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In this paper we establish a Harnack inequality for non-negative solutions of L ⁢ u = f ⁢ ( u ) {Lu=f(u)} where L is a non-divergence structure uniformly elliptic operator and f is a non-decreasing function that satisfies an appropriate growth conditions at infinity.

In this paper, we consider a large class of subordinate random walks $X$ on integer lattice $\mathbb{Z}^d$ via subordinators with Laplace exponents which are complete Bernstein functions satisfying a certain lower scaling condition at zero. We establish estimates for one-step transition probabilities, the Green function and the Green function of a ball, and prove the Harnack inequality for non-negative harmonic functions., 31 pages

In this paper, we consider the Harnack estimates for a nonlinear parabolic equation under the Ricci flow. The gradient estimates for positive solutions as well as Li–Yau type inequalities are also given.

In the first part of this article, we prove two-sided estimates of hitting probabilities of balls, the potential kernel and the Green function for a ball for general isotropic unimodal L\'evy processes. Our bounds are sharp under the absence of the Gaussian component and a mild regularity condition on the density of the L\'{e}vy measure: its radial profile needs to satisfy a scaling-type condition, which is equivalent to $O$-regular variation at zero and at infinity with lower indices greater than $-d - 2$. We also prove a supremum estimate and a regularity result for functions harmonic with respect to a general isotropic unimodal L\'evy process. In the second part we apply the recent results on the boundary Harnack inequality and Martin representation of harmonic functions for the class of isotropic unimodal L\'evy processes characterised by a localised version of the scaling-type condition mentioned above. As a sample application, we provide sharp two-sided estimates of the Green function of a half-space. Our results are expressed in terms of Pruitt's functions $K(r)$ and $L(r)$, measuring local activity and the amount of large jumps of the L\'evy process, respectively., Comment: 35 pages

We prove a priori bounds on the Green function for general integral operators in divergence form in the spirit of Littman, Stampacchia and Weinberger’s result. For general linear integral operators with bounded measurable coefficients, we introduce the so-called fractional harmonic measure and prove several estimates on it. As an application, we prove a new boundary Harnack principle for these operators. Once the bounds on the Green function are known, the proof follows the approach of Caffarelli-Fabes-Mortola-Salsa and K. Bogdan.

It is obtained that there exist strong solutions of Pucci extremal equations with sublinear growth in D u and measurable ingredients. It is proved that a strong maximum principle holds in a local sense in Lemma 4.1 although even the (weak) maximum principle fails. By using this existence result, it is shown that the ABP type maximum principle and the weak Harnack inequality for viscosity solutions hold true. As an application, the Holder continuity for viscosity solutions of possibly singular, quasilinear equations is established.

We study energy functionals obtained by adding a possibly discontinuous potential to an interaction term modeled upon a Gagliardo-type fractional seminorm. We prove that minimizers of such non-differentiable functionals are locally bounded, H\"older continuous, and that they satisfy a suitable Harnack inequality. Hence, we provide an extension of celebrated results of M. Giaquinta and E. Giusti to the nonlocal setting. To do this, we introduce a particular class of fractional Sobolev functions, reminiscent of that considered by E. De Giorgi in his seminal paper of 1957. The flexibility of these classes allows us to also establish regularity of solutions to rather general nonlinear integral equations., Comment: 59 pages

We consider a class of quasilinear elliptic second-order equations of divergence structure admitting uniform degeneration in the domain. We prove that the classical Harnack inequality fails and establish a Harnack inequality corresponding to the equation in question.

In this paper, we focus on a space–time fractional diffusion equation with the generalized Caputo's fractional derivative operator and a general space nonlocal operator (with the fractional Laplace operator as a special case). A weak Harnack's inequality has been established by using a special test function and some properties of the space nonlocal operator. Based on the weak Harnack's inequality, a strong maximum principle has been obtained which is an important characterization of fractional parabolic equations. With these tools, we establish a uniqueness result of an inverse source problem on the determination of the temporal component of the inhomogeneous term, which seems to be the first theoretical result of the inverse problem for such a general fractional diffusion model.

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

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 blowup of the positive solutions and recover a classical Harnack inequality. We also obtain a result of Liouville type for the elliptic equation.

