In this paper we prove refined first-order interpolation inequalities for periodic functions and give applications to various refinements of the Carlson--Landau-type inequalities and to magnetic Schrodinger operators. We also obtain Lieb-Thirring ine
qualities for magnetic Schrodinger operators on multi-dimensional cylinders.
In this paper we study a model of randomly colliding particles interacting with a thermal bath. Collisions between particles are modeled via the Kac master equation while the thermostat is seen as an infinite gas at thermal equilibrium at inverse tem
perature $beta$. The system admits the canonical distribution at inverse temperature $beta$ as the unique equilibrium state. We prove that any initial distribution approaches the equilibrium distribution exponentially fast both by computing the gap of the generator of the evolution, in a proper function space, as well as by proving exponential decay in relative entropy. We also show that the evolution propagates chaos and that the one-particle marginal, in the large-system limit, satisfies an effective Boltzmann-type equation.
The steady state for a system of N particle under the influence of an external field and a Gaussian thermostat and colliding with random virtual scatterers can be obtained explicitly in the limit of small field. We show the sequence of steady state d
istribution, as N varies, forms a chaotic sequence in the sense that the k particle marginal, in the limit of large N, is the k-fold tensor product of the 1 particle marginal. We also show that the chaoticity properties holds in the stronger form of entropic chaoticity.
This article reviews recent work on the Kac master equation and its low dimensional counterpart, the Kac equation.
The sharp trace inequality of Jose Escobar is extended to traces for the fractional Laplacian on R^n and a complete characterization of cases of equality is discussed. The proof proceeds via Fourier transform and uses Liebs sharp form of the Hardy-Littlewood-Sobolev inequality.
We prove a new lower bound on the indirect Coulomb energy in quantum mechanics in terms of the single particle density of the system. The new universal lower bound is an alternative to the classical Lieb--Oxford bound (with a smaller constant, 1.45 <
1.68) but involving an additive kinetic energy term of the single particle density.
We study the effect of a cut-off on the speed of pulled fronts of the one dimensional reaction diffusion equation. We prove rigorous upper and lower bounds on the speed in terms of the cut-off parameter epsilon. From these bounds we estimate the rang
e of validity of the Brunet--Derrida formula for a general class of reaction terms.
We prove a sharp Hardy inequality for fractional integrals for functions that are supported on a general domain. The constant is the same as the one for the half-space and hence our result settles a recent conjecture of Bogdan and Dyda.
We consider the Dirichlet Laplacian with a constant magnetic field in a two-dimensional domain of finite measure. We determine the sharp constants in semi-classical eigenvalue estimates and show, in particular, that Polyas conjecture is not true in the presence of a magnetic field.
