A class of degenerate elliptic equations and a Dido's problem with respect to a measure

In this paper we consider the following class of linear elliptic problems

     

A general two-scale criteria for logarithmic Sobolev inequalities

We present a general criteria to prove that a probability measure satisfies a logarithmic Sobolev inequality, knowing that some of its marginals and associated conditional laws satisfy a logarithmic Sobolev inequality. This is a generalization of a result by N. Grunewald et al. [N. Grunewald, F. Otto, C. Villani, M.G. Westdickenberg, A two-scale approach to logarithmic Sobolev inequalities and the hydrodynamic limit, Ann. Inst. H. Poincaré Probab. Statist., in press].     

The general optimal L-Euclidean logarithmic Sobolev inequality by Hamilton-Jacobi equations

We prove a general optimal L^p-Euclidean logarithmic Sobolev inequality by using Prékopa-Leindler inequality and a special Hamilton-Jacobi equation. In particular we generalize the inequality proved by Del Pino and Dolbeault in (J. Funt. Anal.).     

Classical solutions for the pressure-gradient equations in non-smooth and non-convex domains

We establish the existence of the classical solution for the pressure-gradient equation in a non-smooth and non-convex domain. The equation is elliptic inside the domain, becomes degenerate on the boundary, and is singular at the origin when the origin lies on the boundary. We show the solution is smooth inside the domain and continuous up to the boundary.     

Harnack inequality and smoothness for quasilinear degenerate elliptic equations

We prove local and global regularity for the positive solutions of a quasilinear variational degenerate equation, assuming minimal hypothesis on the coefficients of the lower order terms. As an application we obtain Hölder continuity for the gradient of solutions to nonvariational quasilinear equations.     

A Sobolev-like inequality for the Dirac operator

In this article, we prove a Sobolev-like inequality for the Dirac operator on closed compact Riemannian spin manifolds with a nearly optimal Sobolev constant. As an application, we give a criterion for the existence of solutions to a nonlinear equation with critical Sobolev exponent involving the Dirac operator. We finally specify a case where this equation can be solved.     

The Equivalence of Hypercontractivity and Logarithmic Sobolev Inequality for q ((1 ≤ q ≤ 1) -Ornstein-Uhlenbeck Semigroup

In this paper the author proves the equivalence of hypercontractivity and logarithmic Sobolev inequality for q-Ornstein-Uhlenbeck semigroup U_t⁽(q))= Γ_q(e^-tI)(-1 ≤q≤ 1), where Γ_q is a q-Gaussian functor.     

Cell polarisation model: The 1D case

We study the dynamics of a one-dimensional non-linear and non-local drift-diffusion equation set in the half-line, with the coupling involving the trace value on the boundary. The initial mass M of the density determines the behaviour of the equation: attraction to self-similar profile, to a steady state of finite time, blow-up for supercritical mass. Using the logarithmic Sobolev and the HWI inequalities we obtain a rate of convergence for the sub-critical and critical mass cases. Moreover, we prove a comparison principle on the equation obtained after space integration. This concentration-comparison principle allows proving blow-up of solutions for large initial data without any monotonicity assumption on the initial data.     

直线上函数的Banach空间的Poincaré型不等式的变分公式

基于研究对数Sobolev,Nash和其它泛函不等式的需要,将Poincare不等式 的变分公式拓广到一大类直线上函数的Banach(Orlicz)空间.给出了这些不等式成立 与否的显式判准和显式估计. 作为典型应用,仔细考察了对数Sobolev常数.     

Lipschitz constants and logarithmic Sobolev inequality

We revisit two results of  [3]; they are a logarithmic Sobolev inequality on Rⁿ${\mathbb{R}}^n$ with Lipschitz constants and an expression of Lipschitz constants as the limit of a functional by the entropy. We have two goals in this paper. The first goal is to clarify when the strict inequality holds in this inequality. The second goal is to investigate the asymptotic behavior of this functional by the Abelian and Tauberian theorems of Laplace transforms.     

Log-Sobolev inequalities and regions with exterior exponential cusps

We begin by studying certain semigroup estimates which are more singular than those implied by a Sobolev embedding theorem but which are equivalent to certain logarithmic Sobolev inequalities. We then give a method for proving that such log-Sobolev inequalities hold for Euclidean regions which satisfy a particular Hardy-type inequality. Our main application is to show that domains which have exterior exponential cusps, and hence have no Sobolev embedding theorem, satisfy such heat kernel bounds provided the cusps are not too sharp. Finally, we consider a rotationally invariant domain with an exponentially sharp cusp and prove that ultracontractivity breaks down when the cusp becomes too sharp.     

A very simple proof of the LSI for high temperature spin systems

We present a very simple proof that the $O(n)$ model satisfies a uniform logarithmic Sobolev inequality (LSI) if the difference between the largest and the smallest eigenvalue of the coupling matrix is less than n. This condition applies in particular to the SK spin glass model at inverse temperature $\beta <1/4$. It is the first result of rapid relaxation for the SK model and requires significant cancellations between the ferromagnetic and anti-ferromagnetic spin couplings that cannot be obtained by existing methods to prove Log-Sobolev inequalities. The proof also applies to more general bounded and unbounded spin systems. It uses a single step of zero range renormalisation and Bakry–Emery theory for the renormalised measure.