Harnack Inequality Research Articles

Gradient Estimates for a Weighted Γ-nonlinear Parabolic Equation Coupled with a Super Perelman-Ricci Flow and Implications

This article studies a nonlinear parabolic equation on a complete weighted manifold where the metric and potential evolve under a super Perelman-Ricci flow. It derives elliptic gradient estimates of local and global types for the positive solutions and exploits some of their implications notably to a general Liouville type theorem, parabolic Harnack inequalities and classes of Hamilton type dimension-free gradient estimates. Some examples and special cases are discussed for illustration.

Potential theory for a class of strongly degenerate parabolic operators of Kolmogorov type with rough coefficients

In this paper we develop a potential theory for strongly degenerate parabolic operators of the formL:=∇X⋅(A(X,Y,t)∇X)+X⋅∇Y−∂t, in unbounded domains of the formΩ={(X,Y,t)=(x,xm,y,ym,t)∈Rm−1×R×Rm−1×R×R|xm>ψ(x,y,ym,t)}, where ψ is assumed to satisfy a uniform Lipschitz condition adapted to the dilation structure and the (non-Euclidean) Lie group underlying the operator L. Concerning A=A(X,Y,t) we assume that A is bounded, measurable, symmetric and uniformly elliptic (as a matrix in Rm). Beyond the solvability of the Dirichlet problem and other fundamental properties our results include scale and translation invariant boundary comparison principles, boundary Harnack inequalities and doubling properties of associated parabolic measures. All of our estimates are translation- and scale-invariant with constants only depending on the constants defining the boundedness and ellipticity of A and the Lipschitz constant of ψ. Our results represent a version, for operators of Kolmogorov type with bounded, measurable coefficients, of the by now classical results of Fabes and Safonov, and several others, concerning boundary estimates for uniformly parabolic equations in (time-dependent) Lipschitz type domains.

Global Gaussian Estimates for the Heat Kernel of Homogeneous Sums of Squares

Let ${\mathscr{H}}={\sum }_{j=1}^{m}{X_{j}^{2}}-\partial _{t}$ be a heat-type operator in $\mathbb {R}^{n+1}$ , where X = {X1,…,Xm} is a system of smooth Hörmander’s vector fields in $\mathbb {R}^{n}$ , and every Xj is homogeneous of degree 1 with respect to a family of non-isotropic dilations in $\mathbb {R}^{n}$ , while no underlying group structure is assumed. In this paper we prove global (in space and time) upper and lower Gaussian estimates for the heat kernel Γ(t,x;s,y) of ${\mathscr{H}}$ , in terms of the Carnot-Carathéodory distance induced by X on $\mathbb {R}^{n}$ , as well as global upper Gaussian estimates for the t- or X-derivatives of any order of Γ. From the Gaussian bounds we derive the unique solvability of the Cauchy problem for a possibly unbounded continuous initial datum satisfying exponential growth at infinity. Also, we study the solvability of the ${\mathscr{H}}$ -Dirichlet problem on an arbitrary bounded domain. Finally, we establish a global scale-invariant Harnack inequality for non-negative solutions of ${\mathscr{H}}u=0$ .

