Harnack Inequality Research Articles

Harnack inequality and gradient estimate for G-SDEs with degenerate noise

In this paper, the Harnack inequality for G-SDEs with degenerate noise is derived by the method of coupling by change of measure. Moreover, for any bounded and continuous function f, the gradient estimate $$\left| {\nabla {{\bar P}_t}f} \right|\leqslant c(p,t){(\bar P_t{\left| f \right|^p})^{{1 \over p}}},\;\;\;\;\;\;p > 1,\;\;\;\;\;t> 0$$ is obtained for the associated nonlinear semigroup \({\bar P_t}\). As an application of the Harnack inequality, we prove the existence of the weak solution to degenerate G-SDEs under some integrable condition. Finally, an example is presented. All of the above results extend the existing ones in the linear expectation setting.

Growth theorems for metric spaces with applications to PDE

The paper is devoted to some extensions of the joint results by N. V. Krylov and the author on the Harnack inequalities for second order elliptic and parabolic equations in nondivergence form to metric spaces of homogeneous type. The main tools are special Landis-type growth theorems.

On the Hölder regularity of signed solutions to a doubly nonlinear equation

We establish the interior and boundary Hölder continuity of possibly sign-changing solutions to a class of doubly nonlinear parabolic equations whose prototype is∂t(|u|p−2u)−Δpu=0,p>1. The proof relies on the property of expansion of positivity and the method of intrinsic scaling, all of which are realized by De Giorgi's iteration. Our approach, while emphasizing the distinct roles of sub(super)-solutions, is flexible enough to obtain the Hölder regularity of solutions to initial-boundary value problems of Dirichlet type or Neumann type in a cylindrical domain, up to the parabolic boundary. In addition, based on the expansion of positivity, we are able to give an alternative proof of Harnack's inequality for non-negative solutions. Moreover, as a consequence of the interior estimates, we also obtain a Liouville-type result.

The fractional p-Laplacian evolution equation in $${\mathbb {R}}^N$$ in the sublinear case

We consider the natural time-dependent fractional p-Laplacian equation posed in the whole Euclidean space, with parameter $$1<p<2$$ and fractional exponent $$s\in (0,1)$$ . Rather standard theory shows that the Cauchy Problem for data in the Lebesgue $$L^q$$ spaces is well posed, and the solutions form a family of non-expansive semigroups with regularity and other interesting properties. The superlinear case $$p>2$$ has been dealt with in a recent paper. We study here the “fast” regime $$1<p<2$$ which is more complex. As main results, we construct the self-similar fundamental solution for every mass value M and any p in the subrange $$p_c=2N/(N+s)<p<2$$ , and we show that this is the precise range where they can exist. We also prove that general finite-mass solutions converge towards the fundamental solution having the same mass, and convergence holds in all $$L^q$$ spaces. Fine bounds in the form of global Harnack inequalities are obtained. Another main topic of the paper is the study of solutions having strong singularities. We find a type of singular solution called Very Singular Solution that exists for $$p_c<p<p_1$$ , where $$p_1$$ is a new critical number that we introduce, $$p_1\in (p_c,2)$$ . We extend this type of singular solutions to the “very fast range” $$1<p<p_c$$ . They represent examples of weak solutions having finite-time extinction in that lower p range. We briefly examine the situation in the limit case $$p=p_c$$ . Finally, we show that very singular solutions are related to fractional elliptic problems of the nonlinear eigenvalue type, of interest in their own right.

Hölder continuity of $\omega$-minimizers of functionals with generalized Orlicz growth

We show local H\"older continuity of quasiminimizers of functionals with non-standard (Musielak--Orlicz) growth. Compared with previous results, we cover more general minimizing functionals and need fewer assumptions. We prove Harnack's inequality and a Morrey type estimate for quasiminimizers. Combining this with Ekeland's variational principle, we obtain local H\"older continuity for $\omega$-minimizers.

Discrete versions of the Li-Yau gradient estimate

We study positive solutions to the heat equation on graphs. We prove variants of the Li-Yau gradient estimate and the differential Harnack inequality. For some graphs, we can show the estimates to be sharp. We establish new com- putation rules for differential operators on discrete spaces and introduce a re- laxation function that governs the time dependency in the differential Harnack estimate

On $$\varvec{C^{1/2,1}}$$, $$\varvec{C^{1,2}}$$, and $$\varvec{C^{0,0}}$$ estimates for linear parabolic operators

We show that weak solutions to parabolic equations in divergence form with zero Dirichlet boundary conditions are continuously differentiable up to the boundary when the leading coefficients have Dini mean oscillation and the lower order coefficients verify certain conditions. Similar results are obtained for non-divergence form parabolic operators and their adjoint operators. Under similar conditions, we also establish a Harnack inequality for nonnegative adjoint solutions, together with upper and lower Gaussian bounds for the global fundamental solution.

The Vázquez maximum principle and the Landis conjecture for elliptic PDE with unbounded coefficients

We develop a new, unified approach to the following two classical questions on elliptic PDE:•the strong maximum principle for equations with non-Lipschitz nonlinearities,•the at most exponential decay of solutions in the whole space or exterior domains. Our results apply to divergence and non-divergence operators with locally unbounded lower-order coefficients, in a number of situations where all previous results required bounded ingredients. Our approach, which allows for relatively simple and short proofs, is based on a (weak) Harnack inequality with optimal dependence of the constants in the lower-order terms of the equation and the size of the domain, which we establish.

