THE LOGARITHMIC SOBOLEV INEQUALITY FOR A SUBMANIFOLD IN MANIFOLDS WITH ASYMPTOTICALLY NONNEGATIVE SECTIONAL CURVATURE

In this note,we prove a logarithmic Sobolev inequality which holds for compact submanifolds without a boundary in manifolds with asymptotically nonnegative sectional curvature.Like the Michale-Simon Sobolev inequality,this inequality contains a term involving the mean curvature.

Rearrangement and the weighted logarithmic Sobolev inequality

Here we consider some weighted logarithmic Sobolev inequalities which can be used in the theory of singular Riemanian manifolds.We give the necessary and sufficient conditions such that the 1-dimension weighted logarithmic Sobolev inequality is true and obtain a new estimate on the entropy.

The Logarithmic Sobolev Inequality for a Submanifold in Manifolds with Nonnegative Sectional Curvature

The authors prove a sharp logarithmic Sobolev inequality which holds for compact submanifolds without boundary in Riemannian manifolds with nonnegative sectional curvature of arbitrary dimension and codimension.Like the Michael-Simon Sobolev inequality,this inequality includes a term involving the mean curvature.This extends a recent result of Brendle with Euclidean setting.

Improved Hardy–Littlewood–Sobolev Inequality on S^(n)under Constraints

In this paper,we establish an improved Hardy–Littlewood–Sobolev inequality on Snunder higher-order moments constraint.Moreover,by constructing precise test functions,using improved Hardy–Littlewood–Sobolev inequality on S^(n),we show such inequality is almost optimal in critical case.As an application,we give a simpler proof of the existence of the maximizer for conformal Hardy–Littlewood–Sobolev inequality.

The Logarithmic Sobolev Inequality Along the Ricci Flow:The Caseλ0(g0)=0

A uniform logarithmic Sobolev inequality,a uniform Sobolev inequality and a uniformκ-noncollapsing estimate along the Ricci flow are established in the situation that a certain smallest eigenvalue for the initial metric is zero.

Talagrand's T_2-Transportation Inequality and Log-Sobolev Inequality for Dissipative SPDEs and Applications to Reaction-Diffusion Equations

We establish Talagrand＇s T2-transportation inequalities for infinite dimensional dissipative diffusions with sharp constants, through Galerkin type＇s approximations and the known results in the finite dimensional case. Furthermore in the additive noise case we prove also logarithmic Sobolev inequalities with sharp constants. Applications to Reaction- Diffusion equations are provided.

SOBOLEV INEQUALITY ON RIEMANNIAN MANIFOLDS

Let M be an n dimensional complete Riemannian manifold satisfying the doublingvolume property and an on-diagonal heat kernel estimate. The necessary-sufficientcondition for the Sobolev inequality ‖f‖q ≤ Cn,,v,p,q(‖▽f‖p+‖fp) (2≤p＜q＜∞) is given.

A Stein deficit for the logarithmic Sobolev inequality

We provide some lower bounds on the deficit in the Gaussian logarithmic Sobolev inequality in terms of the so-called Stein characterization of the Gaussian distribution.The techniques are based on the representation of the relative Fisher information along the Ornstein-Uhlenbeck semigroup by the Minimum Mean-Square Error from information theory.

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 Ut(q)=Γq(e-tI)(-1≤q≤1),whereΓq is a q-Gaussian functor.

A Note on Sharp Affine Poincaré-Sobolev Inequalities and Exact in Minimization of Zhang’s Energy on Bounded Variation and Exactness

As for the affine energy, Edir Junior and Ferreira Leite establish the existence of minimizers for particular restricted subcritical and critical variational issues on BV(Ω). Similar functionals exhibit deeper weak* topological traits including lower semicontinuity and affine compactness, and their geometry is non-coercive. Our work also proves the result that extremal functions exist for certain affine Poincaré-Sobolev inequalities.

Verification of the Landau Equation and Hardy’s Inequality

We prove the L estimate for the isotropic version of the homogeneous landau problem, which was explored by M. Gualdani and N. Guillen. As shown in a region of the smooth potentials range under values of the interaction exponent (2), a weighted Poincaré inequality is a natural consequence of the traditional weighted Hardy inequality, which in turn implies that the norms of solutions propagate in the L1 space. Now, the L estimate is based on the work of De Giorgi, Nash, and Moser, as well as a few weighted Sobolev inequalities.

Global Existence of the Solution for a Reduced Model of the Vectorial Quantum Zakharov System

In this paper, we study the global existence of the smooth solution for a reduced quantum Zakharov system in two spatial dimensions. Using energy estimates and the logarithmic type Sobolev inequality, we show the global existence of the solution to this system without any small condition on the initial data.

Logarithmic Sobolev Inequalities for Two-Sided Birth-Death Processes

In this paper, we study the logarithmic Sobolev inequalities for two-sided birth-death processes. An estimate of the logarithmic Sobolev constant α for a two-sided birth-death process is obtained by the Hardy-type inequality and a criteria for a is also presented.

