41 research outputs found
Linear perturbations for the vacuum axisymmetric Einstein equations
In axial symmetry, there is a gauge for Einstein equations such that the
total mass of the spacetime can be written as a conserved, positive definite,
integral on the spacelike slices. This property is expected to play an
important role in the global evolution. In this gauge the equations reduce to a
coupled hyperbolic-elliptic system which is formally singular at the axis. Due
to the rather peculiar properties of the system, the local in time existence
has proved to resist analysis by standard methods. To analyze the principal
part of the equations, which may represent the main source of the difficulties,
we study linear perturbation around the flat Minkowski solution in this gauge.
In this article we solve this linearized system explicitly in terms of integral
transformations in a remarkable simple form. This representation is well suited
to obtain useful estimates to apply in the non-linear case.Comment: 13 pages. We suppressed the statements about decay at infinity. The
proofs of these statements were incomplete. The complete proofs will require
extensive technical analysis. We will studied this in a subsequent work. We
also have rewritten the introduction and slighted changed the titl
On Fourier transforms of radial functions and distributions
We find a formula that relates the Fourier transform of a radial function on
with the Fourier transform of the same function defined on
. This formula enables one to explicitly calculate the
Fourier transform of any radial function in any dimension, provided one
knows the Fourier transform of the one-dimensional function and
the two-dimensional function . We prove analogous
results for radial tempered distributions.Comment: 12 page
De Branges spaces and Krein's theory of entire operators
This work presents a contemporary treatment of Krein's entire operators with
deficiency indices and de Branges' Hilbert spaces of entire functions.
Each of these theories played a central role in the research of both renown
mathematicians. Remarkably, entire operators and de Branges spaces are
intimately connected and the interplay between them has had an impact in both
spectral theory and the theory of functions. This work exhibits the
interrelation between Krein's and de Branges' theories by means of a functional
model and discusses recent developments, giving illustrations of the main
objects and applications to the spectral theory of difference and differential
operators.Comment: 37 pages, no figures. The abstract was extended. Typographical errors
were corrected. The bibliography style was change
Back reaction in the formation of a straight cosmic string
A simple model for the formation of a straight cosmic string, wiggly or
unperturbed is considered. The gravitational field of such string is computed
in the linear approximation. The vacuum expectation value of the stress tensor
of a massless scalar quantum field coupled to the string gravitational field is
computed to the one loop order. Finally, the back-reaction effect on the
gravitational field of the string is obtained by solving perturbatively the
semiclassical Einstein's equations.Comment: 29 pages, LaTeX, no figures. A postcript version can be obtained from
anonymous ftp at ftp://ftp.ifae.es/preprint.f
Wavenumber-explicit continuity and coercivity estimates in acoustic scattering by planar screens
We study the classical first-kind boundary integral equation reformulations
of time-harmonic acoustic scattering by planar sound-soft (Dirichlet) and
sound-hard (Neumann) screens. We prove continuity and coercivity of the
relevant boundary integral operators (the acoustic single-layer and
hypersingular operators respectively) in appropriate fractional Sobolev spaces,
with wavenumber-explicit bounds on the continuity and coercivity constants. Our
analysis is based on spectral representations for the boundary integral
operators, and builds on results of Ha-Duong (Jpn J Ind Appl Math 7:489--513
(1990) and Integr Equat Oper Th 15:427--453 (1992)).Comment: v2 has minor corrections compared to v1. arXiv admin note:
substantial text overlap with arXiv:1401.280