The fractional obstacle problem with drift: Higher regularity of free boundaries

We study the higher regularity of free boundaries in obstacle problems for integro-differential operators with drift, like (−Δ)s+b⋅∇, in the subcritical regime s>12. Our main result states that once the free boundary is C1 then it is C∞, whenever s∉Q.In order to achieve this, we establish a fine boundary expansion for solutions to linear nonlocal equations with drift in terms of the powers of distance function. Quite interestingly, due to the drift term, the powers do not increase by natural numbers and the fact that s is irrational plays al important role. Such expansion still allows us to prove a higher order boundary Harnack inequality, where the regularity holds in the tangential directions only.

The Wiener criterion for elliptic equations with Orlicz growth

We study the boundary continuity of solutions to elliptic equations with Orlicz growth. We first formulate the Wiener criterion which characterizes a regular boundary point by a geometric quantity coming from capacities. Secondly, we develop an estimate for the modulus of continuity at a boundary point, under a geometric condition on operators. The proof relies on capturing the local properties of weak solutions involving the Wolff potential estimate and the weak Harnack inequality.

Qualitative Analysis of a Three-Species Reaction-Diffusion Model with Modified Leslie-Gower Scheme

The qualitative analysis of a three-species reaction-diffusion model with a modified Leslie-Gower scheme under the Neumann boundary condition is obtained. The existence and the stability of the constant solutions for the ODE system and PDE system are discussed, respectively. And then, the priori estimates of positive steady states are given by the maximum principle and Harnack inequality. Moreover, the nonexistence of nonconstant positive steady states is derived by using Poincaré inequality. Finally, the existence of nonconstant positive steady states is established based on the Leray-Schauder degree theory.

New boundary Harnack inequalities with right hand side

We prove new boundary Harnack inequalities in Lipschitz domains for equations with a right hand side. Our main result applies to non-divergence form operators with bounded measurable coefficients and to divergence form operators with continuous coefficients, whereas the right hand side is in Lq with q>n. Our approach is based on the scaling and comparison arguments of [13], and we show that all our assumptions are sharp.As a consequence of our results, we deduce the C1,α regularity of the free boundary in the fully nonlinear obstacle problem and the fully nonlinear thin obstacle problem.

Harnack's inequality for quasilinear elliptic equations with generalized Orlicz growth

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.&#x0D; For more information see https://ejde.math.txstate.edu/Volumes/2021/27/abstr.html

Hamilton inequality for unbounded Laplacians on graphs

In this paper, we study Hamilton inequality for unbounded Laplacians on weighted graphs satisfying CDE′(−K,∞) for some K≥0. This is a generalization of the main result in [8] for bounded Laplacians. For this purpose, we assume a non-degenerated measure and make some modifications to the solutions of the heat equation. As an application of Hamilton inequality, we derive a type of Harnack inequality which compares the heat of two different vertices at the same time.

Functional inequalities for the heat flow on time‐dependent metric measure spaces

We prove that synthetic lower Ricci bounds for metric measure spaces -- both in the sense of Bakry-\'Emery and in the sense of Lott-Sturm-Villani -- can be characterized by various functional inequalities including local Poincar\'e inequalities, local logarithmic Sobolev inequalities, dimension independent Harnack inequality, and logarithmic Harnack inequality. More generally, these equivalences will be proven in the setting of time-dependent metric measure spaces and will provide a characterization of super-Ricci flows of metric measure spaces.

Remarks on parabolic De Giorgi classes

We make several remarks concerning properties of functions in parabolic De Giorgi classes of order p. There are new perspectives including a novel mechanism of propagating positivity in measure, the reservation of membership under convex composition, and a logarithmic type estimate. Based on them, we are able to give new proofs of known properties. In particular, we prove local boundedness and local Hölder continuity of these functions via Moser’s ideas, thus avoiding De Giorgi’s heavy machinery. We also seize this opportunity to give a transparent proof of a weak Harnack inequality for nonnegative members of some super-class of De Giorgi, without any covering argument.

On the Hölder regularity for solutions of integro-differential equations like the anisotropic fractional Laplacian

In this paper we study integro-differential equations like the anisotropic fractional Laplacian. As in Silvestre (Indiana Univ Math J 55:1155–1174, 2006), we adapt the De Giorgi technique to achieve the \(C^{\gamma }\)-regularity for solutions of class \(C^{2}\) and use the geometry found in Caffarelli et al. (Math Ann 360(3–4): 681–714, 2014) to get an ABP estimate, a Harnack inequality and the interior \(C^{1, \gamma }\) regularity for viscosity solutions.

Parabolic Harnack Inequality Implies the Existence of Jump Kernel

We prove that the parabolic Harnack inequality implies the existence of jump kernel for symmetric pure jump process. This allows us to remove a technical assumption on the jumping measure in the recent characterization of the parabolic Harnack inequality for pure jump processes by Chen, Kumagai and Wang. The key ingredients of our proof are the Lévy system formula and estimates on the heat kernel.

Weak Harnack inequalities for eigenvalues and constant rank theorems

We consider convex solutions of nonlinear elliptic equations which satisfy the structure condition of Bian and Guan. We prove a weak Harnack inequality for the eigenvalues of the Hessian of these solutions. This can be viewed as a quantitative version of the constant rank theorem.

Regularity and a Liouville theorem for a class of boundary-degenerate second order equations

We study a class of second-order boundary-degenerate elliptic equations in two dimensions with minimal regularity assumptions. We prove a maximum principle and a Harnack inequality at the degenerate boundary and, assuming local boundedness, continuity. For globally defined non-negative solutions we provide strong constraints on behavior at infinity, and prove a Liouville-type theorem for entire solutions on the closed half-plane. The class of PDE in question includes many from mathematical finance, Keldysh- and Tricomi-type PDE, and the 2nd order reduction of the fully non-linear 4th order Abreu equation from Kähler geometry. We present some possible future research directions.