We define a homogeneous parabolic De Giorgi class of order 2 which suits a mixed type class of evolution equations whose simplest example is \begin{document}$\mu (x) \frac{\partial u}{\partial t} - \Delta u = 0$\end{document} where \begin{document}$\mu$\end{document} can be positive, null and negative. The functions belonging to this class are local bounded and satisfy a Harnack type inequality. Interesting by-products are Holder-continuity, at least in the "evolutionary" part of \begin{document}$\Omega$\end{document} and in particular in the interface \begin{document}$I$\end{document} where \begin{document}$\mu$\end{document} change sign, and an interesting maximum principle.

In this paper, by the method of J. F. Li and X. J. Xu (Differential Harnack inequalities on Riemannian manifolds I: Linear heat equation, Adv. in Math., 226 (2011), 4456–4491), we shall consider the nonlinear parabolic equation Δ−∂∂tu(x,t)+h(x,t)uα(x,t)=0 on Riemannian manifolds with Ricci(M)≥−k, k≥0. First of all, we shall derive the corresponding Li–Xu type gradient estimates of the positive solutions for α≤1 . As applications, we deduce Liouville type theorem and Harnack inequality for some special cases. Besides, when α=1, our results are different from Li and Yau's results. We also extend the results of J. F. Li and X. J. Xu, and the results of Y. Yang.

In this paper, let (Mn, g) be an n-dimensional complete Riemannian manifold with the m-dimensional Bakry–Emery 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 $${u_t} = LF\left( u \right) = \Delta F\left( u \right) - \nabla f \cdot \nabla F\left( u \right),$$ on compact Riemannian manifolds Mn, where F ∈ C2(0,∞), F′ > 0 and f is a C2-smooth function defined on Mn. 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.

A special Harnack inequality is proved for solutions of nonlinear elliptic equations of the p(x)-Laplacian type with a variable exponent p(x) that takes different values on two sides of a hyperplane dividing the domain. Examples are given showing that the classical Harnack inequality does not hold in this case.

In this paper we proof a Harnack inequality and a regularity theorem for quasi-minima of scalar integral functionals with general growth conditions.

We show the sharpness of the modulus of continuity of a function f for which the domain lying above the graph of f satisfies the global boundary Harnack principle, with the aid of precise estimates of the Poisson integrals with respect to the Helmholtz equation in the half space.

We present a new Harnack inequality for non-negative discrete supersolutions of fully nonlinear uniformly elliptic difference equations on rectangular lattices. This estimate applies to all supersolutions and has the Harnack constant depending on the graph distance on lattices. For the proof we modify the proof of the weak Harnack inequality. Applying the same idea to elliptic equations in a Euclidean space, we also derive a Harnack type inequality for non-negative viscosity supersolutions.

We study the regularity theory of quasi-minimizers of functionals with L p ⁢ ( ⋅ ) ⁢ log ⁡ L {L^{p(\,\cdot\,)}\log L} -growth. In particular, we prove the Harnack inequality and, in addition, the local boundedness and the Hölder continuity of the quasi-minimizers. We directly prove our results via De Giorgi’s method.

In this paper, we prove Hamilton’s Harnack inequality and the gradient estimates of the logarithmic heat kernel for the Witten Laplacian on complete Riemannian manifolds. As applications, we prove the W-entropy formula for the Witten Laplacian on complete Riemannian manifolds, and prove a family of logarithmic Sobolev inequalities on complete Riemannian manifolds with natural geometric condition.

Shift Harnack inequality and integration by parts formula are established for semilinear stochastic partial differential equations and stochastic functional partial differential equations by modifying the coupling used by F.-Y. Wang [Ann. Probab., 2012, 42(3): 994–1019]. Log-Harnack inequality is established for a class of stochastic evolution equations with non-Lipschitz coefficients which includes hyperdissipative Navier-Stokes/Burgers equations as examples. The integration by parts formula is extended to the path space of stochastic functional partial differential equations, then a Dirichlet form is defined and the log-Sobolev inequality is established.

In this paper, we prove several Harnack estimates for positive solutions to the heat-type equations with respect to time-dependent Riemannian metric evolving by the geometric flow. In particular, we obtain Li–Yau type estimates and Perelman type differential Harnack inequalities and as an application, we demonstrate how these results can be obtained under various geometric flows.

In this paper we proof an Harnack inequality and a regularity theorem for quasi-minima of scalar integral functionals of the calculus of variations with nearly linear growth conditions. Particularly our results are applicable to functional with L Ln growth conditions.

