190,193 research outputs found
Harmonic fields on the extended projective disc and a problem in optics
The Hodge equations for 1-forms are studied on Beltrami's projective disc
model for hyperbolic space. Ideal points lying beyond projective infinity arise
naturally in both the geometric and analytic arguments. An existence theorem
for weakly harmonic 1-fields, changing type on the unit circle, is derived
under Dirichlet conditions imposed on the non-characteristic portion of the
boundary. A similar system arises in the analysis of wave motion near a
caustic. A class of elliptic-hyperbolic boundary-value problems is formulated
for those equations as well. For both classes of boundary-value problems, an
arbitrarily small lower-order perturbation of the equations is shown to yield
solutions which are strong in the sense of Friedrichs.Comment: 30 pages; Section 3.3 has been revise
Liquid crystal defects in the Landau-de Gennes theory in two dimensions-beyond the one-constant approximation
We consider the two-dimensional Landau-de Gennes energy with several elastic
constants, subject to general -radially symmetric boundary conditions. We
show that for generic elastic constants the critical points consistent with the
symmetry of the boundary conditions exist only in the case . In this case
we identify three types of radial profiles: with two, three of full five
components and numerically investigate their minimality and stability depending
on suitable parameters. We also numerically study the stability properties of
the critical points of the Landau-de Gennes energy and capture the intricate
dependence of various qualitative features of these solutions on the elastic
constants and the physical regimes of the liquid crystal system
SU(2) Cosmological Solitons
We present a class of numerical solutions to the SU(2) nonlinear
-model coupled to the Einstein equations with cosmological constant
in spherical symmetry. These solutions are characterized by the
presence of a regular static region which includes a center of symmetry. They
are parameterized by a dimensionless ``coupling constant'' , the sign of
the cosmological constant, and an integer ``excitation number'' . The
phenomenology we find is compared to the corresponding solutions found for the
Einstein-Yang-Mills (EYM) equations with positive (EYM). If
we choose positive and fix , we find a family of static spacetimes
with a Killing horizon for . As a limiting solution
for we find a {\em globally} static spacetime with
, the lowest excitation being the Einstein static universe. To
interpret the physical significance of the Killing horizon in the cosmological
context, we apply the concept of a trapping horizon as formulated by Hayward.
For small values of an asymptotically de Sitter dynamic region contains
the static region within a Killing horizon of cosmological type. For strong
coupling the static region contains an ``eternal cosmological black hole''.Comment: 20 pages, 6 figures, Revte
SU(2) Cosmological Solitons
We present a class of numerical solutions to the SU(2) nonlinear
-model coupled to the Einstein equations with cosmological constant
in spherical symmetry. These solutions are characterized by the
presence of a regular static region which includes a center of symmetry. They
are parameterized by a dimensionless ``coupling constant'' , the sign of
the cosmological constant, and an integer ``excitation number'' . The
phenomenology we find is compared to the corresponding solutions found for the
Einstein-Yang-Mills (EYM) equations with positive (EYM). If
we choose positive and fix , we find a family of static spacetimes
with a Killing horizon for . As a limiting solution
for we find a {\em globally} static spacetime with
, the lowest excitation being the Einstein static universe. To
interpret the physical significance of the Killing horizon in the cosmological
context, we apply the concept of a trapping horizon as formulated by Hayward.
For small values of an asymptotically de Sitter dynamic region contains
the static region within a Killing horizon of cosmological type. For strong
coupling the static region contains an ``eternal cosmological black hole''.Comment: 20 pages, 6 figures, Revte
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