Harnack estimate for a semilinear parabolic equation

We study the Cauchy problem of a semilinear parabolic equation. We construct an appropriate Harnack quantity and get a differential Harnack inequality. Using this inequality, we prove the finite-time blow-up of the positive solutions and recover a classical Harnack inequality. We also obtain a result of Liouville type for the elliptic equation.     

Harmonic functions on hypergroups

We initiate a study of harmonic functions on hypergroups. In particular, we introduce the concept of a nilpotent hypergroup and show such hypergroup admits an invariant measure as well as a Liouville theorem for bounded harmonic functions. Further, positive harmonic functions on nilpotent hypergroups are shown to be integrals of exponential functions. For arbitrary hypergroups, we derive a Harnack inequality for positive harmonic functions and prove a Liouville theorem for compact hypergroups. We discuss an application to harmonic spherical functions.     

On Harnack estimates for positive solutions of the heat equation on a complete manifold

We establish several new Harnack estimates for the nonnegative solutions of the heat equation on a complete Riemannian manifold with Ricci curvature bounded by a positive or negative constant. This extends to symmetric diffusions whose generator satisfies a “curvature-dimension” inequality.     

Harnack inequalities,exponential separation,and perturbations of principal Floquet bundles for linear parabolic equations

We consider the Dirichlet problem for linear nonautonomous second order parabolic equations with bounded measurable coefficients on bounded Lipschitz domains. Using a new Harnack-type inequality for quotients of positive solutions, we show that each positive solution exponentially dominates any solution which changes sign for all times. We then examine continuity and robustness properties of a principal Floquet bundle and the associated exponential separation under perturbations of the coefficients and the spatial domain.     

Differential Harnack estimates for backward heat equations with potentials under the Ricci flow

In this paper, we derive a general evolution formula for possible Harnack quantities. As a consequence, we prove several differential Harnack inequalities for positive solutions of backward heat-type equations with potentials (including the conjugate heat equation) under the Ricci flow. We shall also derive Perelman's Harnack inequality for the fundamental solution of the conjugate heat equation under the Ricci flow.     

Harmonic functions of subordinate killed Brownian motion

In this paper we study harmonic functions of subordinate killed Brownian motion in a domain D. We first prove that, when the killed Brownian semigroup in D is intrinsic ultracontractive, all nonnegative harmonic functions of the subordinate killed Brownian motion in D are continuous and then we establish a Harnack inequality for these harmonic functions. We then show that, when D is a bounded Lipschitz domain, both the Martin boundary and the minimal Martin boundary of the subordinate killed Brownian motion in D coincide with the Euclidean boundary ∂D. We also show that, when D is a bounded Lipschitz domain, a boundary Harnack principle holds for positive harmonic functions of the subordinate killed Brownian motion in D.     

The equivalence of certain heat kernel and Green function bounds

We show that certain upper and lower bounds on the Green function and heat kernel of a second-order elliptic operator in a bounded region are equivalent, and imply a number of apparently stronger bounds. We also show that these bounds are equivalent to the Harnack inequality except for peculiar regions, and are therefore almost always valid.     

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.     

Differential Harnack Estimates for Time-Dependent Heat Equations with Potentials

In this paper, we prove a differential Harnack inequality for positive solutions of time-dependent heat equations with potentials. We also prove a gradient estimate for the positive solution of the time-dependent heat equation.     

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.     

An Upper Bound for Hessian Matrices of Positive Solutions of Heat Equations

We prove global and local upper bounds for the Hessian of log positive solutions of the heat equation on a Riemannian manifold. The metric is either fixed or evolved under the Ricci flow. These upper bounds seem to be the first general ones that match the well-known lower bounds which have been around for some time. As an application, we discover a local, time reversed Harnack type inequality for bounded positive solutions of the heat equation.     

The Boundary Harnack Principle for Nonlocal Elliptic Operators in Non-divergence Form

Potential Analysis - We prove a boundary Harnack inequality for nonlocal elliptic operators L in non-divergence form with bounded measurable coefficients. Namely, our main result establishes that...     

Harnack's inequality for parabolic nonlocal equations

The main result of this paper is a nonlocal version of Harnack's inequality for a class of parabolic nonlocal equations. We additionally establish a weak Harnack inequality as well as local boundedness of solutions. None of the results require the solution to be globally positive.     

A regularity result for quasilinear parabolic systems

We consider bounded, weak solutions of certain quasilinear parabolic systems of second order. If the solution fulfills a suitable smallness condition, we show that it is H?lder continuous and satisfies an a priori estimate. This is a well known result of Giaquinta and Struwe [3]. Their argument employs the use of Green’s functions, which is completely avoided in our proof. Instead, our crucial tool is a weak Harnack inequality for supersolutions due to Trudinger [7] in connection with a technique developed by L.Caffarelli [1]. Received: 25 September 2006     

Harnack inequality for degenerate and singular operators of <Emphasis Type="Italic">p</Emphasis>-Laplacian type on Riemannian manifolds

We study viscosity solutions to degenerate and singular elliptic equations of p-Laplacian type on Riemannian manifolds. The Krylov–Safonov type Harnack inequality for the p-Laplacian operators with \(1<p<\infty \) is established on the manifolds with Ricci curvature bounded from below based on ABP type estimates. We also prove the Harnack inequality for nonlinear p-Laplacian type operators assuming that a nonlinear perturbation of Ricci curvature is bounded below.     

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).     

Harnack inequalities and Gaussian estimates for a class of hypoelliptic operators

We prove a global Harnack inequality for a class of degenerate evolution operators by repeatedly using an invariant local Harnack inequality. As a consequence we obtain an accurate Gaussian lower bound for the fundamental solution for some meaningful families of degenerate operators.

     

Riemann流形上改进的p-Laplace方程的梯度估计

本文研究Riemann流形上的改进的p-Laplace方程,运用截断函数的估计、Hessian比较定理和Laplace比较定理,得到该方程正解的梯度估计.并应用该结论,得到在Riemann流形上关于改进的p-Laplace方程正解的Harnack不等式和Liouville型定理.     

A new entropy formula for the linear heat equation

We give a monotonicity entropy formula for the linear heat equation on complete manifolds with Ricci curvature bounded from below. As its applications, we get a differential Harnack inequality and a lower bound estimate about the heat kernel.     

Regularity for Certain Nonlinear Parabolic Systems

Abstract

By means of an inequality of Poincaré type, a weak Harnack inequality for the gradient of a solution and an integral inequality of Campanato type, it is shown that a solution to certain degenerate parabolic system is locally Hölder continuous. The system is a generalization of p-Laplacian system. Using a difference quotient method and Moser type iteration it is then proved that the gradient of a solution is locally bounded. Finally using the iteration and scaling it is shown that the gradient of the solution satisfies a Campanato type integral inequality and is locally Hölder continuous.