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.     

A Harnack inequality for weighted degenerate parabolic equations

In this paper we generalize the recent result of DiBenedetto, Gianazza, Vespri on the Harnack inequality for degenerate parabolic equations to the case of a weighted p-Laplacian type operator in the spatial part. The weight is assumed to belong to the suitable Muckenhoupt class.     

Hölder regularity of solutions of degenerate elliptic and parabolic equations

We establish the Alexandroff-Bakelman-Pucci estimate, the Harnack inequality, and the Hölder continuity of solutions to degenerate parabolic equations of the non-divergence form

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Local behaviour of solutions to doubly nonlinear parabolic equations

We give a relatively simple and transparent proof for Harnack’s inequality for certain degenerate doubly nonlinear parabolic equations. We consider the case where the Lebesgue measure is replaced with a doubling Borel measure which supports a Poincaré inequality.     

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.     

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 inequality for a class of second-order degenerate elliptic equations

A second-order degenerate elliptic equation in divergence form with a partially Muckenhoupt weight is studied. In a model case, the domain is divided by a hyperplane into two parts, and in each part the weight is a power function of |x| with the exponent less than the dimension of the space in absolute value. It is well known that solutions of such equations are H?lder continuous, whereas the classical Harnack inequality is missing. In this paper, we formulate and prove the Harnack inequality corresponding to the second-order degenerate elliptic equation under consideration.     

The Harnack inequality for second-order parabolic equations with divergence-free drifts of low regularity

We establish the Harnack inequality for advection-diffusion equations with divergence-free drifts by adapting the classical Moser technique to parabolic equations with drifts with regularity lower than the scale invariant spaces.     

On Harnack’s Inequality for the Linearized Parabolic Monge-Ampère Equation

It is shown that the parabolic Harnack property stands as an intrinsic feature of the Monge-Ampère quasi-metric structure by proving Harnack’s inequality for non-negative solutions to the linearized parabolic Monge-Ampère equation under minimal geometric assumptions.     

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.     

Harnack's inequality for parabolic nonlocal equations

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.     

A Harnack inequality for the parabolic Allen–Cahn equation

We prove a differential Harnack inequality for the solution of the parabolic Allen–Cahn equation \( \frac{\partial f}{\partial t}=\triangle f-(f^3-f)\) on a closed n-dimensional manifold. As a corollary, we find a classical Harnack inequality. We also formally compare the standing wave solution to a gradient estimate of Modica from the 1980s for the elliptic equation.     

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.     

Widder’s Representation Theorem for Symmetric Local Dirichlet Spaces

In classical PDE theory, Widder’s theorem gives a representation for non-negative solutions of the heat equation on \(\mathbb{R }^n\) . We show that an analogous theorem holds for local weak solutions of the canonical “heat equation” on a symmetric local Dirichlet space satisfying a local parabolic Harnack inequality.     

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.     

On A parabolic equation with a singular lower order term

We obtain the existence of the weak Green's functions of parabolic equations with lower order coefficients in the so called parabolic Kato class which is being proposed as a natural generalization of the Kato class in the study of elliptic equations. As a consequence we are able to prove the existence of solutions of some initial boundary value problems. Moreover, based on a lower and an upper bound of the Green's function, we prove a Harnack inequality for the non-negative weak solutions.

     

Harnack inequalities and Gaussian estimates for a class of hypoelliptic operators

We prove a global Harnack inequality for a class of degenerate evolution operators by repeatedly using an invariant local Harnack inequality. As a consequence we obtain an accurate Gaussian lower bound for the fundamental solution for some meaningful families of degenerate operators.

     

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 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.