ON CRITICAL CASES OF SOBOLEV'S INEQUALITIES FOR HEISENBERG GROUPS

We prove some Trudinger-type inequalities and Brezis-Gallouet-Wainger inequality on the Heisenberg group, extending to this context the Euclidean results by T. Ozawa.

L_(p)-Minkowski问题周期解的存在性

L_(p)-Minkowski问题是凸几何分析L_(p)-Brunn-Minkowski理论的核心,其实质是分析给定测度是否为凸体的L_(p)表面积测度问题.这个问题可以简化为二阶微分方程的周期解的存在性,L_(p)-Minkowski问题中周期解的存在性问题如下u″+u=h(t)/u^(ρ),其中h>0是连续的周期函数,常数ρ=1-p.利用了二阶微分方程周期解存在的充分条件,通过建立的一个方程的周期解的存在性判据,再利用Sobolev inequality证明了这个二阶微分方程周期解的存在性,得到在一定条件下周期解存在,这个方法在一定程度上扩大p的取值范围.最后给出一个例子,验证文中所得到的主要结果的可行性.

THE EXISTENCE OF A NONTRIVIAL WEAK SOLUTION TO A DOUBLE CRITICAL PROBLEM INVOLVING A FRACTIONAL LAPLACIAN IN R^N WITH A HARDY TERM

In this paper,we consider the existence of nontrivial weak solutions to a double critical problem involving a fractional Laplacian with a Hardy term:(−Δ)s u−γu|x|2s=|u|2∗s(β)−2 u|x|β+[Iμ∗Fα(⋅,u)](x)fα(x,u),u∈H˙s(R n),(0.1)(1)where s∈(0,1),0≤α,β<2s<n,μ∈(0,n),γ<γH,Iμ(x)=|x|−μ,Fα(x,u)=|u(x)|2#μ(α)|x|δμ(α),fα(x,u)=|u(x)|2#μ(α)−2 u(x)|x|δμ(α),2#μ(α)=(1−μ2n)⋅2∗s(α),δμ(α)=(1−μ2n)α,2∗s(α)=2(n−α)n−2s andγH=4 sΓ2(n+2s4)Γ2(n−2s4).We show that problem(0.1)admits at least a weak solution under some conditions.To prove the main result,we develop some useful tools based on a weighted Morrey space.To be precise,we discover the embeddings H˙s(R n)↪L 2∗s(α)(R n,|y|−α)↪L p,n−2s2 p+pr(R n,|y|−pr),(0.2)(2)where s∈(0,1),0<α<2s<n,p∈[1,2∗s(α))and r=α2∗s(α).We also establish an improved Sobolev inequality,(∫R n|u(y)|2∗s(α)|y|αdy)12∗s(α)≤C||u||θH˙s(R n)||u||1−θL p,n−2s2 p+pr(R n,|y|−pr),∀u∈H˙s(R n),(0.3)(3)where s∈(0,1),0<α<2s<n,p∈[1,2∗s(α)),r=α2∗s(α),0<max{22∗s(α),2∗s−12∗s(α)}<θ<1,2∗s=2nn−2s and C=C(n,s,α)>0 is a constant.Inequality(0.3)is a more general form of Theorem 1 in Palatucci,Pisante[1].By using the mountain pass lemma along with(0.2)and(0.3),we obtain a nontrivial weak solution to problem(0.1)in a direct way.It is worth pointing out that(0.2)and 0.3)could be applied to simplify the proof of the existence results in[2]and[3].

IMPROVED GAGLIARDO-NIRENBERG INEQUALITIES ON HEISENBERG TYPE GROUPS

Motivated by the idea of M. Ledoux who brings out the connection between Sobolev embeddings and heat kernel bounds, we prove an analogous result for Kohn’s sub-Laplacian on the Heisenberg type groups. The main result includes features of an inequality of either Sobolev or Galiardo-Nirenberg type.

INFINITELY MANY SOLUTIONS FOR A CLASS OF DEGENERATE ELLIPTIC EQUATIONS

Let X = （X1, ···, Xm） be an infinitely degenerate system of vector fields. The aim of this paper is to study the existence of infinitely many solutions for the sum of operators X =sum （ ） form j=1 to m Xj Xj.

MULTIPLICITY RESULTS FOR A NONLINEAR ELLIPTIC PROBLEM INVOLVING THE FRACTIONAL LAPLACIAN

In this paper, we consider a class of superlinear elliptic problems involving trac- tional Laplacian （-△）s/2u = λf（u） in a bounded smooth domain with zero Diriehlet bound- ary condition. We use the method on harmonic extension to study the dependence of the number of sign-changing solutions on the parameter λ.

Existence and Multiplicity of Periodic solutions for the Non-autonomous Second-order Hamiltonian Systems

In this paper,we study the existence and multiplicity of periodic solutions of the non-autonomous second-order Hamiltonian systems■where T> 0.Under suitable assumptions on F,some new existence and multiplicity theorems are obtained by using the least action principle and minimax methods in critical point theory.