Elliptic Theory for Sets with Higher Co-dimensional Boundaries

Many geometric and analytic properties of sets hinge on the properties of elliptic measure, notoriously missing for sets of higher co-dimension. The aim of this manuscript is to develop a version of elliptic theory, associated to a linear PDE, which ultimately yields a notion analogous to that of the harmonic measure, for sets of codimension higher than 1. To this end, we turn to degenerate elliptic equations. Let Γ ⊂ R n \Gamma \subset \mathbb {R}^n be an Ahlfors regular set of dimension d > n − 1 d>n-1 (not necessarily integer) and Ω = R n ∖ Γ \Omega = \mathbb {R}^n \setminus \Gamma . Let L = − div ⁡ A ∇ L = - \operatorname {div} A\nabla be a degenerate elliptic operator with measurable coefficients such that the ellipticity constants of the matrix A A are bounded from above and below by a multiple of dist ⁡ ( ⋅ , Γ ) d + 1 − n \operatorname {dist}(\cdot , \Gamma )^{d+1-n} . We define weak solutions; prove trace and extension theorems in suitable weighted Sobolev spaces; establish the maximum principle, De Giorgi-Nash-Moser estimates, the Harnack inequality, the Hölder continuity of solutions (inside and at the boundary). We define the Green function and provide the basic set of pointwise and/or L p L^p estimates for the Green function and for its gradient. With this at hand, we define harmonic measure associated to L L , establish its doubling property, non-degeneracy, change-of-the-pole formulas, and, finally, the comparison principle for local solutions. In another article to appear, we will prove that when Γ \Gamma is the graph of a Lipschitz function with small Lipschitz constant, we can find an elliptic operator L L for which the harmonic measure given here is absolutely continuous with respect to the d d -Hausdorff measure on Γ \Gamma and vice versa. It thus extends Dahlberg’s theorem to some sets of codimension higher than 1.

Non-divergence operators structured on homogeneous Hörmander vector fields: heat kernels and global Gaussian bounds

Let $X_{1},...,X_{m}$ be a family of real smooth vector fields defined in $\mathbb{R}^{n}$, $1$-homogeneous with respect to a nonisotropic family of dilations and satisfying Hörmander's rank condition at $0$ (and therefore at every point of $\mathbb{R}^{n}$). The vector fields are not assumed to be translation invariant with respect to any Lie group structure. Let us consider the nonvariational evolution operator \[ \mathcal{H}:=\sum_{i,j=1}^{m}a_{i,j}(t,x)X_{i}X_{j}-\partial_{t} , \] where $(a_{i,j}(t,x))_{i,j=1}^{m}$ is a symmetric uniformly positive $m\times m$ matrix and the entries $a_{ij}$ are bounded Hölder continuous functions on $\mathbb{R}^{1+n}$, with respect to the ``parabolic'' distance induced by the vector fields. We prove the existence of a global heat kernel $\Gamma(\cdot;s,y)\in C_{X,\mathrm{loc}}^{2,\alpha}(\mathbb{R}^{1+n} \setminus\{(s,y)\})$ for $\mathcal{H}$, such that $\Gamma$ satisfies two-sided Gaussian bounds and $\partial_{t}\Gamma, X_{i}\Gamma,X_{i}X_{j}\Gamma$ satisfy upper Gaussian bounds on every strip $[0,T]\times\mathbb{R}^{n}$. We also prove a scale-invariant parabolic Harnack inequality for $\mathcal{H} $, and a standard Harnack inequality for the corresponding stationary operator \[ \mathcal{L}:=\sum_{i,j=1}^{m}a_{i,j}(x)X_{i}X_{j} \] with Hölder continuous coefficients.

Continuity and Harnack inequality for solutions of the weight parabolic equation with p-Laplasian contained by the singular lower-order term

Problems related to the study of the properties of solutions of differential equations in partial derivatives have attracted the attention of many authors over the past few decades. Many works in recent years have been devoted to the study of the $ p \,-$Laplacian equation, which simulates non-Newtonian elastic filtration and describes many physical processes. When $ p = 1 $ it has a geometric meaning (the equation of the minimum surface) and its study is fundamentally different from the theory for $ p> 1. $ In the case of $ p <{2} $ possible complete attenuation of the solution in a finite time, and when $ p> {2} $ we have a compact solution carrier. This effect is observed on the example of known ``Barenblatt's solutions''. It seems that the question of Harnack's inequality for the solutions of parabolic equations with $ p \,-$Laplacian, which contain a singular junction, still remains open. In this paper, we made the first attempt to study the Harnack inequality for solutions of degenerate equations with $ p \,-$Laplacian perturbed singularity of the lower-order terms. We extend the result to the case of weight parabolic equations with $ p \,-$Laplacian. The proof is based on the corresponding modifications of the de Giorgi iterative technique after the adaptation of the Kilpel\"{a}inen -- Mal\'{y} technique to parabolic equations in combination with the ideas of Liskevich and Skrypnik. This article is a continuation of our study of that devoted to the qualitative properties of the solutions of the parabolic equation with $ p \,-$Laplacian. We summarize the known results in the case of an equation that contains the weights of the Muckenhoupt classes and the singular lower-order terms. Using the weight nonlinear parabolic potential of Wolf of the right-hand side of the equation, we prove a Lemma of De Giorgi type, continuity of solutions, Harnack inequality, show the performance of the Expansion of Positivity.

