1,798 research outputs found
The localization effect for eigenfunctions of the mixed boundary value problem in a thin cylinder with distorted ends
A simple sufficient condition on curved end of a straight cylinder is found
that provides a localization of the principal eigenfunction of the mixed
boundary value for the Laplace operator with the Dirichlet conditions on the
lateral side. Namely, the eigenfunction concentrates in the vicinity of the
ends and decays exponentially in the interior. Similar effects are observed in
the Dirichlet and Neumann problems, too.Comment: 25 pages, 10 figure
Reconciling dark energy models with f(R) theories
Higher order theories of gravity have recently attracted a lot of interest as
alternative candidates to explain the observed cosmic acceleration without the
need of introducing any scalar field. A critical ingredient is the choice of
the function f(R) of the Ricci scalar curvature entering the gravity Lagrangian
and determining the dynamics of the universe. We describe an efficient
procedure to reconstruct f(R) from the Hubble parameter depending on the
redshift z. Using the metric formulation of f(R) theories, we derive a third
order linear differential equation for f(R(z)) which can be numerically solved
after setting the boundary conditions on the basis of physical considerations.
Since H(z) can be reconstructed from the astrophysical data, the method we
present makes it possible to determine, in principle, what is the f(R) theory
which best reproduces the observed cosmological dynamics. Moreover, the method
allows to reconcile dark energy models with f(R) theories finding out what is
the expression of f(R) which leads to the same H(z) of the given quintessence
model. As interesting examples, we consider "quiessence" (dark energy with
constant equation of state) and the Chaplygin gas.Comment: 15 pages, 4 figures, accepted for publication on Physical Review
Water-waves modes trapped in a canal by a body with the rough surface
The problem about a body in a three dimensional infinite channel is
considered in the framework of the theory of linear water-waves. The body has a
rough surface characterized by a small parameter while the
distance of the body to the water surface is also of order . Under a
certain symmetry assumption, the accumulation effect for trapped mode
frequencies is established, namely, it is proved that, for any given and
integer , there exists such that the problem has at
least eigenvalues in the interval of the continuous spectrum in the
case . The corresponding eigenfunctions decay
exponentially at infinity, have finite energy, and imply trapped modes.Comment: 25 pages, 8 figure
Scalar boundary value problems on junctions of thin rods and plates. I. Asymptotic analysis and error estimates
We derive asymptotic formulas for the solutions of the mixed boundary value
problem for the Poisson equation on the union of a thin cylindrical plate and
several thin cylindrical rods. One of the ends of each rod is set into a hole
in the plate and the other one is supplied with the Dirichlet condition. The
Neumann conditions are imposed on the whole remaining part of the boundary.
Elements of the junction are assumed to have contrasting properties so that the
small parameter, i.e. the relative thickness, appears in the differential
equation, too, while the asymptotic structures crucially depend on the
contrastness ratio. Asymptotic error estimates are derived in anisotropic
weighted Sobolev norms.Comment: 34 pages, 4 figure
Cosmography of f(R) gravity
It is nowadays accepted that the universe is undergoing a phase of
accelerated expansion as tested by the Hubble diagram of Type Ia Supernovae
(SNeIa) and several LSS observations. Future SNeIa surveys and other probes
will make it possible to better characterize the dynamical state of the
universe renewing the interest in cosmography which allows a model independent
analysis of the distance - redshift relation. On the other hand, fourth order
theories of gravity, also referred to as gravity, have attracted a lot
of interest since they could be able to explain the accelerated expansion
without any dark energy. We show here how it is possible to relate the
cosmographic parameters (namely the deceleration , the jerk , the
snap and the lerk parameters) to the present day values of
and its derivatives (with ) thus offering
a new tool to constrain such higher order models. Our analysis thus offers the
possibility to relate the model independent results coming from cosmography to
the theoretically motivated assumptions of cosmology.Comment: 18 pages, 4 figure
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