7 research outputs found
The information-theoretic meaning of Gagliardo--Nirenberg type inequalities
Gagliardo--Nirenberg inequalities are interpolation inequalities which were
proved independently by Gagliardo and Nirenberg in the late fifties. In recent
years, their connections with theoretic aspects of information theory and
nonlinear diffusion equations allowed to obtain some of them in optimal form,
by recovering both the sharp constants and the explicit form of the optimizers.
In this note, at the light of these recent researches, we review the main
connections between Shannon-type entropies, diffusion equations and a class of
these inequalities
A concavity property for the reciprocal of Fisher information and its consequences on Costa's EPI
We prove that the reciprocal of Fisher information of a log-concave
probability density in is concave in with respect to the
addition of a Gaussian noise . As a byproduct of this result
we show that the third derivative of the entropy power of a log-concave
probability density in is nonnegative in with respect to
the addition of a Gaussian noise . For log-concave densities this improves
the well-known Costa's concavity property of the entropy power
Interpolation inequalities and spectral estimates for magnetic operators
We prove magnetic interpolation inequalities and Keller-Lieb-Thir-ring
estimates for the principal eigenvalue of magnetic Schr{\"o}dinger operators.
We establish explicit upper and lower bounds for the best constants and show by
numerical methods that our theoretical estimates are accurate
Rényi entropies and nonlinear diffusion equations
Since their introduction in the early sixties, the Rényi entropies have been used in many contexts, ranging from information theory to astrophysics, turbulence phenomena and others. In this note, we enlighten the main connections between Rényi entropies and nonlinear diffusion equations. In particular, it is shown that these relationships allow to prove various functional inequalities in sharp form