Graphs Between the Elliptic and Parabolic Harnack Inequalities

We present graphs that satisfy the uniform elliptic Harnack inequality, for harmonic functions, but not the stronger parabolic one, for solutions of the discrete heat equation. It is known that the parabolic Harnack inequality is equivalent to the conjunction of a volume regularity and a L ² Poincaré inequality. The first example of graph satisfying the elliptic but not the parabolic Harnack inequality is due to M. Barlow and R. Bass. It satisfies the volume regularity and not the Poincaré inequality. We construct another example that does not satisfy the volume regularity.     

完备Khler流形上Khler-Ricci流的局部Harnack估计

本文首先给出非正规化Khler-Ricci流下曲率的发展方程,然后得到了关于曲率的Harnack量在满足曲率局部条件下所产生的一个特殊项CNS.通过对CNS的估计,得到了完备Khler流形上关于Khler-Ricci流的局部Harnack不等式.最后,作为主要定理的应用,我们将结果推广到数量曲率的情形.     

Curvature, Harnack's Inequality, and a Spectral Characterization of Nilmanifolds

For closed n-dimensional Riemannian manifolds M with almostnonnegative Ricci curvature, the Laplacian on one-forms is known toadmit at most n small eigenvalues. If there are n small eigenvalues, or if M is orientable and has n – 1 small eigenvalues, then M isdiffeomorphic to a nilmanifold, and the metric is almost left invariant.We show that our results are optimal for n 4.     

A HARNACK TYPE INEQUALITY FOR SOME CONFORMALLY INVARIANT EQUATIONS ON HALF EUCLIDEAN SPACE

We establish a Harnack type inequality on half Euclidean space for general conformally invariant fully nonlinear elliptic equations of second order.     

A Harnack inequality for Dirichlet eigenvalues

We prove a Harnack inequality for Dirichlet eigenfunctions of abelian homogeneous graphs and their convex subgraphs. We derive lower bounds for Dirichlet eigenvalues using the Harnack inequality. We also consider a randomization problem in connection with combinatorial games using Dirichlet eigenvalues. © 2000 John Wiley & Sons, Inc. J Graph Theory 34: 247–257, 2000     

The Harnack inequality for second-order parabolic equations with divergence-free drifts of low regularity

We establish the Harnack inequality for advection-diffusion equations with divergence-free drifts by adapting the classical Moser technique to parabolic equations with drifts with regularity lower than the scale invariant spaces.     

Global properties of invariant measures

We study global regularity properties of invariant measures associated with second order differential operators in R^N. Under suitable conditions, we prove global boundedness of the density, Sobolev regularity, a Harnack inequality and pointwise upper and lower bounds.     

Generalized Liouville Theorem in Nonnegatively Curved Alexandrov Spaces

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 dimensional and strong Liouville theorem holds in Alexandrov spaces with nonnegative curvature.     

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

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 0σ1 is a real constant,and b(x,t) is a function which is C~2 in the x-variable and C~1 in the t-variable.     

A Singular Quasilinear Anisotropic Elliptic Boundary Value Problem. II

Let with . We consider the equations

with and . We show that if is a convex bounded region in , there exists at least one classical solution to this boundary value problem. If the region is not convex, we show the existence of a weak solution. Partial results for the existence of classical solutions for non-convex domains in are also given.

     

Ratio limit theorem for parabolic horn-shaped domains

We prove that for horn-shaped domains of parabolic type, the ratio of the heat kernel at different fixed points has a limit when the time tends to infinity. We also give an explicit formula for the limit in terms of the harmonic functions.

     

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

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.     

Differential Harnack inequalities on Riemannian manifolds I: Linear heat equation

In the first part of this paper, we get new Li–Yau type gradient estimates for positive solutions of heat equation on Riemannian manifolds with Ricci(M)?−k, k∈R. As applications, several parabolic Harnack inequalities are obtained and they lead to new estimates on heat kernels of manifolds with Ricci curvature bounded from below. In the second part, we establish a Perelman type Li–Yau–Hamilton differential Harnack inequality for heat kernels on manifolds with Ricci(M)?−k, which generalizes a result of L. Ni (2004, 2006) [20] and [21]. As applications, we obtain new Harnack inequalities and heat kernel estimates on general manifolds. We also obtain various entropy monotonicity formulas for all compact Riemannian manifolds.     

On a Singular Semilinear Elliptic Boundary Value Problem and the Boundary Harnack Principle

On a bounded C ²-domain we consider the singular boundary-value problem 1/2u=f(u) in D, u _D =, where d3, f:(0,)(0,) is a locally Hölder continuous function such that f(u) as u0 at the rate u ^–, for some (0,1), and is a non-negative continuous function satisfying certain growth assumptions. We show existence of solutions bounded below by a positive harmonic function, which are smooth in D and continuous in . Such solutions are shown to satisfy a boundary Harnack principle.     

Harnack inequality for some classes of Markov processes

In this paper we establish a Harnack inequality for nonnegative harmonic functions of some classes of Markov processes with jumps. Mathematics Subject Classification (2000): Primary 60J45, 60J75, Secondary 60J25.This work was completed while the authors were in the Research in Pairs program at the Mathematisches Forschungsinstitut Oberwolfach. We thank the Institute for the hospitality.The research of this author is supported in part by NSF Grant DMS-9803240.The research of this author is supported in part by MZT grant 0037107 of the Republic of Croatia.     

Lower bounds for transition probabilities on graphs

The paper presents two results. The first one provides separate conditions for the upper and lower estimates of the distribution of the time of exit from balls of a random walk on a weighted graph. The main result of the paper is that the lower estimate follows from the elliptic Harnack inequality. The second result is an off-diagonal lower bound for the transition probability of the random walk.