Nonlocal Harnack inequalities for nonlocal Schrödinger operators with A1-Muckenhoupt potentials

In this paper, by applying the De Giorgi-Nash-Moser theory we obtain nonlocal Harnack inequalities for (locally nonnegative in Ω) weak solutions of the nonlocal Schrödinger equations{LKu+Vu=0 in Ω,u=g in Rn∖Ω where V=V+−V− with V−∈Lloc1(Rn) and V+∈Llocq(Rn) (q>n2s>1, 0<s<1) is a potential such that (V−,V+b,k) belongs to the (A1,A1)-Muckenhoupt class and V+b,k is in the A1-Muckenhoupt class for all k∈N (here, V+b,k:=V+max⁡{b,1/k}/b for a nonnegative bounded function b on Rn with V+/b∈Llocq(Rn)), LK is an integro-differential operator, Ω⊂Rn is a bounded domain with Lipschitz boundary and g∈Hs(Rn). Interestingly, this result implies the classical Harnack inequalities for globally nonnegative weak solutions of the equations. In addition, we obtain nonlocal weak Harnack inequalities of its weak supersolutions. In particular, we note that all the above results are still working for any nonnegative potential in Llocq(Rn) (q>n2s>1, 0<s<1).

Fractional elliptic equations in nondivergence form: definition, applications and Harnack inequality

We define the fractional powers Ls=(−aij(x)∂ij)s, 0<s<1, of nondivergence form elliptic operators L=−aij(x)∂ij in bounded domains Ω⊂Rn, under minimal regularity assumptions on the coefficients aij(x) and on the boundary ∂Ω. We show that these fractional operators appear in several applications such as fractional Monge–Ampère equations, elasticity, and finance. The solution u to the nonlocal Poisson problem{(−aij(x)∂ij)su=finΩu=0on∂Ω is characterized by a local degenerate/singular extension problem. We develop the method of sliding paraboloids in the Monge–Ampère geometry and prove the interior Harnack inequality and Hölder estimates for solutions to the extension problem when the coefficients aij(x) are bounded, measurable functions. This in turn implies the interior Harnack inequality and Hölder estimates for solutions u to the fractional problem.

Wolff Potential Estimates for Supersolutions of Equations with Generalized Orlicz Growth

In this paper, we establish pointwise estimates for supersolutions of quasilinear elliptic equations with structural conditions involving a generalized Orlicz growth in terms of a Wolff type potential. As a consequence, under an extra assumption, we obtain that the supersolutions satisfy a Harnack inequality and local Hölder continuity.

Scale invariant regularity estimates for second order elliptic equations with lower order coefficients in optimal spaces

We show local and global scale invariant regularity estimates for subsolutions and supersolutions to the equation −div(A∇u+bu)+c∇u+du=−divf+g, assuming that A is elliptic and bounded. In the setting of Lorentz spaces, under the assumptions b,f∈Ln,1, d,g∈Ln2,1 and c∈Ln,q for q≤∞, we show that, with the surprising exception of the reverse Moser estimate, scale invariant estimates with “good” constants (that is, depending only on the norms of the coefficients) do not hold in general. On the other hand, assuming a necessary smallness condition on b,d or c,d, we show a maximum principle and Moser's estimate for subsolutions with “good” constants. We also show the reverse Moser estimate for nonnegative supersolutions with “good” constants, under no smallness assumptions when q<∞, leading to the Harnack inequality for nonnegative solutions and local continuity of solutions. Finally, we show that, in the setting of Lorentz spaces, our assumptions are the sharp ones to guarantee these estimates.

