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harnack AND MEAN VALUE INEQUALITIES ON GRAPHS

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Acta Mathematica Scientia 2018年第6期38卷 1751-1758页

作者： Yong LIN Hongye SONG Department of Mathematics Renmin University of China School of General Education Beijing International Studies University

We prove a harnack inequality for positive harmonic functions on graphs whichis similar to a classical result of Yau on Riemannian manifolds. Also, we prove a mean valueinequality of nonnegative subharmonic functions ... 详细信息

关键词： harmonic function subharmonic function harnack inequality mean valueinequality graph

Global Gradient Estimate on Graph and Its Applications

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Acta Mathematica Sinica,English Series 2016年第11期32卷 1350-1356页

作者： Yong LIN Shuang LIU Yun Yan YANG Department of Mathematics Renmin University of China

Continuing our previous work （arXiv：1509.07981vl）, 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.

关键词： Gradient estimate global gradient estimate harnack inequality locally finite graph

A sharp gradient estimate for the weighted p-Laplacian

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Applied Mathematics(A Journal of Chinese Universities) 2012年第4期27卷 462-474页

作者： WANG Lin-feng ZHU Yue-ping School of Science Nantong UniversityNantong 226007China

Let M be an n-dimensional complete noncompact Riemannian manifold with sectional curvature bounded from below, dμ = e^h（x） dV（x） the weighted measure and △μ,p the weighted p-Laplacian. In this paper we consider the non-linear elliptic equation △μ,pu=-λμ,p｜u｜^p-2ufor p ∈ （1, 2）. We derive a sharp gradient estimate for positive smooth solutions of this equation. As applications, we get a harnack inequality and a Liouville type theorem..

关键词： weighted p-Laplacian gradient estimate harnack inequality Liouville theorem.

Gradient Estimates for a Nonlinear Parabolic Equation with Diffusion on Complete Noncompact Manifolds

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Chinese Annals of Mathematics,Series B 2015年第1期36卷 57-66页

作者： Liang ZHAO Zongwei MA Department of Mathematics Nanjing University of Aeronautics and Astronautics College of Mathematics Physics and Information Engineering Jiaxing University

The authors obtain some gradient estimates for positive solutions to the following nonlinear parabolic equation:αu/αt=△u-b(x,t)u~σ on complete noncompact manifolds with Ricci curvature bounded from below,where 0<... 详细信息

关键词： Gradient estimates Positive solutions harnack inequality

LOWER BOUNDS FOR SUP+INF AND SUP*INF AND AN EXTENSION OF CHEN-LIN RESULT IN DIMENSION 3

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Acta Mathematica Scientia 2008年第4期28卷 749-758页

作者： Samy Skander Bahoura Department of Mathematics Patras University

We give two results about harnack type inequalities. First, on Riemannian surfaces, we have an estimate of type sup ＋ inf. The second result concern the solutions of prescribed scalar curvature equation on the unit ball of Rn with Dirichlet condition. Next, we give an inequality of type （supK ^u）^2s-1 × infπu ≤ c for positive solutions of △u = V u^5 on Ω belong toR^3, where K is a compact set of Ω and V is s-Holderian, s ∈] - 1/2, 1]. For the case s = 1/2 and Ω = S3, we prove that, if minΩ u 〉 m 〉 0 （for some particular constant m 〉 0）, and the H¨olderian constant A of V tends to 0 （in certain meaning）, we have the uniform boundedness of the supremum of the solutions of the previous equation on any compact set of Ω.

关键词： sup × inf sup + inf harnack inequality moving-plane method

Existence and Concentration of Bound States of a Class of Nonlinear Schrdinger Equations in R^2 with Potential Tending to Zero at Infinity

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Acta Mathematica Sinica,English Series 2012年第11期28卷 2243-2274页

作者： Da Cheng CUI Ji Hui ZHANG Ming Wen FEI School of Mathematics and Information Technology Nanjing Xiaozhuang University School of Mathematical Sciences Nanjing Normal University Academy of Mathematics and Systems Science China Academy of Science School of Mathematical and Computer Sciences Anhui Normal University

In this paper, we establish the existence and concentration of solutions of a class of non- linear SchrSdinger equation

关键词： Bound state nonlinear SchrSdinger equation harnack inequality concentration-compact-ness

Generalized Liouville Theorem in Nonnegatively Curved Alexandrov Spaces

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Chinese Annals of Mathematics,Series B 2009年第2期30卷 111-128页

作者： Bobo HUA School of Mathematical Sciences Fudan University Shanghai 200433 China

In this paper, Yau’s conjecture on harmonic functions in Riemannian manifolds is generalized to Alexandrov spaces. It is proved that the space of harmonic functions with polynomial growth of a fixed rate is finite di... 详细信息

关键词： Alexandrov space Harmonic function harnack inequality

Results on entire solutions for a degenerate critical elliptic equation with anisotropic coefficients

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Science China Mathematics 2011年第2期54卷 221-242页

作者： CHEN ShaoWei LIN LiShan School of Mathematical Sciences Capital Normal University Beijing 100048 China institute of Applied Mathematics Academy of Mathematics and Systems Sciences Chinese Academy of Sciences Beijing 100190 China

In this paper, we study the following degenerate critical elliptic equation with anisotropic coefficients-div(|x N | 2α▽u) = K(x)|x N | α·2 * (s)-s |u| 2 * (s)-2 u in R N ,where x = (x 1 , . . . , x N ) ∈ R N , N≥3, α > 1/2, 0≤ s ≤2 and 2 * (s) = 2(N-s)/(N-2). Some basic properties of the degenerate elliptic operator -div(|x N |2α▽u) are investigated and some regularity, symmetry and uniqueness results for entire solutions of this equation are obtained. We also get some variational identities for solutions of this equation. As consequences, we obtain some nonexistence results for this equation.

关键词： weighted Sobolev inequalities harnack inequality moving sphere method variational identities

ON　THE　UPPER　ESTIMATES　OF　FUNDAMENTAL　SOLUTIONS　OF　PARABOLIC　EQUATIONS　ON　RIEMANNIAN　MANIFOLDS

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Chinese Annals of Mathematics,Series B 1995年第1期16卷 119-130页

作者： LI JIAYU SHAO XIN (Departmellt of Mathematics, Anhui University, Hefei 230039, China.Basic Department, Anhui Agricultural College, Hefei 230036, China.) ANHUI UNIV DEPT MATHHEFEI 230039PEOPLES R CHINA ANHUI AGR COLL DEPT BASICHEFEI 230036PEOPLES R CHINA

The authors first derive the gradient estimates and harnack inequalties for positive solutions of the equation on(x, t)An(x, t) + b(x, t)' ac(￣, t) + h(x, t)u(x, t) ￣ m ￣ 0on complete Riemannian manifolds, and then derive the upper bounds of any positive L2fundamental solution of the equation when h(x, t) and b(x, t) are independent of t.

关键词： Parabolic equation Gradient estimate harnack inequality Fundamentalsolution Riemannian manifolds.