45 research outputs found
From Poincar\'e to logarithmic Sobolev inequalities: a gradient flow approach
We use the distances introduced in a previous joint paper to exhibit the
gradient flow structure of some drift-diffusion equations for a wide class of
entropy functionals. Functional inequalities obtained by the comparison of the
entropy with the entropy production functional reflect the contraction
properties of the flow. Our approach provides a unified framework for the study
of the Kolmogorov-Fokker-Planck (KFP) equation
Infinite-horizon problems under periodicity constraint
We study so{\`u}e infinite-horizon optimization problems on spaces of
periodic functions for non periodic Lagrangians. The main strategy relies on
the reduction to finite horizon thanks in the introduction of an avering
operator.We then provide existence results and necessary optimality conditions
in which the corresponding averaged Lagrangian appears
Weighted interpolation inequalities: a perturbation approach
We study optimal functions in a family of Caffarelli-Kohn-Nirenberg
inequalities with a power-law weight, in a regime for which standard
symmetrization techniques fail. We establish the existence of optimal
functions, study their properties and prove that they are radial when the power
in the weight is small enough. Radial symmetry up to translations is true for
the limiting case where the weight vanishes, a case which corresponds to a
well-known subfamily of Gagliardo-Nirenberg inequalities. Our approach is based
on a concentration-compactness analysis and on a perturbation method which uses
a spectral gap inequality. As a consequence, we prove that optimal functions
are explicit and given by Barenblatt-type profiles in the perturbative regime
Geodesics for a class of distances in the space of probability measures
In this paper, we study the characterization of geodesics for a class of
distances between probability measures introduced by Dolbeault, Nazaret and
Savar e. We first prove the existence of a potential function and then give
necessary and suffi cient optimality conditions that take the form of a coupled
system of PDEs somehow similar to the Mean-Field-Games system of Lasry and
Lions. We also consider an equivalent formulation posed in a set of probability
measures over curves
Optimal transportation for the determinant
Among -valued triples of random vectors having fixed marginal
probability laws, what is the best way to jointly draw in such a way
that the simplex generated by has maximal average volume? Motivated
by this simple question, we study optimal transportation problems with several
marginals when the objective function is the determinant or its absolute value
Weighted fast diffusion equations (Part I): Sharp asymptotic rates without symmetry and symmetry breaking in Caffarelli-Kohn-Nirenberg inequalities
In this paper we consider a family of Caffarelli-Kohn-Nirenberg interpolation
inequalities (CKN), with two radial power law weights and exponents in a
subcritical range. We address the question of symmetry breaking: are the
optimal functions radially symmetric, or not ? Our intuition comes from a
weighted fast diffusion (WFD) flow: if symmetry holds, then an explicit entropy
- entropy production inequality which governs the intermediate asymptotics is
indeed equivalent to (CKN), and the self-similar profiles are optimal for
(CKN). We establish an explicit symmetry breaking condition by proving the
linear instability of the radial optimal functions for (CKN). Symmetry breaking
in (CKN) also has consequences on entropy - entropy production inequalities and
on the intermediate asymptotics for (WFD). Even when no symmetry holds in
(CKN), asymptotic rates of convergence of the solutions to (WFD) are determined
by a weighted Hardy-Poincar{\'e} inequality which is interpreted as a
linearized entropy - entropy production inequality. All our results rely on the
study of the bottom of the spectrum of the linearized diffusion operator around
the self-similar profiles, which is equivalent to the linearization of (CKN)
around the radial optimal functions, and on variational methods. Consequences
for the (WFD) flow will be studied in Part II of this work
Weighted fast diffusion equations (Part II): Sharp asymptotic rates of convergence in relative error by entropy methods
This paper is the second part of the study. In Part~I, self-similar solutions
of a weighted fast diffusion equation (WFD) were related to optimal functions
in a family of subcritical Caffarelli-Kohn-Nirenberg inequalities (CKN) applied
to radially symmetric functions. For these inequalities, the linear instability
(symmetry breaking) of the optimal radial solutions relies on the spectral
properties of the linearized evolution operator. Symmetry breaking in (CKN) was
also related to large-time asymptotics of (WFD), at formal level. A first
purpose of Part~II is to give a rigorous justification of this point, that is,
to determine the asymptotic rates of convergence of the solutions to (WFD) in
the symmetry range of (CKN) as well as in the symmetry breaking range, and even
in regimes beyond the supercritical exponent in (CKN). Global rates of
convergence with respect to a free energy (or entropy) functional are also
investigated, as well as uniform convergence to self-similar solutions in the
strong sense of the relative error. Differences with large-time asymptotics of
fast diffusion equations without weights will be emphasized