321 research outputs found
Convex Sobolev inequalities and spectral gap
This note is devoted to the proof of convex Sobolev (or generalized
Poincar\'{e}) inequalities which interpolate between spectral gap (or
Poincar\'{e}) inequalities and logarithmic Sobolev inequalities. We extend to
the whole family of convex Sobolev inequalities results which have recently
been obtained by Cattiaux and Carlen and Loss for logarithmic Sobolev
inequalities. Under local conditions on the density of the measure with respect
to a reference measure, we prove that spectral gap inequalities imply all
convex Sobolev inequalities with constants which are uniformly bounded in the
limit approaching the logarithmic Sobolev inequalities. We recover the case of
the logarithmic Sobolev inequalities as a special case
Bifurcation diagrams and multiplicity for nonlocal elliptic equations modeling gravitating systems based on Fermi-Dirac statistics
This paper is devoted to multiplicity results of solutions to nonlocal
elliptic equations modeling gravitating systems. By considering the case of
Fermi-Dirac statistics as a singular perturbation of Maxwell-Boltzmann one, we
are able to produce multiplicity results. Our method is based on cumulated mass
densities and a logarithmic change of coordinates that allows us to describe
the set of all solutions by a non-autonomous perturbation of an autonomous
dynamical system. This has interesting consequences in terms of bifurcation
diagrams, which are illustrated by a some numerical computations. More
specifically, we study a model based on the Fermi function as well as a
simplified one for which estimates are easier to establish. The main difficulty
comes from the fact that the mass enters in the equation as a parameter which
makes the whole problem non-local
Heterogeneous social interactions and the COVID-19 lockdown outcome in a multi-group SEIR model
We study variants of the SEIR model for interpreting some qualitative
features of the statistics of the Covid-19 epidemic in France. Standard SEIR
models distinguish essentially two regimes: either the disease is controlled
and the number of infected people rapidly decreases, or the disease spreads and
contaminates a significant fraction of the population until herd immunity is
achieved. After lockdown, at first sight it seems that social distancing is not
enough to control the outbreak. We discuss here a possible explanation, namely
that the lockdown is creating social heterogeneity: even if a large majority of
the population complies with the lockdown rules, a small fraction of the
population still has to maintain a normal or high level of social interactions,
such as health workers, providers of essential services, etc. This results in
an apparent high level of epidemic propagation as measured through
re-estimations of the basic reproduction ratio. However, these measures are
limited to averages, while variance inside the population plays an essential
role on the peak and the size of the epidemic outbreak and tends to lower these
two indicators. We provide theoretical and numerical results to sustain such a
view
Best matching Barenblatt profiles are delayed
The growth of the second moments of the solutions of fast diffusion equations
is asymptotically governed by the behavior of self-similar solutions. However,
at next order, there is a correction term which amounts to a delay depending on
the nonlinearity and on a distance of the initial data to the set of
self-similar Barenblatt solutions. This distance can be measured in terms of a
relative entropy to the best matching Barenblatt profile. This best matching
Barenblatt function determines a scale. In new variables based on this scale,
which are given by a self-similar change of variables if and only if the
initial datum is one of the Barenblatt profiles, the typical scale is monotone
and has a l
Stability results for logarithmic Sobolev and Gagliardo-Nirenberg inequalities
This paper is devoted to improvements of functional inequalities based on
scalings and written in terms of relative entropies. When scales are taken into
account and second moments fixed accordingly, deficit functionals provide
explicit stability measurements, i.e., bound with explicit constants distances
to the manifold of optimal functions. Various results are obtained for the
Gaussian logarithmic Sobolev inequality and its Euclidean counterpart, for the
Gaussian generalized Poincar{\'e} inequalities and for the Gagliardo-Nirenberg
inequalities. As a consequence, faster convergence rates in diffusion equations
(fast diffusion, Ornstein-Uhlenbeck and porous medium equations) are obtained
Nonlinear diffusions: extremal properties of Barenblatt profiles, best matching and delays
In this paper, we consider functionals based on moments and non-linear
entropies which have a linear growth in time in case of source-type so-lutions
to the fast diffusion or porous medium equations, that are also known as
Barenblatt solutions. As functions of time, these functionals have convexity
properties for generic solutions, so that their asymptotic slopes are extremal
for Barenblatt profiles. The method relies on scaling properties of the
evo-lution equations and provides a simple and direct proof of sharp
Gagliardo-Nirenberg-Sobolev inequalities in scale invariant form. The method
also gives refined estimates of the growth of the second moment and, as a
consequence, establishes the monotonicity of the delay corresponding to the
best matching Barenblatt solution compared to the Barenblatt solution with same
initial sec-ond moment. Here the notion of best matching is defined in terms of
a relative entropy
Improved Poincar\'e inequalities
Although the Hardy inequality corresponding to one quadratic singularity,
with optimal constant, does not admit any extremal function, it is well known
that such a potential can be improved, in the sense that a positive term can be
added to the quadratic singularity without violating the inequality, and even a
whole asymptotic expansion can be build, with optimal constants for each term.
This phenomenon has not been much studied for other inequalities. Our purpose
is to prove that it also holds for the gaussian Poincar\'e inequality. The
method is based on a recursion formula, which allows to identify the optimal
constants in the asymptotic expansion, order by order. We also apply the same
strategy to a family of Hardy-Poincar\'e inequalities which interpolate between
Hardy and gaussian Poincar\'e inequalities
Extremal functions for Caffarelli-Kohn-Nirenberg and logarithmic Hardy inequalities
We consider a family of Caffarelli-Kohn-Nirenberg interpolation inequalities
and weighted logarithmic Hardy inequalities which have been obtained recently
as a limit case of the first ones. We discuss the ranges of the parameters for
which the optimal constants are achieved by extremal functions. The comparison
of these optimal constants with the optimal constants of Gagliardo-Nirenberg
interpolation inequalities and Gross' logarithmic Sobolev inequality, both
without weights, gives a general criterion for such an existence result in some
particular cases.Comment: Proc. Edinburgh A (2012) To appea
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