带权流形上热方程的Harnack估计

利用两种不同的方法讨论了带权流形上热方程和Schrodinger方程解的Harnack估计,先利用最大模原理证明热方程解的梯度估计,从而得到解的Harnack估计,另外利用算子半群的方法证明位势函数为常数的Schrodinger方程解的Harnack估计.     

带权流形上的纽曼特征值估计

讨论一类光滑紧致带权黎曼流形上的纽曼特征值估计问题,假定这类流形具有光滑边界,边界是凸的,而且流形上的Bakery-Emery Ricci曲率具有正的下界.利用了极大模原理去证明热方程解的梯度估计,然后得到热核上界估计.再利用热核与特征值的关系,得到了特征值的下界估计.     

芬斯勒流形上与梯度向量场和Laplacian有关的若干重要不等式（英文）

本文首先刻画了Randers流形上任一光滑函数的梯度向量场并得到了一个梯度估计.其次,本文在Ric_N≥K> 0的条件下获得了芬斯勒Laplacian的非零特征值的一个下界估计.最后,本文在Ric_∞≥K> 0的条件下,给出了紧致芬斯勒流形上的对数Sobolev不等式的一个全新且简单的证明.     

Bakry-Emery型Ricci曲率下两类抛物方程正解的梯度估计

本文考虑完备黎曼流形上,在Bakry-Emery型Ricci曲率有下界的条件下两类抛物方程?u/?t=△Vu+au log u 和(△v-?/?t)u(x,t)+p(x,t)uβ(x,t)+q(x,t)u(x,t)=0正解的梯度估计,这里α,β ∈(R),△V(·):=△+(V,▽(·)).由于引入了 △V,相应地,在...     

具p-Laplacian算子型奇异边值问题的正解

考虑一类p-Laplacian算子型泛函微分方程的奇异边值问题,利用锥不动点定理,得到了其正解及多个正解存在的充分条件.     

A Gradient Estimate for Solutions of the Heat Equation II

The author obtains an estimate for the spatial gradient of solutions of the heat equation, subject to a homogeneous Neumann boundary condition, in terms of the gradient of the initial data. The proof is accomplished via the maximum principle; the main assumption is that the sufficiently smooth boundary be convex.     

Sharp Gradient Estimate and Yau's Liouville Theorem for the Heat Equation on Noncompact Manifolds

We derive a sharp, localized version of elliptic type gradientestimates for positive solutions (bounded or not) to the heatequation. These estimates are related to the Cheng–Yauestimate for the Laplace equation and Hamilton's estimate forbounded solutions to the heat equation on compact manifolds.As applications, we generalize Yau's celebrated Liouville theoremfor positive harmonic functions to positive ancient (includingeternal) solutions of the heat equation, under certain growthconditions. Surprisingly this Liouville theorem for the heatequation does not hold even in Rⁿ without such a condition.We also prove a sharpened long-time gradient estimate for thelog of the heat kernel on noncompact manifolds. 2000 MathematicsSubject Classification 35K05, 58J35.     

p-Laplace方程正解的存在性和多重性

这篇文章讨论边值问题-(| u′|p-2u′)′=λf(t ,u) ,t∈(0,1) ,p >1,u(0) =u(1) =0,其中f(t ,u)≥-M( M是正常数) ,对(t ,u)∈0,1×0,∞) .我们利用度理论和锥上的不动点定理得到方程存在两个正解.     

Gradient estimates for positive solutions of the Laplacian with drift

Let be a complete Riemannian manifold of dimension without boundary and with Ricci curvature bounded below by where If is a vector field such that and on for some nonnegative constants and then we show that any positive solution of the equation satisfies the estimate

on , for all In particular, for the case when this estimate is advantageous for small values of and when it recovers the celebrated Liouville theorem of Yau (Comm. Pure Appl. Math. 28 (1975), 201-228).

     

一类具p-Laplacian算子三阶m点边值问题的三个正解

考察了一类具p-Laplacian算子三阶m点边值问题的三个正解.首先利用二阶m点边值问题的Green函数把该类问题转化为一个等价的积分方程,在适当的锥上应用Avery-Peteron不动点定理讨论该类积分方程的正解存在性,从而得到了正解存在的充分条件.     

Sharp spectral bounds for complex perturbations of the indefinite Laplacian

We derive quantitative bounds for eigenvalues of complex perturbations of the indefinite Laplacian on the real line. Our results substantially improve existing results even for real potentials. For $L^1$-potentials, we obtain optimal spectral enclosures which accommodate also embedded eigenvalues, while our result for $L^p$-potentials yield sharp spectral bounds on the imaginary parts of eigenvalues of the perturbed operator for all $p\in [1,\infty )$. The sharpness of the results are demonstrated by means of explicit examples.     

一类拟线性椭圆型方程弱解梯度的一致估计

在区域Ω的边界满足一致p厚条件下,利用一致p厚的边界Sobolev不等式、一些容量不等式和一个精确的逆Hlder不等式,我们给出了一类拟线性椭圆型方程divAp(x,Du) Bp(x,u,Du)=f(x)弱解梯度的一致估计.     

Solutions for the Kirchhoff type equations with fractional Laplacian

Due to the singularity and nonlocality of the fractional Laplacian, the classical tools such as Sturm comparison, Wronskians, Picard--Lindel\"{o}f iteration, and shooting arguments (which are all purely local concepts) are not{\ applicable} when analyzing solutions in the setting of the nonlocal operator $\left( -\Delta \right) ^{s}$. Furthermore, the nonlocal term of the Kirchhoff type equations will also cause some mathematical difficulties. The present work is motivated by the method of semi-classical problems which show that the existence of solutions of the Kirchhoff type equations are equivalent to the corresponding associated fractional differential and algebraic system. In such case, the existence of the fractional Kirchhoff equation can be obtained by using the corresponding fractional elliptic equation. Therefore some qualitative properties of solutions for the associated problems can be inherited. In particular, the classical uniqueness results can be applied to this equation.     

Sharp lower bounds for the first eigenvalues of the bi-drifting Laplacian

In the present paper we study some kinds of the problems for the bi-drifting Laplacian operator and get some sharp lower bounds for the first eigenvalue for these eigenvalue problems on compact manifolds with boundary (also called a smooth metric measure space) and weighted Ricci curvature bounded inferiorly.