2,946 research outputs found
A remark on Einstein warped products
We prove triviality results for Einstein warped products with non-compact
bases. These extend previous work by D.-S. Kim and Y.-H. Kim. The proof, from
the viewpoint of "quasi-Einstein manifolds" introduced by J. Case, Y.-S. Shu
and G. Wei, rely on maximum principles at infinity and Liouville-type theorems.Comment: 12 pages. Corrected typos. Final version: to appear on Pacific J.
Mat
Practically linear analogs of the Born-Infeld and other nonlinear theories
I discuss theories that describe fully nonlinear physics, while being
practically linear (PL), in that they require solving only linear differential
equations. These theories may be interesting in themselves as manageable
nonlinear theories. But, they can also be chosen to emulate genuinely nonlinear
theories of special interest, for which they can serve as approximations. The
idea can be applied to a large class of nonlinear theories, exemplified here
with a PL analogs of scalar theories, and of Born-Infeld (BI) electrodynamics.
The general class of such PL theories of electromagnetism are governed by a
Lagrangian L=-(1/2)F_mnQ^mn+ S(Q_mn), where the electromagnetic field couples
to currents in the standard way, while Qmn is an auxiliary field, derived from
a vector potential that does not couple directly to currents. By picking a
special form of S(Q_mn), we can make such a theory similar in some regards to a
given fully nonlinear theory, governed by the Lagrangian -U(F_mn). A
particularly felicitous choice is to take S as the Legendre transform of U. For
the BI theory, this Legendre transform has the same form as the BI Lagrangian
itself. Various matter-of-principle questions remain to be answered regarding
such theories. As a specific example, I discuss BI electrostatics in more
detail. As an aside, for BI, I derive an exact expression for the
short-distance force between two arbitrary point charges of the same sign, in
any dimension.Comment: 20 pages, Version published in Phys. Rev.
On existence of matter outside a static black hole
It is expected that matter composed of a perfect fluid cannot be at rest
outside of a black hole if the spacetime is asymptotically flat and static
(non-rotating). However, there has not been a rigorous proof for this
expectation without assuming spheical symmetry. In this paper, we provide a
proof of non-existence of matter composed of a perfect fluid in static black
hole spacetimes under certain conditions, which can be interpreted as a
relation between the stellar mass and the black hole mass.Comment: 4pages, final version accepted for publication in Journal of
Mathematical Physic
Spherical linear waves in de Sitter spacetime
We apply Christodoulou's framework, developed to study the Einstein-scalar
field equations in spherical symmetry, to the linear wave equation in de Sitter
spacetime, as a first step towards the Einstein-scalar field equations with
positive cosmological constant. We obtain an integro-differential evolution
equation which we solve by taking initial data on a null cone. As a corollary
we obtain elementary derivations of expected properties of linear waves in de
Sitter spacetime: boundedness in terms of (characteristic) initial data, and a
Price law establishing uniform exponential decay, in Bondi time, to a constant.Comment: 9 pages, 1 figure; v2: minor changes, references added, matches final
published versio
Suspension and levitation in nonlinear theories
I investigate stable equilibria of bodies in potential fields satisfying a
generalized Poisson equation: divergence[m(grad phi) grad phi]= source density.
This describes diverse systems such as nonlinear dielectrics, certain flow
problems, magnets, and superconductors in nonlinear magnetic media; equilibria
of forced soap films; and equilibria in certain nonlinear field theories such
as Born-Infeld electromagnetism. Earnshaw's theorem, totally barring stable
equilibria in the linear case, breaks down. While it is still impossible to
suspend a test, point charge or dipole, one can suspend point bodies of finite
charge, or extended test-charge bodies. I examine circumstances under which
this can be done, using limits and special cases. I also consider the analogue
of magnetic trapping of neutral (dipolar) particles.Comment: Five pages, Revtex, to appear in Physics Letters
Directional approach to spatial structure of solutions to the Navier-Stokes equations in the plane
We investigate a steady flow of incompressible fluid in the plane. The motion
is governed by the Navier-Stokes equations with prescribed velocity
at infinity. The main result shows the existence of unique solutions for
arbitrary force, provided sufficient largeness of . Furthermore a
spacial structure of the solution is obtained in comparison with the Oseen
flow. A key element of our new approach is based on a setting which treats the
directino of the flow as \emph{time} direction. The analysis is done in
framework of the Fourier transform taken in one (perpendicular) direction and a
special choice of function spaces which take into account the inhomogeneous
character of the symbol of the Oseen system. From that point of view our
technique can be used as an effective tool in examining spatial asymptotics of
solutions to other systems modeled by elliptic equations
Nonclassical rotational inertia for a supersolid under rotation
As proposed by Leggett [4], the supersolidity of a crystal is characterized
by the Non Classical Rotational Inertia (NCRI) property. Using a model of
quantum crystal introduced by Josserand, Pomeau and Rica [5], we prove that
NCRI occurs. This is done by analyzing the ground state of the aforementioned
model, which is related to a sphere packing problem, and then deriving a
theoretical formula for the inertia momentum. We infer a lower estimate for the
NCRI fraction, which is a landmark of supersolidity
The plastikstufe - a generalization of the overtwisted disk to higher dimensions
In this article, we give a first prototype-definition of overtwistedness in
higher dimensions. According to this definition, a contact manifold is called
"overtwisted" if it contains a "plastikstufe", a submanifold foliated by the
contact structure in a certain way. In three dimensions the definition of the
plastikstufe is identical to the one of the overtwisted disk. The main
justification for this definition lies in the fact that the existence of a
plastikstufe implies that the contact manifold does not have a (semipositive)
symplectic filling.Comment: This is the version published by Algebraic & Geometric Topology on 15
December 200
Results for a turbulent system with unbounded viscosities: weak formulations, existence of solutions, boundedness, smoothness'
We consider a circulation system arising in turbulence modelling in fluid
dynamics with unbounded eddy viscosities. Various notions of weak solutions are
considered and compared. We establish existence and regularity results. In
particular we study the boundedness of weak solutions. We also establish an
existence result for a classical solutio
Local Asymmetry and the Inner Radius of Nodal Domains
Let M be a closed Riemannian manifold of dimension n. Let f be an
eigenfunction of the Laplace-Beltrami operator corresponding to an eigenvalue
\lambda. We show that the volume of {f>0} inside any ball B whose center lies
on {f=0} is > C|B|/\lambda^n. We apply this result to prove that each nodal
domain contains a ball of radius > C/\lambda^n.Comment: 12 pages, 1 figure; minor corrections; to appear in Comm. PDE
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