Harnack Inequality Research Articles

Differential Harnack Inequality for a Parabolic Equation Under the Finsler-Geometric Flow

Differential Harnack Inequality for a Parabolic Equation Under the Finsler-Geometric Flow

Aronson–Bénilan estimates for weighted porous medium equations under the geometric flow

In this paper, we study Aronson–Bénilan gradient estimates for positive solutions of weighted porous medium equations coupled with the geometric flow on a complete measure space . As an application, by integrating the gradient estimates, we derive the corresponding 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 method of stochastic characteristics for linear second-order hypoelliptic equations

We study hypoelliptic stochastic differential equations (SDEs) and their connection to degenerate-elliptic boundary value problems on bounded or unbounded domains. In particular, we provide probabilistic conditions that guarantee that the formal stochastic representation of a solution is smooth on the interior of the domain and continuously approaches the prescribed boundary data at a given boundary point. The main general results are proved using fine properties of the process stopped at the boundary of the domain combined with hypoellipticity of the operators associated to the SDE. The main general results are then applied to deduce properties of the associated Green’s functions and to obtain a generalization of Bony’s Harnack inequality. We moreover revisit the transience and recurrence dichotomy for hypoelliptic diffusions and its relationship to invariant measures.

Harnack inequalities and Gaussian estimates for random walks on metric measure spaces

Harnack inequalities and Gaussian estimates for random walks on metric measure spaces

Exponential quasi-ergodicity for processes with discontinuous trajectories

This paper tackles the issue of establishing an upper-bound on the asymptotic ratio of survival probabilities between two different initial conditions, asymptotically in time for a given Markov process with extinction. Such a comparison is a crucial step in recent techniques for proving exponential convergence to a quasi-stationary distribution. We introduce a weak form of the Harnack’s inequality as the essential ingredient for such a comparison. This property is actually a consequence of the convergence property that we intend to prove. Its complexity appears as the price to pay for the level of flexibility required by our applications, notably for processes with jumps on a multidimensional state-space. We show in our illustrations how simply and efficiently it can be used nonetheless. As illustrations, we consider two continuous-time processes on ℝd that do not satisfy the classical Harnack’s inequality, even in a local version. The first one is a piecewise deterministic process while the second is a pure jump process with restrictions on the directions of its jumps.

Spatial patterns for a predator-prey system with Beddington-DeAngelis functional response and fractional cross-diffusion

<abstract><p>In this paper, we investigate a predator-prey system with fractional type cross-diffusion incorporating the Beddington-DeAngelis functional response subjected to the homogeneous Neumann boundary condition. First, by using the maximum principle and the Harnack inequality, we establish a priori estimate for the positive stationary solution. Second, we study the non-existence of non-constant positive solutions mainly by employing the energy integral method and the Poincaré inequality. Finally, we discuss the existence of non-constant positive steady states for suitable large self-diffusion $ d_2 $ or cross-diffusion $ d_4 $ by using the Leray-Schauder degree theory, and the results reveal that the diffusion $ d_2 $ and the fractional type cross-diffusion $ d_4 $ can create spatial patterns.</p></abstract>

Mean-field stochastic differential equations driven by <inline-formula><tex-math id="M1">$ G $</tex-math></inline-formula>-Brownian motion

The main purpose of this paper is to analyze the mean-field stochastic differential equations driven by $ G $-Brownian motion (mean-field $ G $-SDEs). We first investigate the existence and uniqueness of the solution for the mean-field $ G $-SDEs by utilizing the Banach fixed point theorem. Moreover, by the method of coupling by change of measures, the Harnack and log-Harnack inequalities for mean-field $ G $-SDEs with additive noise are established.

Local Hessian estimates of solutions to nonlinear parabolic equations along Ricci flow

In the paper, we first obtain some Hessian estimates for any positive solution to the nonlinear parabolic equations$ \begin{equation*} \partial_t u(x, t) = \Delta_{g(t)} u(x, t)+\lambda u^{\alpha}(x, t) \end{equation*} $on a Riemannian manifold with a fixed metric and along the Ricci flow by constructing a new auxiliary function. Moreover, we only need the curvature tensor is bounded. These estimates imply some local, time reversed Harnack type inequalities. And our conclusion generalizes the Han and Zhang's result, but also our proof process is different and has been slightly simplified. As its applications, we derive the Hessian estimate of harmonic function and eigenfunction.

