Because of the importance of Harnack inequalities, when the notion of (elliptic) Q-minima was founded in [1], it is asked whether these inequalities hold for it. The Harnack inequality for (elliptic)Q-minima then is p...Because of the importance of Harnack inequalities, when the notion of (elliptic) Q-minima was founded in [1], it is asked whether these inequalities hold for it. The Harnack inequality for (elliptic)Q-minima then is proved in [2]. Wieser extended the notion of Q-minima to the parabolic case, and obtained the Hlder continuity. In this note, under those conditions, by which the Hlder continuity of the parabolic Q-minima was obtained in [ 3], we prove that the Harnack inequalities hold for Q-minima.展开更多
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...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..展开更多
This paper will develop a Li-Yau-Hamilton type differential Harnack estimate for positive solutions to the Newell-Whitehead-Segel equation on R^n.We then use our LYH-differential Harnack inequality to prove several pr...This paper will develop a Li-Yau-Hamilton type differential Harnack estimate for positive solutions to the Newell-Whitehead-Segel equation on R^n.We then use our LYH-differential Harnack inequality to prove several properties about positive solutions to the equation,including deriving a classical Harnack inequality and characterizing standing solutions and traveling wave solutions.展开更多
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 ...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.展开更多
基金Project supported by the National Natural Science Foundation of China
文摘Because of the importance of Harnack inequalities, when the notion of (elliptic) Q-minima was founded in [1], it is asked whether these inequalities hold for it. The Harnack inequality for (elliptic)Q-minima then is proved in [2]. Wieser extended the notion of Q-minima to the parabolic case, and obtained the Hlder continuity. In this note, under those conditions, by which the Hlder continuity of the parabolic Q-minima was obtained in [ 3], we prove that the Harnack inequalities hold for Q-minima.
基金Supported by the National Natural Science Foundation of China (11171254, 11271209)
文摘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..
基金supported by NSF through the Research Experience for Undergraduates Program at Cornell University, grant-1156350supported by Cornell University Summer Program for Undergraduate Researchpartially supported by a grant from the Simons Foundation (#280161)
文摘This paper will develop a Li-Yau-Hamilton type differential Harnack estimate for positive solutions to the Newell-Whitehead-Segel equation on R^n.We then use our LYH-differential Harnack inequality to prove several properties about positive solutions to the equation,including deriving a classical Harnack inequality and characterizing standing solutions and traveling wave solutions.
基金supported by National Natural Science Foundation of China(Grant No.11271011)supported by National Natural Science Foundation of China(Grant Nos.11171347 and 11471014)
文摘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.
基金Supported by National Natural Science Foundation of China (Grant No. 10871096)China Postdoctoral Science Foundation Funded Project (Grant No. 200904501112)Planned Projects for Postdoctoral Research Funds of Jiangsu Province (Grant No. 0901046C)
文摘In this paper, we establish the existence and concentration of solutions of a class of non- linear SchrSdinger equation