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

A well known theorem of Delmotte is that Gaussian bounds, parabolic Harnack inequality, and the combination of volume doubling and Poincare inequality are equivalent for graphs. In this paper we consider graphs for which these conditions hold, but only for sufficiently large balls, and prove a similar equivalence.

In this paper, we consider the gradient estimates for a positive solution of the nonlinear parabolic equation ∂ t u = Δ t u + h u p on a Riemannian manifold whose metrics evolve under the Ricci flow. Two Harnack inequalities and other interesting results are obtained.

In this paper we establish gradient estimates for positive solutions to the equation (0.1)△fup=−λu on any smooth metric measure space whose m-Bakry–Emery curvature is bounded from below by −(m−1)K with K≥0. These estimates imply Liouville theorems for (0.1). When p→1, our main theorem reduces to the gradient estimate of Wang (2010) [9]. As applications, several Harnack inequalities are obtained.

The dimension-free Harnack inequality and uniform heat kernel upper/lower bounds are derived for a class of infinite-dimensional GEM processes, which was introduced in Feng and Wang (J. Appl. Probab. 44 938–949 2007) to simulate the two-parameter GEM distributions. In particular, the associated Dirichlet form satisfies the super log-Sobolev inequality which strengthens the log-Sobolev inequality derived in Feng and Wang (J. Appl. Probab. 44 938–949 2007). To prove the main results, explicit Harnack inequality and super Poincare inequality are established for the one-dimensional Wright-Fisher diffusion processes. The main tool of the study is the coupling by change of measures.

We show that shift Harnack type inequalities (in the sense of F.-Y.~Wang \cite{Wan14}) are preserved under Bochner's subordination. The proofs are based on two types of moment estimates for subordinators. As a by-product we establish moment estimates for general L\'evy processes., Comment: to appear in Stochastic Process. Appl

It is shown that the parabolic Harnack property stands as an intrinsic feature of the Monge-Ampere quasi-metric structure by proving Harnack’s inequality for non-negative solutions to the linearized parabolic Monge-Ampere equation under minimal geometric assumptions.

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.

We prove a Harnack inequality for minimisers of a class of non-autonomous functionals with non-standard growth conditions. They are characterised by the fact that their energy density switches between two types of different degenerate phases.

In this paper we deal with quasilinear singular parabolic equations with L ∞ coefficients, whose prototypes are the p -Laplacian ( 2 N N + 1 p 2 ) equations. In this range of the parameters, we are in the so called fast diffusion case. Extending a recent result (Ragnedda et al. 2013), we are able to prove Harnack estimates at large, i.e. starting from the value attained in a point by the solution, we are able to give explicit and sharp pointwise estimates, from below by using the Barenblatt solutions. In the last section we briefly show how these results can be adapted to equations of porous medium type in the fast diffusion range i.e. ( N − 2 N ) + m 1 .

The dimension-free Harnack inequality for the heat semigroup is established on the \(\mathrm{{RCD}}(K,\infty )\) space, which is a non-smooth metric measure space having the Ricci curvature bounded from below in the sense of Lott–Sturm–Villani plus the Cheeger energy being quadratic. As its applications, the heat semigroup entropy-cost inequality and contractivity properties of the semigroup are studied, and a strong-enough Gaussian concentration implying the log-Sobolev inequality is also shown as a generalization of the one on the smooth Riemannian manifold.

In this paper, we prove logarithmic Harnack inequalities for homogeneous graphs. As a consequence, we derive lower estimates for the log-Sobolev constant, extending previous results for Ricci flat graphs.

We prove a differential Harnack inequality for the Endangered Species Equation, a nonlinear parabolic equation. Our derivation relies on an idea related to the parabolic maximum principle. As an application of this inequality, we will show that positive solutions to this equation must blowup in finite time. We also use this inequality to partially answer a question of Hamilton, 2011.

In this work, we give a sufficient and necessary condition in which the (parabolic) Harnack inequality of the original space would imply that of quasi-conformal spaces. The result can be applied to a large class of functions with discrete singularities.

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.

By using coupling by change of measures, the Driver-type integration by parts formula is established for a class of stochastic differential equations driven by fractional Brownian motions. As applications, (log) shift Harnack inequalities and estimates on the distribution density of the solutions are presented.