J. D. Fernandes, J. Groisman, S. T. Melo
We prove a Harnack inequality for a class of two-weight degenerate elliptic operators. The metric distance is induced by continuous Grushin-type vector fields. It is not know whether there exist cutoffs fitting the metric balls. This obstacle is bypassed by means of a covering argument that allows the use of rectangles in the Moser iteration.
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In this paper we firstly derive the weak elliptic Harnack inequality from the generalized capacity condition, the tail estimate of jump measure and the Poincaré inequality, for any regular Dirichlet form without killing part on a measure metric space, by using the lemma of growth and the John-Nirenberg inequality. We secondly show several equivalent characterizations of the weak elliptic Harnack inequality for any (not necessarily regular) Dirichlet form. We thirdly present some consequences of the weak elliptic Harnack inequality.
In this paper we establish a scale invariant Harnack inequality for the fractional powers of parabolic operators $(\partial_t - \mathscr{L})^s$, $0<s<1$, where $\mathscr{L}$ is the infinitesimal generator of a class of symmetric semigroups. As a by-product we also obtain a similar result for the nonlocal operators $(-\mathscr{L})^s$. Our focus is on non-Euclidean situations.
Elliptic Harnack inequality and Poincaré inequality for $p$-energies on metric measure spaces
本文在体积加倍度量测度空间上，证明椭圆Harnack不等式和双侧容量界蕴含Poincaré不等式，并建立与截断Sobolev不等式的等价性。
The fractional nonlocal linearized Monge--Ampère equation is introduced. A Harnack inequality for nonnegative solutions to the Poisson problem on Monge--Ampère sections is proved.
The main result of this paper is a nonlocal version of Harnack's inequality for a class of parabolic nonlocal equations. We additionally establish a weak Harnack inequality as well as local boundedness of solutions. None of the results require the solution to be globally positive.
We prove a Harnack inequality for functions which, at points of large gradient, are solutions of elliptic equations with unbounded drift.
We establish a version of the Harnack inequality for the Jordan-Kinderlehrer-Otto scheme of the heat equation on the flat torus.
We give a direct analytic proof of the classical Boundary Harnack inequality for solutions to linear uniformly elliptic equations in either divergence or non-divergence form.
This paper will develop a Li-Yau-Hamilton type differential Harnack estimate for positive solutions to the Newell-Whitehead equation on $\mathbb{R}^n$. We then use our LYH-differential Harnack inequality to prove several properties about positive solutions to the equation, including deriving a classical Harnack inequality, and characterizing standing solutions and traveling wave solutions.
We establish a Harnack inequality of fractional Laplace equations without imposing sign condition on the coefficient of zero order term via the Moser's iteration and John-Nirenberg inequality.
We give a sup $\times$ inf inequality for an elliptic equation.
In 1957, De Giorgi [3] proved the Hölder continuity for elliptic equations in divergence form and Moser [7] gave a new proof in 1960. Next year, Moser [8] obtained the Harnack inequality. In this note, we point out that the Harnack inequality was hidden in [3].
We prove Harnack's inequality for bounded weak solutions to quasilinear second order elliptic equations with generalized Orlicz growth conditions. Our approach covers new cases of variable exponent and (p,q) growth conditions.
This paper is concerned with nonlinear elliptic equations in nondivergence form where the operator has a first order drift term which is not Lipschitz continuous. Under this condition the equations are nonhomogeneous and nonnegative solutions do not satisfy the classical Harnack's inequality. This paper presents a new type of generalization of the classical Harnack's inequality for such equations. As a corollary we obtain the optimal Harnack type of estimate for p(x)-harmonic functions which quantifies the strong minimum principle.
A Differential Harnack Inequality for the FitzHugh-Nagumo Equation
本文针对ℝⁿ上FitzHugh-Nagumo方程的正解建立了Li-Yau-Hamilton型微分Harnack估计，并借此证明了该方程正解的经典Harnack不等式及行波解速度下界等性质。
We prove an invariant Harnack's inequality for operators in non-divergence form structured on Heisenberg vector fields when the coefficient matrix is uniformly positive definite, continuous, and symplectic. The method consists in constructing appropriate barriers to obtain pointwise-to-measure estimates for supersolutions in small balls, and then invoking the axiomatic approach from [DGL08] to obtain Harnack's inequality.
