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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

THE gradient estimate OF SUBELLIPTIC HARMONIC MAPS WITH A POTENTIAL

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Acta Mathematica Scientia 2024年第4期44卷 1189-1199页

作者： Han LUO School of Mathematical Sciences Fudan UniversityShanghai 200433China

In this paper,we investigate subelliptic harmonic maps with a potential from noncompact complete sub-Riemannian manifolds corresponding to totally geodesic Riemannian *** some suitable conditions,we give the gradient ... 详细信息

关键词： sub-Riemannian manifolds subelliptic harmonic maps with potential gradient estimate Liouville Theorem

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.

The gradient estimate of a Neumann Eigenfunction on a Compact Manifold with Boundary

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Chinese Annals of Mathematics,Series B 2015年第6期36卷 991-1000页

作者： Jingchen HU Yiqian SHI Bin XU Wu Wen-Tsun Key Laboratory of Mathematics USTCChinese Academy of Sciences School of Mathematical Sciences University of Science and Technology of China

Let eλ（x） be a Neumann eigenfunction with respect to the positive Laplacian λ on a compact Riemannian manifold M with boundary such that A eλ = λ2eλ in the interior of M and the normal derivative of ex vanishes on the boundary of M. Let xλ be the unit band spectral projection operator associated with the Neumann Laplacian and f be a square integrable function on M. The authors show the following gradient estimate

关键词： Neumann eigenfunction gradient estimate

gradient estimate for an Exponentially Harmonic Type Heat Equation on Riemannian Manifolds

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Journal of Mathematical Research with Applications 2024年第4期44卷 495-501页

作者： Jianyi MAO Xinrong JIANG School of Mathematics and Statistics Jiangxi Normal University

In this paper, we derive a Hamilton-Souplet-Zhang type gradient estimate for exponentially harmonic type heat equation on Riemannian manifolds. As its application, we obtain a Liouville type theorem.

关键词： gradient estimate exponentially harmonic type heat equation Riemannian manifold Liouville theorem

Harnack inequality and gradient estimate for G-SDEs with degenerate noise

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Science China Mathematics 2022年第4期65卷 813-826页

作者： Xing Huang Fen-Fen Yang Center for Applied Mathematics Tianjin UniversityTianjin300072China

In this paper,the Harnack inequality for G-SDEs with degenerate noise is derived by the method of coupling by change of the ***,for any bounded and continuous function f,the gradient estimate∣∇P_(t)f∣≤c(p,t)(Pt|f|p)^(1/p),p>1,t>0 is obtained for the associated nonlinear semigroup P¯*** an application of the Harnack inequality,we prove the existence of the weak solution to degenerate G-SDEs under some integrable ***,an example is *** of the above results extend the existing ones in the linear expectation setting.

关键词： Harnack inequality degenerate noise G-SDE gradient estimate weak solution invariant expectation

Harnack inequality and gradient estimate for functional G-SDEs with degenerate noise

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Probability, Uncertainty and Quantitative Risk 2022年第2期7卷 119-132页

作者： Fen-Fen Yang Department of Mathematics Shanghai UniversityShanghai 2004China

In this paper,the Harnack and shift Harnack inequalities for functional G-SDEs with degenerate noise are derived by the method of coupling by change of ***,the gradient estimate for the associated nonlinear semigroup ... 详细信息

关键词： Harnack inequality gradient estimate Degenerate noise Functional G-SDEs

gradient estimateS FOR POSITIVE SMOOTH f-HARMONIC FUNCTIONS

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Acta Mathematica Scientia 2010年第5期30卷 1614-1618页

作者：陈立陈文艺 Academy of Mathematics and Systems Science Chinese Academy of Sciences School of Mathematics and Statistics Wuhan University

For Riemannian manifolds with a measure, we study the gradient estimates for positive smooth f-harmonic functions when the ∞-Bakry-Emery Ricci tensor and Ricci tensor are bounded from below, generalizing the classica... 详细信息

关键词： gradient estimate f-harmonic function Bakry-Emery Ricci tensor

gradient estimates for the Equation Δu+cu^(-α)=0 on Riemannian Manifolds

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Acta Mathematica Sinica,English Series 2010年第6期26卷 1177-1182页

作者： Yun Yan YANG Department of Mathematics Information School Renmin University of China Beijing 100872 P. R. China

Let （M, g） be a complete non-compact Riemannian manifold without boundary. In this paper, we give the gradient estimates on positive solutions to the following elliptic equation with singular nonlinearity：△u（x）＋cu^-a=0 in M,where a 〉 0, c are two real constants. When c 〈 0 and M is a bounded smooth domain in R^n, the above equation is known as the thin film equation, which describes a steady state of the thin film （see Guo-Wei [Manuscripta Math., 120, 193-209 （2006）]）. The results in this paper can be viewed as an supplement of that of J. Li [J. Funct. Anal., 100, 233-256 （1991）], where the nonlinearity is the positive power of u.

关键词： positive solution gradient estimate thin film equation