3,042 research outputs found
A Non-Local Mean Curvature Flow and its semi-implicit time-discrete approximation
We address in this paper the study of a geometric evolution, corresponding to
a curvature which is non-local and singular at the origin. The curvature
represents the first variation of the energy recently proposed as a variant of
the standard perimeter penalization for the denoising of nonsmooth curves.
To deal with such degeneracies, we first give an abstract existence and
uniqueness result for viscosity solutions of non-local degenerate Hamiltonians,
satisfying suitable continuity assumption with respect to Kuratowsky
convergence of the level sets. This abstract setting applies to an approximated
flow. Then, by the method of minimizing movements, we also build an "exact"
curvature flow, and we illustrate some examples, comparing the results with the
standard mean curvature flow
A differential method for bounding the ground state energy
For a wide class of Hamiltonians, a novel method to obtain lower and upper
bounds for the lowest energy is presented. Unlike perturbative or variational
techniques, this method does not involve the computation of any integral (a
normalisation factor or a matrix element). It just requires the determination
of the absolute minimum and maximum in the whole configuration space of the
local energy associated with a normalisable trial function (the calculation of
the norm is not needed). After a general introduction, the method is applied to
three non-integrable systems: the asymmetric annular billiard, the many-body
spinless Coulombian problem, the hydrogen atom in a constant and uniform
magnetic field. Being more sensitive than the variational methods to any local
perturbation of the trial function, this method can used to systematically
improve the energy bounds with a local skilled analysis; an algorithm relying
on this method can therefore be constructed and an explicit example for a
one-dimensional problem is given.Comment: Accepted for publication in Journal of Physics
Geometric approach to nonvariational singular elliptic equations
In this work we develop a systematic geometric approach to study fully
nonlinear elliptic equations with singular absorption terms as well as their
related free boundary problems. The magnitude of the singularity is measured by
a negative parameter , for , which reflects on
lack of smoothness for an existing solution along the singular interface
between its positive and zero phases. We establish existence as well sharp
regularity properties of solutions. We further prove that minimal solutions are
non-degenerate and obtain fine geometric-measure properties of the free
boundary . In particular we show sharp
Hausdorff estimates which imply local finiteness of the perimeter of the region
and a.e. weak differentiability property of
.Comment: Paper from D. Araujo's Ph.D. thesis, distinguished at the 2013 Carlos
Gutierrez prize for best thesis, Archive for Rational Mechanics and Analysis
201
Repeated games for eikonal equations, integral curvature flows and non-linear parabolic integro-differential equations
The main purpose of this paper is to approximate several non-local evolution
equations by zero-sum repeated games in the spirit of the previous works of
Kohn and the second author (2006 and 2009): general fully non-linear parabolic
integro-differential equations on the one hand, and the integral curvature flow
of an interface (Imbert, 2008) on the other hand. In order to do so, we start
by constructing such a game for eikonal equations whose speed has a
non-constant sign. This provides a (discrete) deterministic control
interpretation of these evolution equations. In all our games, two players
choose positions successively, and their final payoff is determined by their
positions and additional parameters of choice. Because of the non-locality of
the problems approximated, by contrast with local problems, their choices have
to "collect" information far from their current position. For integral
curvature flows, players choose hypersurfaces in the whole space and positions
on these hypersurfaces. For parabolic integro-differential equations, players
choose smooth functions on the whole space
Enhancement of Wigner crystallization in quasi low-dimensional solids
The crystallization of electrons in quasi low-dimensional solids is studied
in a model which retains the full three-dimensional nature of the Coulomb
interactions. We show that restricting the electron motion to layers (or
chains) gives rise to a rich sequence of structural transitions upon varying
the particle density. In addition, the concurrence of low-dimensional electron
motion and isotropic Coulomb interactions leads to a sizeable stabilization of
the Wigner crystal, which could be one of the mechanisms at the origin of the
charge ordered phases frequently observed in such compounds
Energy solutions to one-dimensional singular parabolic problems with data are viscosity solutions
We study one-dimensional very singular parabolic equations with periodic
boundary conditions and initial data in , which is the energy space. We
show existence of solutions in this energy space and then we prove that they
are viscosity solutions in the sense of Giga-Giga.Comment: 15 page
Dynamical response of the "GGG" rotor to test the Equivalence Principle: theory, simulation and experiment. Part I: the normal modes
Recent theoretical work suggests that violation of the Equivalence Principle
might be revealed in a measurement of the fractional differential acceleration
between two test bodies -of different composition, falling in the
gravitational field of a source mass- if the measurement is made to the level
of or better. This being within the reach of ground based
experiments, gives them a new impetus. However, while slowly rotating torsion
balances in ground laboratories are close to reaching this level, only an
experiment performed in low orbit around the Earth is likely to provide a much
better accuracy.
We report on the progress made with the "Galileo Galilei on the Ground" (GGG)
experiment, which aims to compete with torsion balances using an instrument
design also capable of being converted into a much higher sensitivity space
test.
In the present and following paper (Part I and Part II), we demonstrate that
the dynamical response of the GGG differential accelerometer set into
supercritical rotation -in particular its normal modes (Part I) and rejection
of common mode effects (Part II)- can be predicted by means of a simple but
effective model that embodies all the relevant physics. Analytical solutions
are obtained under special limits, which provide the theoretical understanding.
A simulation environment is set up, obtaining quantitative agreement with the
available experimental data on the frequencies of the normal modes, and on the
whirling behavior. This is a needed and reliable tool for controlling and
separating perturbative effects from the expected signal, as well as for
planning the optimization of the apparatus.Comment: Accepted for publication by "Review of Scientific Instruments" on Jan
16, 2006. 16 2-column pages, 9 figure
The Tychonoff uniqueness theorem for the G-heat equation
In this paper, we obtain the Tychonoff uniqueness theorem for the G-heat
equation
The Cosmological Time Function
Let be a time oriented Lorentzian manifold and the Lorentzian
distance on . The function is the cosmological
time function of , where as usual means that is in the causal
past of . This function is called regular iff for all
and also along every past inextendible causal curve. If the
cosmological time function of a space time is regular it has
several pleasant consequences: (1) It forces to be globally hyperbolic,
(2) every point of can be connected to the initial singularity by a
rest curve (i.e., a timelike geodesic ray that maximizes the distance to the
singularity), (3) the function is a time function in the usual sense, in
particular (4) is continuous, in fact locally Lipschitz and the second
derivatives of exist almost everywhere.Comment: 19 pages, AEI preprint, latex2e with amsmath and amsth
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