A “Sup + C Inf” inequality for Liouville‐type equations with singular potentials

We generalize a Harnack‐type inequality (I. Shafrir, C. R. Acad. Sci. Paris, 315 (1992), 159–164), for solutions of Liouville equations to the case where the weight function may admit zeroes or singularities of power‐type |x|^2α, with α ∈ (?1, 1). © 2011 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim     

Regularity theory for fully nonlinear integro‐differential equations

We consider nonlinear integro‐differential equations like the ones that arise from stochastic control problems with purely jump Lévy processes. We obtain a nonlocal version of the ABP estimate, Harnack inequality, and interior C^{1, α} regularity for general fully nonlinear integro‐differential equations. Our estimates remain uniform as the degree of the equation approaches 2, so they can be seen as a natural extension of the regularity theory for elliptic partial differential equations. © 2008 Wiley Periodicals, Inc.     

Carleson measures and conformal self‐mappings in the real unit ball

Bounded and compact Carleson measures in the unit ball B of R ⁿ , n ≥ 2, are characterized by means of global Dirichlet integrals of the conformal self‐map T_a taking a ∈ B to the origin. The same proof applies in the unit ball of C ⁿ . It is also proved that the powers of the Jacobian of T_a satisfy the weak Harnack inequality and even Harnack's inequality with a constant independent of a. As an application of these results it is shown that the two different definitions for Carleson measures in the existing literature are equivalent for a certain range of parameter values. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)     

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.     

A lower‐epiperimetric inequality for area‐minimizing surfaces

The epiperimetric inequality introduced by E. R. Reifenberg in [3] gives a rate of decay at a point for the decreasing k‐density of area of an area‐minimizing integral k‐cycle. While dilating the cycle at that point, this rate of decay holds once the configuration is close to a tangent cone configuration and above the limiting density corresponding to that configuration. This is why we propose to call the Reifenberg epiperimetric inequality an upper‐epiperimetric inequality. A direct consequence of this upper‐epiperimetric inequality is the statement that any point possesses a unique tangent cone. The upper‐epiperimetric inequality was proven by B. White in [5] for area‐minimizing 2‐cycles in ?ⁿ. In the present paper we introduce the notion of a lower‐epiperimetric inequality. This inequality gives this time a rate of decay for the decreasing k‐density of area of an area‐minimizing integral k‐cycle, while dilating the cycle at a point once the configuration is close to a tangent cone configuration and below the limiting density corresponding to that configuration. Our main result in the present paper is to prove the lower‐epiperimetric inequality for area‐minimizing 2‐cycles in ?ⁿ. As a consequence of this inequality we prove the “splitting before tilting” phenomenon for calibrated 2‐rectifiable cycles, which plays a crucial role in the proof of the regularity of 1‐1 integral currents in [4]. © 2004 Wiley Periodicals, Inc.     

On a Harnack inequality for the elliptic (<Emphasis Type="Italic">p</Emphasis>, <Emphasis Type="Italic">q</Emphasis>)-Laplacian

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.     

Harnack inequality for the solutions of the <Emphasis Type="Italic">p</Emphasis>-Laplacian with a partially Muckenhoupt weight

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.     

A Minkowski Inequality for Hypersurfaces in the Anti‐de Sitter‐Schwarzschild Manifold

We prove a sharp inequality for hypersurfaces in the n‐dimensional anti‐de Sitter‐Schwarzschild manifold for general n ≥ 3. This inequality generalizes the classical Minkowski inequality for surfaces in the three‐dimensional euclidean space and has a natural interpretation in terms of the Penrose inequality for collapsing null shells of dust. The proof relies on a new monotonicity formula for inverse mean curvature flow and uses a geometric inequality established by the first author in [3].© 2015 Wiley Periodicals, Inc.     

Schröndiger Type and Relaxed Dirichlet Problems for the Subelliptic p-Laplacian

We study at first the solutions of a Schrödinger type problem relative to the subelliptic p-Laplacian: we prove, for potentials that are in the Kato space, an Harnack inequality on enough small intrinsic balls; the continuity of the solutions to the homogeneous Dirichlet problem follows from some estimates in the proof of the Harnack inequality. In the second part of the paper we study a relaxed Dirichlet problem for the subelliptic p-Laplacian and we prove a Wiener type criterion for the regularity of a point (with respect to our problem).     

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).     

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.     

