Li-Yau Multiplier Set and Optimal Li-Yau Gradient Estimate on Hyperbolic Spaces

Potential Analysis - In this paper, motivated by finding sharp Li-Yau-type gradient estimate for positive solution of heat equations on complete Riemannian manifolds with negative Ricci curvature...     

Gradient estimates for the heat equation under the Ricci flow

The paper considers a manifold M evolving under the Ricci flow and establishes a series of gradient estimates for positive solutions of the heat equation on M. Among other results, we prove Li-Yau-type inequalities in this context. We consider both the case where M is a complete manifold without boundary and the case where M is a compact manifold with boundary. Applications of our results include Harnack inequalities for the heat equation on M.     

Global gradient estimate on graph and its applications

Continuing our previous work(ar Xiv:1509.07981v1),we derive another global gradient estimate for positive functions,particularly for positive solutions to the heat equation on finite or locally finite graphs.In general,the gradient estimate in the present paper is independent of our previous one.As applications,it can be used to get an upper bound and a lower bound of the heat kernel on locally finite graphs.These global gradient estimates can be compared with the Li–Yau inequality on graphs contributed by Bauer et al.[J.Differential Geom.,99,359–409(2015)].In many topics,such as eigenvalue estimate and heat kernel estimate(not including the Liouville type theorems),replacing the Li–Yau inequality by the global gradient estimate,we can get similar results.     

Invariant elliptic estimates

In this paper, we revisit elliptic estimates invariant under domain expansion. We improve the invariant elliptic estimates in the previous paper [Y. Cho, T. Ozawa, Y. Shim, Calc. Var. PDE 34 (2009) 321-339] via gradient estimate and discuss an application to the Lamé system.     

Addenda to “The entropy formula for linear heat equation”

We add two sections to [8] and answer some questions asked there. In the first section we give another derivation of Theorem 1.1 of [8], which reveals the relation between the entropy formula, (1.4) of [8], and the well-known Li-Yau ’s gradient estimate. As a by-product we obtain the sharp estimates on ‘Nash’s entropy’ for manifolds with nonnegative Ricci curvature. We also show that the equality holds in Li-Yau’s gradient estimate, for some positive solution to the heat equation, at some positive time, implies that the complete Riemannian manifold with nonnegative Ricci curvature is isometric to ℝ ⁿ .In the second section we derive a dual entropy formula which, to some degree, connects Hamilton’s entropy with Perelman ’s entropy in the case of Riemann surfaces.     

紧致黎曼流形上一类非线性热方程的梯度估计

In this paper,we study gradient estimates for the nonlinear heat equation ut-△u =au log u,on compact Riemannian manifold with or without boundary.We get a Hamilton type gradient estimate for the positi...     

Hypoelliptic heat kernel inequalities on the Heisenberg group

We study the existence of “L^p-type” gradient estimates for the heat kernel of the natural hypoelliptic “Laplacian” on the real three-dimensional Heisenberg Lie group. Using Malliavin calculus methods, we verify that these estimates hold in the case p>1. The gradient estimate for p=2 implies a corresponding Poincaré inequality for the heat kernel. The gradient estimate for p=1 is still open; if proved, this estimate would imply a logarithmic Sobolev inequality for the heat kernel.     

Harnack inequalities for SDEs driven by subordinate Brownian motions

By using regularization approximations of the underlying subordinator and a gradient estimate approach, the dimension-independent Harnack inequalities are established for the inhomogeneous semigroup associated with a class of SDEs with Lévy noise containing a subordinate Brownian motion. Our estimates in Harnack type inequalities improve the corresponding ones in the recent paper by Wang and Wang (2014) [10].     

Nonexistence of nonnegative solutions of elliptic systems of divergence type

In this paper we deal with noncoercive elliptic systems of divergence type, that include both the p-Laplacian and the mean curvature operator and whose right-hand sides depend also on a gradient factor. We prove that any nonnegative entire (weak) solution is necessarily constant. The main argument of our proofs is based on previous estimates, given in Filippucci (2009) [12] for elliptic inequalities. Actually, the main technique for proving the central estimate has been developed by Mitidieri and Pohozaev (2001) [23] and relies on the method of test functions. No use of comparison and maximum principles or assumptions on symmetry or behavior at infinity of the solutions are required.     

On the first eigenvalue of the Witten–Laplacian and the diameter of compact shrinking solitons

We prove a lower bound estimate for the first non-zero eigenvalue of the Witten–Laplacian on compact Riemannian manifolds. As an application, we derive a lower bound estimate for the diameter of compact gradient shrinking Ricci solitons. Our results improve some previous estimates which were obtained by the first author and Sano (Asian J Math, to appear), and by Andrews and Ni (Comm Partial Differential Equ, to appear). Moreover, we extend the diameter estimate to compact self-similar shrinkers of mean curvature flow.     

双线性非协调有限元逼近的梯度恢复后验误差估计

给出了二阶椭圆方程的双线性非协调有限元逼近的梯度恢复后验误差估计.该误差估计是在Q_1非协调元上得到的,并给出了误差的上下界.进一步证明该误差估计在拟一致网格上是渐进精确地.证明依赖于clement插值和Helmholtz分解,数值结果验证了理论的正确性.     