Harnack inequality for doubly nonlinear mixed local and nonlocal parabolic equations
本文针对双重非线性混合局部与非局部抛物型方程，利用比较原理、局部有界性估计和正性扩张方法，建立了非负弱解的Harnack不等式。
This paper is concerned with semilinear equations in divergence form \[ \diver(A(x)Du) = f(u) \] where $f :\R \to [0,\infty)$ is nondecreasing. We prove a sharp Harnack type inequality for nonnegative solutions which is closely connected to the classical Keller-Osserman condition for the existence of entire solutions.
The weak Harnack inequality for $L^p$-viscosity supersolutions of fully nonlinear second-order uniformly parabolic partial differential equations with unbounded coefficients and inhomogeneous terms is proved. It is shown that Hölder continuity of $L^p$-viscosity solutions is derived from the weak Harnack inequality for $L^p$-viscosity supersolutions. The local maximum principle for $L^p$-viscosity subsolutions and the Harnack inequality for $L^p$-viscosity solutions are also obtained. Several further remarks are presented when equations have superlinear growth in the first space derivatives.
Integral Harnack inequality and expansion of positivity for fractional porous medium equation
本文在最优尾部条件下，证明了快速扩散分数多孔介质方程非负弱解的积分Harnack不等式与正性扩张性质。
For a general class of divergence type quasi-linear degenerate parabolic equations with measurable coeffcients and lower order terms from non-linear Kato-type classes, we prove local boundedness and continuity of solutions, and the intrinsic Harnack inequality for positive solutions.
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.
We prove that the elliptic Harnack inequality (on a manifold, graph, or suitably regular metric measure space) is stable under bounded perturbations, as well as rough isometries.
We consider a class of generalized nonlocal $p$-Laplacian equations. We find some proper structural conditions to establish a version of nonlocal Harnack inequalities of weak solutions to such nonlocal problems by using the expansion of positivity and energy estimates.
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 $κ\geq 0$. In the elliptic case, Wang and Zhang \cite{WZ} recently extended the results of \cite{Ca} 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 {\it viscosity solutions} to nonlinear uniformly {\it parabolic equations} in nondivergence form on $M$. The Harnack inequali…
We give some estimates of type sup $\times$ inf on Riemannian manifold of dimension 5.
The Parabolic Harnack Inequality on Weighted Riemannian Manifolds
基于Grigor'yan的方法，对一大类抛物型微分算子建立了加权黎曼流形上的抛物型Harnack不等式。
We establish weak Harnack inequalities for positive, weak supersolutions to certain doubly degenerate parabolic equations. The prototype of this kind of equations is $$\partial_tu-\operatorname{div}|u|^{m-1}|Du|^{p-2}Du=0,\quad p>2,\quad m+p>3.$$ Our proof is based on Caccioppoli inequalities, De Giorgi's estimates and Moser's iterative method.
Harnack inequality for nonlocal problems with non-standard growth
本文基于作者此前相关研究，证明了具有非标准增长的非局部问题的局部极小值与弱解的完整Harnack不等式，核心辅助结果为对应De Giorgi类函数的局部有界性及弱Harnack不等式。
We consider weak solutions to $$-Δ_pu+a(x,u)|\nabla u|^q=f(x,u),$$ with $p>1$, $q\geq\max\,\{p-1,1\}$. We exploit the Moser iteration technique to prove a Harnack comparison inequality for $C^1$ weak solutions. As a consequence we deduce a strong comparison principle.
We show that the quotient of two positive harmonic functions vanishing on the boundary of a $C^{k,α}$ domain is of class $C^{k,α}$ up to the boundary.
We prove the Harnack inequality for general nonlocal elliptic equations with zero order terms. As an application we prove the existence of the principal eigenvalue in general domains. Furthermore, we study the eigenvalue problem associated to the existence of self-similar solutions to the parabolic problem and provide estimates on the decay rate.
In this paper we prove an invariant Harnack inequality on Carnot-Carathéodory balls for fractional powers of sub-Laplacians in Carnot groups. The proof relies on an "abstract" formulation of a technique recently introduced by Caffarelli and Silvestre. In addition, we write explicitly the Poisson kernel for a class of degenerate subelliptic equations in product-type Carnot groups.