Harnack Inequalities and Bôcher‐Type Theorems for Conformally Invariant,Fully Nonlinear Degenerate Elliptic Equations

We give a generalization of a theorem of Bôcher for the Laplace equation to a class of conformally invariant fully nonlinear degenerate elliptic equations. We also prove a Harnack inequality for locally Lipschitz viscosity solutions and a classification of continuous radially symmetric viscosity solutions. © 2014 Wiley Periodicals, Inc.     

Harnack inequalities and sub-Gaussian estimates for random walks

We show that the -parabolic Harnack inequality for random walks on graphs is equivalent, on one hand, to the sub-Gaussian estimate for the transition probability and, on the other hand, to the conjunction of the elliptic Harnack inequality, the doubling volume property, and the fact that the mean exit time in any ball of radius R is of the order . The latter condition can be replaced by a certain estimate of the resistance of annuli. Received: 15 November 2001 / Revised version: 21 February 2002 / Published online: 6 August 2002     

Wang's Harnack inequalities for space–time white noises driven SPDEs with two reflecting walls and their applications

In this paper, we establish Wang's Harnack inequalities for Gaussian space–time white noises driven the stochastic partial differential equation with double reflecting walls, which is of the infinite dimensional Skorokhod equation. We first establish both the Harnack inequality with power and the log-Harnack inequality for the special case of additive noises by the coupling approach. Then we investigate the log-Harnack inequality for the Markov semigroup associated with the reflected SPDE driven by multiplicative noises using the penalization method and the comparison principle for SPDEs. As their applications, we study the strong Feller property, uniqueness of invariant measures, the entropy-cost inequality, and some other important properties of the transition density.     

Extended Harnack inequalities with exceptional sets and a boundary Harnack principle

Harnack’s inequality is one of the most fundamental inequalities for positive harmonic functions and has been extended to positive solutions of general elliptic equations and parabolic equations. This article gives a different generalization; namely, we generalize Harnack chains rather than equations. More precisely, we allow a small exceptional set and yet obtain a similar Harnack inequality. The size of an exceptional set is measured by capacity. The results are new even for classical harmonic functions. Our extended Harnack inequality includes information about the boundary behavior of positive harmonic functions. It yields a boundary Harnack principle for a very nasty domain whose boundary is given locally by the graph of a function with modulus of continuity worse than Hölder continuity.     

A new formulation of Harnackʼs inequality for nonlocal operators

We provide a new formulation of Harnack?s inequality for nonlocal operators. In contrast to previous versions we do not assume harmonic functions to have a sign. The version of Harnack?s inequality given here generalizes Harnack?s classical result from 1887 to nonlocal situations. As a consequence we derive Hölder regularity estimates by an extension of Moser?s method. The inequality that we propose is equivalent to Harnack?s original formulation but seems to be new even for the Laplace operator.     

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.     

Harnack estimate for curvature flows depending on mean curvature

The maximum principle is applied to prove the Harnack estimate of curvature flows of hypersurfaces in Rn＋1,where the normal velocity is given by a smooth function f depending only on the mean curvature.By use of the estimate,some corollaries are obtained including the integral Harnack inequality.In particular,the conditions are given with which the solution to the flows is a translation soliton or an expanding soliton.     

A modified log-Harnack inequality and asymptotically strong Feller property

We introduce a new Harnack type inequality, which is a modification of the log-Harnack inequality established by R?ckner and Wang and prove that it implies the asymptotically strong Feller property (ASF). This inequality generalizes the criterion for ASF introduced by Hairer and Mattingly. As an example, we show by an asymptotic coupling that the 2D stochastic Navier-Stokes equation driven by highly degenerate but essentially elliptic noise satisfies our modified log-Harnack inequality.     

On the well‐posedness of the Euler equations in the Triebel‐Lizorkin spaces

We prove local‐in‐time unique existence and a blowup criterion for solutions in the Triebel‐Lizorkin space for the Euler equations of inviscid incompressible fluid flows in ?ⁿ, n ≥ 2. As a corollary we obtain global persistence of the initial regularity characterized by the Triebel‐Lizorkin spaces for the solutions of two‐dimensional Euler equations. To prove the results, we establish the logarithmic inequality of the Beale‐Kato‐Majda type, the Moser type of inequality, as well as the commutator estimate in the Triebel‐Lizorkin spaces. The key methods of proof used are the Littlewood‐Paley decomposition and the paradifferential calculus by J. M. Bony. © 2002 John Wiley & Sons, Inc.