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Exact relativistic treatment of stationary counter-rotating dust disks I: Boundary value problems and solutions
This is the first in a series of papers on the construction of explicit
solutions to the stationary axisymmetric Einstein equations which describe
counter-rotating disks of dust. These disks can serve as models for certain
galaxies and accretion disks in astrophysics. We review the Newtonian theory
for disks using Riemann-Hilbert methods which can be extended to some extent to
the relativistic case where they lead to modular functions on Riemann surfaces.
In the case of compact surfaces these are Korotkin's finite gap solutions which
we will discuss in this paper. On the axis we establish for general genus
relations between the metric functions and hence the multipoles which are
enforced by the underlying hyperelliptic Riemann surface. Generalizing these
results to the whole spacetime we are able in principle to study the classes of
boundary value problems which can be solved on a given Riemann surface. We
investigate the cases of genus 1 and 2 of the Riemann surface in detail and
construct the explicit solution for a family of disks with constant angular
velocity and constant relative energy density which was announced in a previous
Physical Review Letter.Comment: 32 pages, 1 figure, to appear in Phys. Rev.
Gravity on codimension 2 brane worlds
We compute the matching conditions for a general thick codimension 2 brane, a
necessary previous step towards the investigation of gravitational phenomena in
codimension 2 braneworlds. We show that, provided the brane is weakly curved,
they are specified by the integral in the extra dimensions of the brane
energy-momentum, independently of its detailed internal structure. These
general matching conditions can then be used as boundary conditions for the
bulk solution. By evaluating Einstein equations at the brane boundary we are
able to write an evolution equation for the induced metric on the brane
depending only on physical brane parameters and the bulk energy-momentum
tensor. We particularise to a cosmological metric and show that a realistic
cosmology can be obtained in the simplest case of having just a non-zero
cosmological constant in the bulk. We point out several parallelisms between
this case and the codimension 1 brane worlds in an AdS space.Comment: 24 page
On the solvability of third-order three point systems of differential equations with dependence on the first derivative
This paper presents sufficient conditions for the solvability of the third
order three point boundary value problem \begin{equation*} \left\{
\begin{array}{c} -u^{\prime \prime \prime }(t)=f(t,\,v(t),\,v^{\prime }(t)) \\
-v^{\prime \prime \prime }(t)=h(t,\,u(t),\,u^{\prime }(t)) \\ u(0)=u^{\prime
}(0)=0,u^{\prime }(1)=\alpha u^{\prime }(\eta ) \\ v(0)=v^{\prime
}(0)=0,v^{\prime }(1)=\alpha v^{\prime }(\eta ).
\end{array}
\right. \end{equation*} The arguments apply Green's function associated to
the linear problem and the Guo--Krasnosel'ski\u{\i} theorem of
compression-expansion cones. The dependence on the first derivatives is
overcome by the construction of an adequate cone and suitable conditions of
superlinearity/sublinearity near and Last section contains an
example to illustrate the applicability of the theorem.Comment: 21 page
The Unified Method: I Non-Linearizable Problems on the Half-Line
Boundary value problems for integrable nonlinear evolution PDEs formulated on
the half-line can be analyzed by the unified method introduced by one of the
authors and used extensively in the literature. The implementation of this
general method to this particular class of problems yields the solution in
terms of the unique solution of a matrix Riemann-Hilbert problem formulated in
the complex -plane (the Fourier plane), which has a jump matrix with
explicit -dependence involving four scalar functions of , called
spectral functions. Two of these functions depend on the initial data, whereas
the other two depend on all boundary values. The most difficult step of the new
method is the characterization of the latter two spectral functions in terms of
the given initial and boundary data, i.e. the elimination of the unknown
boundary values. For certain boundary conditions, called linearizable, this can
be achieved simply using algebraic manipulations. Here, we present an effective
characterization of the spectral functions in terms of the given initial and
boundary data for the general case of non-linearizable boundary conditions.
This characterization is based on the analysis of the so-called global
relation, on the analysis of the equations obtained from the global relation
via certain transformations leaving the dispersion relation of the associated
linearized PDE invariant, and on the computation of the large asymptotics
of the eigenfunctions defining the relevant spectral functions.Comment: 39 page
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