Harnack and mean value inequalities on graphs

We prove a Harnack inequality for positive harmonic functions on graphs which is similar to a classical result of Yau on Riemannian manifolds. Also, we prove a mean value inequality of nonnegative subharmonic functions on graphs.     

On Harnack inequality for degenerate nonlinear higher-order elliptic equations

In this article we establish Harnack type inequality and derive a result of local Hölder continuity for solutions of degenerate nonlinear higher-order elliptic equations where the degeneracy is of Muckenhoupt kind.     

The boundary Harnack inequality for infinity harmonic functions in Lipschitz domains satisfying the interior ball condition

In this note, we give a short proof for the boundary Harnack inequality for infinity harmonic functions in a Lipschitz domain satisfying the interior ball condition. Our argument relies on the use of quasiminima and the notion of comparison with cones.     

Harnack Inequalities for Jump Processes

We consider a class of pure jump Markov processes in R ^d whose jump kernels are comparable to those of symmetric stable processes. We establish a Harnack inequality for nonnegative functions that are harmonic with respect to these processes. We also establish regularity for the solutions to certain integral equations.     

Harnack inequality and continuity of solutions to quasi-linear degenerate parabolic equations with coefficients from Kato-type classes

For a general class of divergence type quasi-linear degenerate parabolic equations with measurable coefficients and lower order terms from nonlinear Kato-type classes, we prove local boundedness and continuity of solutions, and the intrinsic Harnack inequality for positive solutions.     

A Harnack Inequality for Degenerate Parabolic Equations

In this paper we apply the Moser iteration method to degenerate parabolic divergence structure equations. Under some conditions we get a Harnack inequality for weak solutions and from it derive Hölder estimates for weak solutions of uniformly degenerate parabolic equations and the continuity of weak solutions of non-uniformly degenerate parabolic equations.     

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.     

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.     

Harnack inequalities for stochastic equations driven by Lévy noise

By using coupling argument and regularization approximations of the underlying subordinator, dimension-free Harnack inequalities are established for a class of stochastic equations driven by a Lévy noise containing a subordinate Brownian motion. The Harnack inequalities are new even for linear equations driven by Lévy noise, and the gradient estimate implied by our log-Harnack inequality considerably generalizes some recent results on gradient estimates and coupling properties derived for Lévy processes or linear equations driven by Lévy noise. The main results are also extended to semilinear stochastic equations in Hilbert spaces.     

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.     

Harnack inequality and continuity of solutions to elliptic equations with nonstandard growth conditions and lower order terms

In this paper, we prove the Harnack inequality and continuity of solutions for a general class of divergence-type elliptic equations with nonstandard growth measurable coefficients in the main part and lower order terms from nonlinear Kato-type classes.     

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.     

Harnack Inequality for Quasilinear Elliptic Equations in Generalized Orlicz-Sobolev Spaces

Potential Analysis - In this paper we prove, by a new method, the Harnack inequality for positive solutions of quasilinear elliptic equations in the generalized Orlicz-Sobolev space setting. Our...     

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.     

Parabolic Harnack inequality of viscosity solutions on Riemannian manifolds

We consider viscosity solutions to nonlinear uniformly parabolic equations in nondivergence form on a Riemannian manifold M with the sectional curvature bounded from below by −κ   for κ≥0$\kappa \ge 0$. In the elliptic case, Wang and Zhang [24] recently extended the results of [5] to nonlinear elliptic equations in nondivergence form on such M, where they obtained the Harnack inequality for classical solutions. We establish the Harnack inequality for nonnegative viscosity solutions to nonlinear uniformly parabolic equations in nondivergence form on M. The Harnack inequality of nonnegative viscosity solutions to the elliptic equations is also proved.     

A remark on the Harnack inequality for non-self-adjoint evolution equations

In this paper we consider a non-self-adjoint evolution equation on a compact Riemannian manifold with boundary. We prove a Harnack inequality for a positive solution satisfying the Neumann boundary condition. In particular, the boundary of the manifold may be nonconvex and this gives a generalization to a theorem of Yau.

     

Intrinsic Harnack inequalities for parabolic equations with variable exponents

In this paper, we discuss a class of parabolic equations with variable exponents. By using the Steklov average and Young’s inequality, we establish energy estimates for solutions to these equations. Then based on measure theoretical arguments, we derive intrinsic Harnack inequalities for solutions from functions and cylinders constructed here.     

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.     

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.