Elliptic Harnack Inequality Research Articles

Stability of elliptic Harnack inequalities

Stability of elliptic Harnack inequalities

Parabolic Harnack Estimates for anisotropic slow diffusion

We prove a Harnack inequality for positive solutions of a parabolic equation with slow anisotropic spatial diffusion. After identifying its natural scalings, we reduce the problem to a Fokker—Planck equation and construct a self-similar Barenblatt solution. We exploit translation invariance to obtain positivity near the origin via a self-iteration method and deduce a sharp anisotropic expansion of positivity. This eventually yields a scale invariant Harnack inequality in an anisotropic geometry dictated by the speed of the diffusion coefficients. As a corollary, we infer Hölder continuity, an elliptic Harnack inequality and a Liouville theorem.

The weak elliptic Harnack inequality revisited

The weak elliptic Harnack inequality revisited

Elliptic Harnack inequality for ℤd

We prove the scale invariant Elliptic Harnack Inequality (EHI) for non-negative harmonic functions on ${\mathbb{Z}}^d$. The purpose of this note is to provide a simplified self-contained probabilistic proof of EHI in ${\mathbb{Z}}^d$ that is accessible at the undergraduate level. We use the Local Central Limit Theorem for simple symmetric random walks on ${\mathbb{Z}}^d$ to establish Gaussian bounds for the $n$-step probability function. The uniform Green inequality and the classical Balayage formula then imply the EHI.

On the conformal walk dimension: quasisymmetric uniformization for symmetric diffusions

We introduce the notion of conformal walk dimension, which serves as a bridge between elliptic and parabolic Harnack inequalities. The importance of this notion is due to the fact that, for a given strongly local, regular symmetric Dirichlet space in which every metric ball has compact closure (MMD space), the finiteness of the conformal walk dimension characterizes the conjunction of the metric doubling property and the elliptic Harnack inequality. Roughly speaking, the conformal walk dimension of an MMD space is defined as the infimum over all possible values of the walk dimension with which the parabolic Harnack inequality can be made to hold by a time change of the associated diffusion and by a quasisymmetric change of the metric. We show that the conformal walk dimension of any MMD space satisfying the metric doubling property and the elliptic Harnack inequality is two, and provide a necessary condition for a pair of such changes to attain the infimum defining the conformal walk dimension when it is attained by the original pair. We also prove a necessary condition for the existence of such a pair attaining the infimum in the setting of a self-similar Dirichlet form on a self-similar set, and apply it to show that the infimum fails to be attained for the Vicsek set and the N-dimensional Sierpiński gasket with Nge 3, in contrast to the attainment for the two-dimensional Sierpiński gasket due to Kigami (Math Ann 340(4):781–804, 2008).

Elliptic Harnack's inequality for a singular nonlinear parabolic equation in non‐divergence form

We prove an elliptic Harnack's inequality for a general form of a parabolic equation that generalizes both the standard parabolic p $p$ -Laplace equation and the normalized version that has been proposed in stochastic game theory. This version of the inequality does not require the intrinsic waiting time and we get the estimate with the same time level on both sides of the inequality.

An elliptic Harnack inequality for difference equations with random balanced coefficients

We prove an elliptic Harnack inequality at large scale on the lattice Zd for nonnegative solutions of a difference equation with balanced i.i.d. coefficients which are not necessarily elliptic. We also identify the optimal constant in the Harnack inequality. Our proof relies on a quantitative homogenization result of the corresponding invariance principle to Brownian motion and on percolation estimates. As a corollary of our main theorem, we derive an almost optimal Hölder estimate.

Random conductance models with stable-like jumps: Heat kernel estimates and Harnack inequalities

We establish two-sided heat kernel estimates for random conductance models with non-uniformly elliptic (possibly degenerate) stable-like jumps on graphs. These are long range counterparts of the well known two-sided Gaussian heat kernel estimates by M.T. Barlow for nearest neighbor (short range) random walks on the supercritical percolation cluster. Unlike the cases of nearest neighbor conductance models, we cannot use parabolic Harnack inequalities since even elliptic Harnack inequalities do not hold in the present setting. As an application, we establish the local limit theorem for the models.

Spectrally positive Bakry-Émery Ricci curvature on graphs

We investigate analytic and geometric implications of non-constant Ricci curvature bounds. We prove a Lichnerowicz eigenvalue estimate and finiteness of the fundamental group assuming that L+2Ric is a positive operator where L is the graph Laplacian. Assuming that the negative part of the Ricci curvature is small in Kato sense, we prove diameter bounds, elliptic Harnack inequality and Buser inequality. This article seems to be the first one establishing these results while allowing for some negative curvature.

Boundary Harnack principle and elliptic Harnack inequality

We prove a scale-invariant boundary Harnack principle for inner uniform domains over a large family of Dirichlet spaces. A novel feature of our work is that we do not assume volume doubling property for the symmetric measure.

Stability of the elliptic Harnack inequality

Stability of the elliptic Harnack inequality

Elliptic Harnack inequalities for symmetric non-local Dirichlet forms

We study relations and characterizations of various elliptic Harnack inequalities for symmetric non-local Dirichlet forms on metric measure spaces. We allow the scaling function be state-dependent and the state space possibly disconnected. Stability of elliptic Harnack inequalities is established under certain regularity conditions and implication for a priori Hölder regularity of harmonic functions is explored. New equivalent statements for parabolic Harnack inequalities of non-local Dirichlet forms are obtained in terms of elliptic Harnack inequalities.