一种带参数的不完全Cholesky分解共轭梯度法

共轭梯度法在解高阶稀疏线性方程组方面有许多其它经典的迭代法所没有的优点,但当线性方程组相当病态、系数矩阵条件数很坏时,共轭梯度法的收敛速度很慢.因此,又产生了预条件处理共轭梯度法. 我们用预条件处理共轭梯度法求解线性方程组Ax=b(这里A是对称正定稀疏阵且条件数很大).预条件处理共轭梯度法旨在寻找一适当的正定矩阵C,C通常写成     

Heat Kernel Estimates for Non-symmetric Finite Range Jump Processes

In this paper,we first establish the sharp two-sided heat kernel estimates and the gradient estimate for the truncated fractional Laplacian under gradient perturbation ■,where ■ is the truncated fractional Laplacian,α∈(1,2) and b ∈ K_d~(α-1).In the second part,for a more general finite range jump process,we present some sufficient conditions to allow that the two sided estimates of the heat kernel are comparable to the Poisson type function for large distance |x-y|in short time.     

Elliptic gradient estimates for diffusion operators on complete riemannian manifolds

In this note, we obtain the elliptic estimate for diffusion operator L = Δ + ∇??∇ on complete, noncompact Riemannian manifolds, under the curvature condition CD(K,m), which generalizes B. L. Kotschwar's work [5]. As an application, we get estimate on the heat kernel. The Bernstein-type gradient estimate for Schrödinger-type gradient is also derived.     

Moment bounds on the corrector of stochastic homogenization of non-symmetric elliptic finite difference equations

We consider the corrector equation from the stochastic homogenization of uniformly elliptic finite difference equations with random, possibly non symmetric coe?cients. Under the assumption that the coe?cients are stationary and ergodic in the quantitative form of a logarithmic Sobolev inequality (LSI), we obtain optimal bounds on the corrector and its gradient in dimensions d≥2. Similar estimates have recently been obtained in the special case of diagonal coe?cients making extensive use of the maximum principle and scalar techniques. Our new method only invokes arguments that are also available for elliptic systems and does not use the maximum principle. In particular, our proof relies on the LSI to quantify ergodicity and on regularity estimates on the derivative of the discrete Green’s function in weighted spaces. In the critical case d = 2, our argument for the estimate on the gradient of the elliptic Green’s function uses a Calderón–Zygmund estimate in discrete weighted spaces, which we state and prove. As applications, we provide a quantitative two-scale expansion and a quantitative approximation of the homogenized coe?cients.     

Beurling-Ahlfors扩张的伸张函数的增长阶的估计

本文研究了一般情形下的Beurling-Ahlfors扩张的伸张函数的增长阶的估计.根据伸张函数的估计公式,利用分段估计的方法,获得了伸张函数的更精确的估计,改进了[6]的结果.     

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本文综述混合效应模型参数估计方面的若干新进展. 平衡混合效应方差分析模型的协方差阵具有一定结构. 对这类模型, 文献[1]提出了参数估计的一种新方法, 称为谱分解法. 新方法的突出特点是, 能同时给出固定效应和方差分量的估计, 前者是线性的, 后者是二次的,且相互独立. 而后, 文献[2--9]证明了谱分解估计的进一步的统计性质, 同时给出了协方差阵对应的估计, 它不仅是正定阵, 而且可获得它的风险函数, 这些文献还研究了谱分解估计与方差分析估计, 极大似然估计, 限制极大似然估计以及最小范数二次无偏估计的关系. 本文综述这一方向的部分研究成果, 并提出一些待进一步研究的问题.     

OPTIMAL ERROR ESTIMATES OF THE PARTITION OF UNITY METHOD WITH LOCAL POLYNOMIAL APPROXIMATION SPACES

In this paper, we provide a theoretical analysis of the partition of unity finite elementmethod (PUFEM), which belongs to the family of meshfree methods. The usual erroranalysis only shows the order of error estimate to the same as the local approximations[12].Using standard linear finite element base functions as partition of unity and polynomials aslocal approximation space, in 1-d case, we derive optimal order error estimates for PUFEMinterpolants. Our analysis show that the error estimate is of one order higher than thelocal approximations. The interpolation error estimates yield optimal error estimates forPUFEM solutions of elliptic boundary value problems.     

Gradient estimates via linear and nonlinear potentials

We prove new potential and nonlinear potential pointwise gradient estimates for solutions to measure data problems, involving possibly degenerate quasilinear operators whose prototype is given by −Δpu=μ. In particular, no matter the nonlinearity of the equations considered, we show that in the case p?2 a pointwise gradient estimate is possible using standard, linear Riesz potentials. The proof is based on the identification of a natural quantity that on one hand respects the natural scaling of the problem, and on the other allows to encode the weaker coercivity properties of the operators considered, in the case p?2. In the case p>2 we prove a new gradient estimate employing nonlinear potentials of Wolff type.     

A survey of some estimates of eigenvalues and condition numbers for certain preconditioned matrices

Eigenvalue and condition number estimates for preconditioned iteration matrices provide the information required to estimate the rate of convergence of iterative methods, such as preconditioned conjugate gradient methods. In recent years various estimates have been derived for (perturbed) modified (block) incomplete factorizations. We survey and extend some of these and derive new estimates. In particular we derive upper and lower estimates of individual eigenvalues and of condition number. This includes a discussion that the condition number of preconditioned second order elliptic difference matrices is O(h⁻¹). Some of the methods are applied to compute certain parameters involved in the computation of the preconditioner.