The weak elliptic Harnack inequality revisited

The weak elliptic Harnack inequality revisited

Weak Harnack Inequalities for Eigenvalues and the Monotonicity of Hessian's Rank

Weak Harnack Inequalities for Eigenvalues and the Monotonicity of Hessian's Rank

Green function estimates for second order elliptic operators in non-divergence form with Dini continuous coefficients

Two-sided sharp Green function estimates are obtained for second order uniformly elliptic operators in non-divergence form with Dini continuous coefficients in bounded C1,1 domains, which are shown to be comparable to that of the Dirichlet Laplace operator in the domain. The first and second order derivative estimates of the Green functions are also derived. Moreover, boundary Harnack inequality with an explicit boundary decay rate and interior Schauder’s estimates for these differential operators are established, which may be of independent interest.

Weak Harnack inequalities for eigenvalues and a constant rank theorem for level sets

We consider convex level sets of solutions of a class of quasilinear elliptic equations. We prove a weak Harnack inequality for the eigenvalues of the Weingarten tensor of level sets and then obtain a quantitative version of constant rank theorem.

Differential Harnack inequalities via Concavity of the arrival time

Differential Harnack inequalities via Concavity of the arrival time

Harnack inequality for polyharmonic equations

Some new types of mean value formulas for the polyharmonic functions were established. Based on the formulas, the Harnack inequality for the nonnegative solutions to the polyharmonic equations was proved.

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.

Strong Maximum Principle and Boundary Estimates for Nonhomogeneous Elliptic Equations

We give a simple proof of the strong maximum principle for viscosity subsolutions of fully nonlinear nonhomogeneous degenerate elliptic equations on the formF(x,u,Du,D2u)=0\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ F(x,u,Du,D^{2}u) = 0 $$\\end{document} under suitable assumptions allowing for non-Lipschitz growth in the gradient term. In case of smooth boundaries, we also prove a Hopf lemma, a boundary Harnack inequality, and that positive viscosity solutions vanishing on a portion of the boundary are comparable with the distance function near the boundary. Our results apply, e.g., to weak solutions of an eigenvalue problem for the variable exponent p-Laplacian.

$$C^{2,\alpha }$$ regularity of free boundaries in parabolic non-local obstacle problems

We study the regularity of the free boundary in the parabolic obstacle problem for the fractional Laplacian \((-\Delta )^s\) (and more general integro-differential operators) in the regime \(s>\frac{1}{2}\). We prove that once the free boundary is \(C^1\) it is actually \(C^{2,\alpha }\). To do so, we establish a boundary Harnack inequality in \(C^1\) and \(C^{1,\alpha }\) (moving) domains, providing that the quotient of two solutions of the linear equation, that vanish on the boundary, is as smooth as the boundary. As a consequence of our results we also establish for the first time optimal regularity of such solutions to nonlocal parabolic equations in moving domains.

Regularity of Lipschitz free boundaries for a p(x)-Laplacian problem with right hand side

We continue our study in [24] on viscosity solutions to a one-phase free boundary problem for the p(x)-Laplacian with non-zero right hand side. We first prove that viscosity solutions are locally Lipschitz continuous, which is the optimal regularity for the problem. Then we prove that Lipschitz free boundaries of viscosity solutions are C1,α. We also present some applications of our results.Moreover, we obtain new results for the operator under consideration that are of independent interest, such as a Harnack inequality.

Non-negative Ollivier curvature on graphs, reverse Poincaré inequality, Buser inequality, Liouville property, Harnack inequality and eigenvalue estimates

We prove that for combinatorial graphs with non-negative Ollivier curvature, one has‖Ptμ−Ptν‖1≤W1(μ,ν)t for all probability measures μ,ν where Pt is the heat semigroup and W1 is the ℓ1-Wasserstein distance. This turns out to be an equivalent formulation of a version of reverse Poincaré inequality. Furthermore, this estimate allows us to prove Buser inequality, Liouville property and the eigenvalue estimate λ1≥log⁡(2)/diam2.