Gradient estimates for a nonlinear elliptic equation under geometric flow

We investigate the parabolic Allen-Cahn equation ∂u∂t=Δu+(1−u2)u under general geometric flow on complete noncompact manifolds. We first prove a local gradient estimate for the positive solution u to this equation, then we obtain a Harnack inequality as an application.

On the maximum principles and the quantitative version of the Hopf lemma for uniformly elliptic integro-differential operators

In the present paper we prove estimates on subsolutions of the equation − A v + c ( x ) v = 0 , x ∈ D , where D ⊂ R d is a domain (i.e. an open and connected set) and A is an integro-differential operator of the Waldenfels type, whose differential part satisfies the uniform ellipticity condition on compact sets. In general, the coefficients of the operator need not be continuous but only bounded and Borel measurable. Some of our results may be considered “quantitative” versions of the Hopf lemma, as they provide the lower bound on the outward normal directional derivative at the maximum point of a subsolution in terms of its value at the point. We shall also show lower bounds on the subsolution around its maximum point by the principal eigenfunction associated with A and the domain. Additional results, among them the weak and strong maximum principles, the weak Harnack inequality are also proven.

A Zvonkin's transformation for stochastic differential equations with singular drift and applications

In this paper, by establishing the localized L p - L q estimate and Sobolev estimates for parabolic partial differential equations with a singular first order term and a Lipschitz first order term, a new Zvonkin-type transformation is given for stochastic differential equations with singular and Lipschitz drifts. The associated Krylov's estimate is established. As applications, dimension-free Harnack inequalities are established for stochastic equations with Hölder continuous diffusion coefficient and singular drift term without regularity assumption.

Gradient Estimates and Harnack Inequalities of a Nonlinear Heat Equation for the Finsler-Laplacian

Gradient Estimates and Harnack Inequalities of a Nonlinear Heat Equation for the Finsler-Laplacian

Stability analysis and optimal harvesting control of a cross-diffusion prey-predator system

In this paper, we consider a cross-diffusion prey-predator system with fear effect and prey refuge. The upper and lower bounds of the system are obtained by using priori estimates and Harnack Inequality. Then sufficient conditions for the local stability and global stability of the system are established. We obtain that the cross-diffusion coefficients can affect the stability of the original system, meanwhile the fear effect and prey refuge suppress the formation of Turing instability. By using the Leray-Schauder degree theory, we study the existence and nonexistence of the non-constant steady states. Moreover, we discuss the effects of the fear effect and prey refuge on the optimal harvesting. Finally, we obtain the optimal harvesting strategies under different fear effect values and prey refuge values, the different maximum sustainable yields (MSY) are correspondingly given. Numerical simulations are carried out to verify and illustrate these theoretical results.

Harnack inequalities for a class of heat flows with nonlinear reaction terms

A class of semilinear heat flows with general nonlinear reaction terms is considered on complete Riemannian manifolds with Ricci curvature bounded from below. Two types of (space-time and space only) gradient estimates are established for positive solutions to the flow, and the corresponding Harnack inequalities are obtained to allow for comparison of solutions. Some specific examples of the reaction term such as logarithmic reaction, Fisher-KPP and Allen-Cahn equations are discussed as applications of the estimates so derived. Referring to logarithmic nonlinearities, some discussions are made on Liouville type properties of bounded solutions.