Nonlocal Harnack inequality in a disconnected region
本文在不连通区域中建立非局部方程弱解的Harnack不等式，比较解在不同连通分支上的值，并给出基于局部化极大值原理和Poisson核估计的两种证明方法。
We prove interior Harnack's inequalities for solutions of fractional nonlocal equations. Our examples include fractional powers of divergence form elliptic operators with potentials, operators arising in classical orthogonal expansions and the radial Laplacian. To get the results we use an analytic method based on a generalization of the Caffarelli--Silvestre extension problem, the Harnack's inequality for degenerate Schrödinger operators proved by C. E. Gutiérrez, and a transference method. In this manner we apply local PDE techniques to nonlocal operators. On the way a maximum principle and …
Harnack inequality for degenerate fully nonlinear parabolic equations
该研究将p>2的p-抛物方程推广至非散度形式的退化完全非线性抛物方程，证明了非负解的内蕴Harnack不等式与非负上解的弱Harnack不等式，给出了已有散度形式相关结果的非散度形式对应结论。
We present some results on a fully nonlinear version of the Yamabe problem and a Harnack type inequality for general conformally invariant fully nonlinear second order elliptic equations.
Mixed Local-Nonlocal Parabolic Equations in Heisenberg Group : Harnack Inequality with an Optimal Tail
本文通过正性扩展方法和聚类引理，证明了海森堡群上线性混合局部-非局部抛物方程具有最优尾部的哈纳克不等式，避免了Moser迭代和对数估计。
We study unbounded weak supersolutions of elliptic partial differential equations with generalized Orlicz (Musielak--Orlicz) growth. We show that they satisfy the weak Harnack inequality with optimal exponent provided that they belong to a suitable Lebesgue or Sobolev space. Furthermore, we establish the sharpness of our central assumptions.
We discuss the problem how "bad" may be lower-order coefficients in elliptic and parabolic second order equations to ensure some qualitative properties of solution such as strong maximum principle, Harnack's inequality, Liouville's theorem. The answers are given in terms of the Lebesgue spaces and the Morrey spaces.
In this paper, we establish a scale invariant Harnack inequality for some inhomogeneous parabolic equations in a suitable intrinsic geometry dictated by the nonlinearity. The class of equations that we consider correspond to the parabolic counterpart of the equations studied by Julin in [10] where a generalized Harnack inequality was obtained which quantifies the strong maximum principle. Our version of parabolic Harnack (see Theorem 1.2) when restricted to the elliptic case is however quite different from that in [10]. The key new feature of this work is an appropriate modification of the sta…
Geometric aspects of the Harnack Inequality for a nonlocal heat equation
本文揭示分数热方程正解的精确双边 Harnack 界与高维圆形几何的关联，并推广了前人结果。
We prove Harnack inequality and local regularity results for weak solutions of a quasilinear degenerate equation in divergence form under natural growth conditions. The degeneracy is given by a suitable power of a strong $A_\infty$ weight. Regularity results are achieved under minimal assumptions on the coefficients and, as an application, we prove $C^{1,α}$ local estimates for solutions of a degenerate equation in non divergence form.
In this paper we give both an historical and technical overview of the theory of Harnack inequalities for nonlinear parabolic equations in divergence form. We start reviewing the elliptic case with some of its variants and geometrical consequences. The linear parabolic Harnack inequality of Moser is discussed extensively, together with its link to two-sided kernel estimates and to the Li-Yau differential Harnack inequality. Then we overview the more recent developements of the theory for nonlinear degenerate/singular equations, highlighting the differences with the quadratic case and introduci…
The weak Harnack inequality and the rigidity of the anisotropic Trudinger's equation
该研究针对各向异性特鲁丁格方程非负弱超解证明了局部弱哈纳克不等式，给出了绕过克里洛夫-萨丰诺夫覆盖论证的哈纳克不等式证明，并确定了巴伦布拉特型剖面。
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 our assumptions are robust to time changes of the corresponding diffusions. In particular, we do not assume volume doubling property for the symmetric measure.
We study local regularity properties for solutions of linear, non-uniformly elliptic equations. Assuming certain integrability conditions on the coefficient field, we prove local boundedness and Harnack inequality. The assumed integrability assumptions are essentially sharp and improve upon classical results by Trudinger [ARMA 1971]. We then apply the deterministic regularity results to the corrector equation in stochastic homogenization and establish sublinearity of the corrector.