On the elliptic Harnack inequality

A brief exposition on some tools for proving the elliptic Harnack inequality is presented.

Generalized capacity, Harnack inequality and heat kernels of Dirichlet forms on metric measure spaces

We give necessary and sufficient conditions for sub-Gaussian estimates of the heat kernel of a strongly local regular Dirichlet form on a metric measure space. The conditions for two-sided estimates are given in terms of the generalized capacity inequality and the Poincare inequality. The main difficulty lies in obtaining the elliptic Harnack inequality under these assumptions. The conditions for upper bound alone are given in terms of the generalized capacity inequality and the Faber–Krahn inequality.

Interior Harnack Estimates: The State-of-the-Art for Quasilinear Singular Parabolic Equations

In this paper we give some historical information about elliptic and parabolic Harnack inequalities. Then we state the main results known for Harnack inequalities of solutions to quasilinear degenerate parabolic equations. Lastly we focus our attention on Harnack inequalities of solutions to quasilinear singular parabolic equations where the theory did important steps forward in the last few years but still there are some points to be fully understood.

Heat Kernels and Green Functions on Metric Measure Spaces

AbstractWe prove that, in a setting of local Dirichlet forms on metric measure spaces, a two-sided sub-Gaussian estimate of the heat kernel is equivalent to the conjunction of the volume doubling property, the elliptic Harnack inequality, and a certain estimate of the capacity between concentric balls. The main technical tool is the equivalence between the capacity estimate and the estimate of a mean exit time in a ball that uses two-sided estimates of a Green function in a ball.

A stability theorem for elliptic Harnack inequalities

We prove a stability theorem for the elliptic Harnack inequality: if two weighted graphs are equivalent, then the elliptic Harnack inequality holds for harmonic functions with respect to one of the graphs if and only if it holds for harmonic functions with respect to the other graph. As part of the proof, we give a characterization of the elliptic Harnack inequality.

Invariance principle for the random conductance model

We study a continuous time random walk X in an environment of i.i.d. random conductances $${\mu_{e} \in [0,\infty)}$$ in $${\mathbb{Z}^d}$$ . We assume that $${\mathbb{P}(\mu_{e} > 0) > p_c}$$ , so that the bonds with strictly positive conductances percolate, but make no other assumptions on the law of the μ e . We prove a quenched invariance principle for X, and obtain Green’s functions bounds and an elliptic Harnack inequality.

Forward, backward and elliptic Harnack inequalities for non-negative solutions to certain singular parabolic partial differential equations

Forward, backward and elliptic Harnack inequalities for non-negative solutions of a class of singular, quasi-linear, parabolic equations, are established. These classes of singular equations include the p-Laplacean equation and equations of the porous medium type. Key novel points include form of a Harnack estimate backward in time, that has never been observed before, and measure theoretical proofs, as opposed to comparison principles. These Harnack estimates are established in the super-critical range (1.5) below. Such a range is optimal for a Harnack estimate to hold. Mathematics Subject Classification (2010): 35K65 (primary); 35B65, 35B45 (secondary). 1. Main results Let E be an open set in RN and for T > 0 let ET = E × (0, T ]. Let u be a weak solution u ∈ Cloc ( 0, T ; Lloc(E) ) ∩ L p loc(0, T ; W 1,p loc (E)) 1 < p < 2 (1.1) of a quasi-linear, singular parabolic equation of the type ut − div A(x, t, u, Du) = B(x, t, u, Du) weakly in ET (1.2) where the functions A : ET × RN+1 → RN and B : ET × RN+1 → R are only assumed to be measurable and subject to the structure conditions   A(x, t, u, Du) · Du ≥ Co|Du|p − C p |A(x, t, u, Du)| ≤ C1|Du|p−1 + C p−1 |B(x, t, u, Du)| ≤ C |Du|p−1 + C p a.e. in ET (1.3) This work has been partially supported by I.M.A.T.I. – C.N.R. – Pavia. DiBenedetto’s work partially supported by National Science Foundation grant DMS-0652385. Received March 9, 2009; accepted in revised form May 31, 2009.

Intrinsic ultracontractivity of nonsymmetric diffusions with measure-valued drifts and potentials

Recently, in [Preprint (2006)], we extended the concept of intrinsic ultracontractivity to nonsymmetric semigroups. In this paper, we study the intrinsic ultracontractivity of nonsymmetric diffusions with measure-valued drifts and measure-valued potentials in bounded domains. Our process Y is a diffusion process whose generator can be formally written as L + μ ∇ ¯ ν with Dirichlet boundary conditions, where L is a uniformly elliptic second-order differential operator and μ = (μ 1 ,... μ d ) is such that each component μ i , i = 1,..., d, is a signed measure belonging to the Kato class K d , and v is a (nonnegative) measure belonging to the Kato class K d,2 . We show that scale-invariant parabolic and elliptic Harnack inequalities are valid for Y. In this paper, we prove the parabolic boundary Harnack principle and the intrinsic ultracontractivity for the killed diffusion Y D with measure-valued drift and potential when D is one of the following types of bounded domains: twisted Holder domains of order a e (1/3, 1], uniformly Holder domains of order a ∈ (0, 2) and domains which can be locally represented as the region above the graph of a function. This extends the results in [J. Funct. Anal. 100 (1991) 181-206] and [Probab. Theory Related Fields 91 (1992) 405-443]. As a consequence of the intrinsic ultracontractivity, we get that the supremum of the expected conditional lifetimes of Y D is finite.