Centered densities on Lie groups of polynomial volume growth

 We study the asymptotic behavior of the convolution powers of a centered density on a connected Lie group G of polynomial volume growth. The main tool is a Harnack inequality which is proved by using ideas from Homogenization theory and by adapting the method of Krylov and Safonov. Applying this inequality we prove that the positive -harmonic functions are constant. We also characterise the -harmonic functions which grow polynomially. We give Gaussian estimates for , as well as for the differences and . We give estimates, similar to the ones given by the classical Berry-Esseen theorem, for and . We use these estimates to study the associated Riesz transforms. Received: 5 July 1999 / Revised version: 8 April 2002 / Published online: 22 August 2002     

Harnack Differential Inequalities for the Parabolic Equation ut=LF(u) on Riemannian Manifolds and Applications

In this paper, let(M~n, g) be an n-dimensional complete Riemannian manifold with the mdimensional Bakry–mery Ricci curvature bounded below. By using the maximum principle, we first prove a Li–Yau type Harnack differential inequality for positive solutions to the parabolic equation u_t= LF(u)=ΔF(u)-f·F(u),on compact Riemannian manifolds Mn, where F∈C~2(0, ∞), F0 and f is a C~2-smooth function defined on M~n. As application, the Harnack differential inequalities for fast diffusion type equation and porous media type equation are derived. On the other hand, we derive a local Hamilton type gradient estimate for positive solutions of the degenerate parabolic equation on complete Riemannian manifolds. As application, related local Hamilton type gradient estimate and Harnack inequality for fast dfiffusion type equation are established. Our results generalize some known results.     

Harnack estimate for curvature flows depending on mean curvature

The maximum principle is applied to prove the Harnack estimate of curvature flows of hypersurfaces in Rn＋1,where the normal velocity is given by a smooth function f depending only on the mean curvature.By use of the estimate,some corollaries are obtained including the integral Harnack inequality.In particular,the conditions are given with which the solution to the flows is a translation soliton or an expanding soliton.     

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.     

Schröndiger Type and Relaxed Dirichlet Problems for the Subelliptic p-Laplacian

We study at first the solutions of a Schrödinger type problem relative to the subelliptic p-Laplacian: we prove, for potentials that are in the Kato space, an Harnack inequality on enough small intrinsic balls; the continuity of the solutions to the homogeneous Dirichlet problem follows from some estimates in the proof of the Harnack inequality. In the second part of the paper we study a relaxed Dirichlet problem for the subelliptic p-Laplacian and we prove a Wiener type criterion for the regularity of a point (with respect to our problem).     

An Alexandroff–Bakelman–Pucci estimate on Riemannian manifolds

In Cabré (1997) [2], Cabré established an Alexandroff–Bakelman–Pucci (ABP) estimate on Riemannian manifolds with non-negative sectional curvatures and applied it to establish the Krylov–Safonov Harnack inequality on manifolds with non-negative sectional curvatures. In the present paper, we generalize the results of [2]. We obtain an ABP estimate on manifolds with Ricci curvatures bounded from below and apply this estimate to prove the Krylov–Safonov Harnack inequality on manifolds with sectional curvatures bounded from below. We also use this ABP estimate to study Minkowski-type inequalities.     

Harnack inequality and exponential separation for oblique derivative problems on Lipschitz domains

We consider the oblique derivative problem for linear nonautonomous second order parabolic equations with bounded measurable coefficients on bounded Lipschitz cylinders. We derive an optimal elliptic-type Harnack inequality for positive solutions of this problem and use it to show that each positive solution exponentially dominates any solution which changes sign for all times. We show several nontrivial applications of both the exponential estimate and the derived Harnack inequality.     

Harnack estimates for quasi-linear degenerate parabolic differential equations

We establish the intrinsic Harnack inequality for non-negative solutions of a class of degenerate, quasilinear, parabolic equations, including equations of the p-Laplacian and porous medium type. It is shown that the classical Harnack estimate, while failing for degenerate parabolic equations, it continues to hold in a space-time geometry intrinsic to the degeneracy. The proof uses only measure-theoretical arguments, it reproduces the classical Moser theory, for non-degenerate equations, and it is novel even in that context. Hölder estimates are derived as a consequence of the Harnack inequality. The results solve a long standing problem in the theory of degenerate parabolic equations.     