In this paper, we study fully non-linear elliptic equations in non-divergence form which can be degenerate when "the gradient is small". Typical examples are either equations involving the $m$-Laplace operator or Bellman-Isaacs equations from stochastic control problems. We establish an Alexandroff-Bakelman-Pucci estimate and we prove a Harnack inequality for viscosity solutions of such degenerate elliptic equations.
We consider weak solutions of degenerate second order partial differential equations of Kolmogorov-Fokker-Planck type with measurable coefficients in divergence form. We give a geometric statement of the Harnack inequality recently proven by Golse, Imbert, Mouhot and Vasseur. As a corollary we obtain a strong maximum principle.
In the context of a metric measure Dirichlet space satisfying volume doubling and Poincaré inequality, we prove the parabolic Harnack inequality for weak solutions of the heat equation associated with local nonsymmetric bilinear forms. In particular, we show that these weak solutions are locally bounded.
In [10], R. Hamilton established a differential Harnack inequality for solutions to the Ricci flow with nonnegative curvature operator. We show that this inequality holds under the weaker condition that M x R^2 has nonnegative isotropic curvature.
In this paper we consider a class of prescribing curvature type equations on half Euclidean balls. Under suitable assumptions on the scalar curvature function and boundary mean curvature function we prove a min-max type inequality and the corresponding energy estimates.
This article proves a weak Harnack inequality with a tail term for sign changing supersolutions of a mixed local and nonlocal parabolic equation. Our argument is purely analytic. It is based on energy estimates and the Moser iteration technique. Instead of the parabolic John-Nirenberg lemma, we adopt a lemma of Bombieri to the mixed local and nonlocal parabolic case. To this end, we prove an appropriate reverse Hölder inequality and a logarithmic estimate for weak supersolutions.
ABP estimate and Harnack inequality for a class of degenerate fully nonlinear pseudo-$p$-Laplacian equations
本文研究了一类退化完全非线性伪p-拉普拉斯方程的ABP估计与Harnack不等式，采用滑动抛物面方法与各向异性函数相结合的方法进行证明。
Elliptic Harnack's inequality for a singular nonlinear parabolic equation in non-divergence form
本文证明了一类推广标准p-Laplace方程及随机博弈归一化版本的一般抛物方程的椭圆型Harnack不等式，无需内在等待时间，且两侧时间层相同。
We extend some classical results dealing with boundary Harnack inequatilities to a class of quasilinear elliptic equations and derive some new estimates for solutions of such equations with an isolated singularity on the boundary of a domain.
We consider second-order linear parabolic operators in non-divergence form that are intrinsically defined on Riemannian manifolds. In the elliptic case, Cabré proved a global Krylov-Safonov Harnack inequality under the assumption that the sectional curvature of the underlying manifold is nonnegative. Later, Kim improved Cabré's result by replacing the curvature condition by a certain condition on the distance function. Assuming essentially the same condition introduced by Kim, we establish Krylov-Safonov Harnack inequality for nonnegative solutions of the non-divergent parabolic equation. This…
In this paper, we obtain boundary Harnack estimates and comparison theorem for nonnegative solutions to the linearized Monge-Ampère equations under natural assumptions on the domain, Monge-Ampère measures and boundary data. Our results are boundary versions of Caffarelli and Gutiérrez's interior Harnack inequality for the linearized Monge-Ampère equations. As an application, we obtain sharp upper bound and global $L^p$-integrability for the Green's function of the linearized Monge-Ampère operator.
This paper deals with two separate but related results. First we consider weak solutions to a parabolic operator with Hörmander vector fields. Adapting the iteration scheme of Jürgen Moser for elliptic and parabolic equations in $\mathbb{R}^n$ we show a parabolic Harnack inequality. Then, after proving the Harnack inequality for weak solutions to equations of the form $u_t = \sum X_i (a_{ij} X_j u)$ we use this to show Hölder continuity. We assume the coefficients are bounded and elliptic. The iteration scheme is a tool that may be adapted to many settings and we extend this to nonlinear parab…
Backward Harnack inequality and Hamilton estimates for heat type equations
本文基于Hamilton梯度估计，在紧致流形上建立了无需额外限制的正解时间后向Harnack不等式，实现了时空点间双向比较，拓展了经典Harnack不等式的应用前景。
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