then the mass is
non-negative for every such metric on every closed manifold of the same
dimension as .Comment: 39 page
Harmonic spinors and local deformations of the metric
Let (M,g) be a compact Riemannian spin manifold. The Atiyah-Singer index
theorem yields a lower bound for the dimension of the kernel of the Dirac
operator. We prove that this bound can be attained by changing the Riemannian
metric g on an arbitrarily small open set.Comment: minor changes, to appear in Mathematical Research Letter
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