阿贝尔齐性图上一种改进的关于Dirichlet特征值的Harnack型不等式

在本文中, 我们证明了在阿贝尔齐性图上一种改进的关于Dirichlet特征值的Harnack 型不等式,由此, 利用此Harnack 型不等式得到Dirichlet特征值的一个下界估计, 推广了 Chung 和 Yau 关于齐性图的一些结果.     

Gradient estimates and Harnack inequalities on non-compact Riemannian manifolds

A gradient-entropy inequality is established for elliptic diffusion semigroups on arbitrary complete Riemannian manifolds. As applications, a global Harnack inequality with power and a heat kernel estimate are derived.     

Harnack inequality for symmetric stable processes on fractalsL'inégalité de Harnack pour les processus symétriques stables sur les fractals

We study nonnegative harmonic functions of symmetric α-stable processes on d-sets F. We prove the Harnack inequality for such functions when α∈(0,2/d_w)∪(d_s,2). Furthermore, we investigate the decay rate of harmonic functions and the Carleson estimate near the boundary of a region in F. In the particular case of natural cells in the Sierpiński gasket we also prove the boundary Harnack principle. To cite this article: K. Bogdan et al., C. R. Acad. Sci. Paris, Ser. I 335 (2002) 59–63.     

A Harnack inequality for the parabolic Allen–Cahn equation

We prove a differential Harnack inequality for the solution of the parabolic Allen–Cahn equation \( \frac{\partial f}{\partial t}=\triangle f-(f^3-f)\) on a closed n-dimensional manifold. As a corollary, we find a classical Harnack inequality. We also formally compare the standing wave solution to a gradient estimate of Modica from the 1980s for the elliptic equation.     

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

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

Harnack inequality for two‐weight subelliptic p ‐Laplace operators

In this note, we prove a Harnack inequality for two‐weight subelliptic p ‐Laplace operators together with an upper bound of the Harnack constant associated with such inequality. (© 2006 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

Interpolating Between Li-Yau's and Hamilton's Harnack Inequalities on a Surface

We establish a one-parameter family of Harnack inequalities connecting Li and Yau's differential Harnack inequality for the heat equation to Hamilton's Harnack inequality for the Ricci flow on a 2-dimensional manifold with positive scalar curvature.     

Interpolating between constrained Li–Yau and Chow–Hamilton Harnack inequalities on a surface

We establish a one-parameter family of Harnack inequalities connecting the constrained trace Li–Yau differential Harnack inequality for the heat equation to the constrained trace Chow–Hamilton Harnack inequality for the Ricci flow on a 2-dimensional closed manifold with positive scalar curvature, and thereby generalize Chow’s interpolated Harnack inequality (J. Partial Diff. Eqs. 11 (1998), 137–140).     

A new formulation of Harnackʼs inequality for nonlocal operators

We provide a new formulation of Harnack?s inequality for nonlocal operators. In contrast to previous versions we do not assume harmonic functions to have a sign. The version of Harnack?s inequality given here generalizes Harnack?s classical result from 1887 to nonlocal situations. As a consequence we derive Hölder regularity estimates by an extension of Moser?s method. The inequality that we propose is equivalent to Harnack?s original formulation but seems to be new even for the Laplace operator.     

Harnack type inequalities for conformal scalar curvature equation

We derive a Harnack type inequality for the conformal scalar curvature equation on B _3R . If the positive scalar curvature function K(x) is sub-harmonic in a neighborhood of each critical point and the maximum of u over B _R is comparable to its maximum over B _3R , then the Harnack type inequality can be obtained. Zhang is supported by NSF-DMS-0600275.     

Extended Harnack inequalities with exceptional sets and a boundary Harnack principle

Harnack’s inequality is one of the most fundamental inequalities for positive harmonic functions and has been extended to positive solutions of general elliptic equations and parabolic equations. This article gives a different generalization; namely, we generalize Harnack chains rather than equations. More precisely, we allow a small exceptional set and yet obtain a similar Harnack inequality. The size of an exceptional set is measured by capacity. The results are new even for classical harmonic functions. Our extended Harnack inequality includes information about the boundary behavior of positive harmonic functions. It yields a boundary Harnack principle for a very nasty domain whose boundary is given locally by the graph of a function with modulus of continuity worse than Hölder continuity.     

Large time estimates for heat kernels in nilpotent Lie groups

The aim of this paper is to obtain some estimate for large time for the Heat kernel corresponding to a sub-Laplacian with drift term on a nilpotent Lie group. We also obtain a uniform Harnack inequality for a “bounded” family of sub-Laplacians with drift in the first commutator of the Lie algebra of the nilpotent group.