Regularity properties for quasiminimizers of a (p, q)-Dirichlet integral

Using a variational approach we study interior regularity for quasiminimizers of a (p, q)-Dirichlet integral, as well as regularity results up to the boundary, in the setting of a metric space equipped with a doubling measure and supporting a Poincaré inequality. For the interior regularity, we use De Giorgi type conditions to show that quasiminimizers are locally Hölder continuous and they satisfy Harnack inequality, the strong maximum principle and Liouville’s Theorem. Furthermore, we give a pointwise estimate near a boundary point, as well as a sufficient condition for Hölder continuity and a Wiener type regularity condition for continuity up to the boundary. Finally, we consider (p, q)-minimizers and we give an estimate for their oscillation at boundary points.

Qualitative properties of singular solutions to fractional elliptic equations

In this paper, by the moving spheres method, Caffarelli-Silvestre extension formula and blow-up analysis, we study the local behaviour of nonnegative solutions to fractional elliptic equations \begin{align*} (-\Delta)^{\alpha} u =f(u),~~ x\in \Omega\backslash \Gamma, \end{align*} where $0&lt;\alpha &lt;1$, $\Omega = \mathbb {R}^{N}$ or $\Omega$ is a smooth bounded domain, $\Gamma$ is a singular subset of $\Omega$ with fractional capacity zero, $f(t)$ is locally bounded and positive for $t\in [0,\,\infty )$, and $f(t)/t^{({N+2\alpha })/({N-2\alpha })}$ is nonincreasing in $t$ for large $t$, rather than for every $t&gt;0$. Our main result is that the solutions satisfy the estimate \begin{align*} f(u(x))/ u(x)\leq C d(x,\Gamma)^{{-}2\alpha}. \end{align*} This estimate is new even for $\Gamma =\{0\}$. As applications, we derive the spherical Harnack inequality, asymptotic symmetry, cylindrical symmetry of the solutions.

On Potential Theory of Markov Processes with Jump Kernels Decaying at the Boundary

Motivated by some recent potential theoretic results on subordinate killed L\'evy processes in open subsets of the Euclidean space, we study processes in an open set $D\subset {\mathbb R}^d$ defined via Dirichlet forms with jump kernels of the form $J^D(x,y)=j(|x-y|)\mathcal{B}(x,y)$ and critical killing functions. Here $j(|x-y|)$ is the L\'evy density of an isotropic stable process (or more generally, a pure jump isotropic unimodal L\'evy process) in $\mathbb{R}^d$. The main novelty is that the term $\mathcal{B}(x,y)$ tends to 0 when $x$ or $y$ approach the boundary of $D$. Under some general assumptions on $\mathcal{B}(x,y)$, we construct the corresponding process and prove that non-negative harmonic functions of the process satisfy the Harnack inequality and Carleson's estimate. We give several examples of boundary terms satisfying those assumptions. The examples depend on four parameters, $\beta_1, \beta_2, \beta_3$, $\beta_4$, roughly governing the decay of the boundary term near the boundary of $D$. In the second part of this paper, we specialise to the case of the half-space $D=\mathbb{R}_+^d=\{x=(\widetilde{x},x_d):\, x_d>0\}$, the $\alpha$-stable kernel $j(|x-y|)=|x-y|^{-d-\alpha}$ and the killing function$\kappa(x)=c x_d^{-\alpha}$, $\alpha\in (0,2)$, where $c$ is a positive constant. Our main result in this part is a boundary Harnack principle which says that, for any $p>(\alpha-1)_+$, there are values of the parameters $\beta_1, \beta_2, \beta_3$, $\beta_4$, and the constant $c$ such that non-negative harmonic functions of the process must decay at the rate $x_d^p$ if they vanish near a portion of the boundary. We further show that there are values of the parameters $\beta_1, \beta_2, \beta_3$, $\beta_4$, for which the boundary Harnack principle fails despite the fact that Carleson's estimate is valid.

An interpolating Harnack inequality for nonlinear heat equation on a surface

An interpolating Harnack inequality for nonlinear heat equation on a surface