Harnack type estimates and Hölder continuity for non-negative solutions to certain sub-critically singular parabolic partial differential equations

DiBenedetto, Emmanuele; Gianazza, Ugo; Vespri, Vincenzo

doi:10.1007/s00229-009-0317-9

Cited by 38 publications

(54 citation statements)

References 8 publications

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“…Let p as in (1.5) be given and suppose that u is a nonnegative and continuous weak solution to (1.1) in O, (x 0 ,t 0 ) ∈ O, and assume that u(x 0 ,t 0 ) > 0. The following result has been proved in [DBGV1]. There are positive constants c i ≡ c i (n, p, ν, L), i ∈ {1, 2, 3}, such that if…”

Section: Introduction and Resultsmentioning

confidence: 99%

A boundary Harnack inequality for singular equations of 𝑝-parabolic type

Kuusi

Mingione

Nyström

2014

Proc. Amer. Math. Soc.

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Abstract. We prove a boundary Harnack type inequality for non-negative solutions to singular equations of p-parabolic type, 2n/(n + 1) < p < 2, in time-independent cylinder whose base is C 1,1 -regular. Simple examples show, using the corresponding estimates valid for the heat equation as a point of reference, that this type of inequalities can not, in general, be expected to hold in the degenerate case (2 < p < ∞).2000 Mathematics Subject Classification.

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Section: Introduction and Resultsmentioning

confidence: 99%

A boundary Harnack inequality for singular equations of 𝑝-parabolic type

Kuusi

Mingione

Nyström

2014

Proc. Amer. Math. Soc.

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“…In [6] a detailed discussion will be given to show that the range of p in (1.5) is optimal for the Harnack estimate (1.7)-(1.8) to hold. Indeed, for p in the subcritical range 1 < p ≤ p * , explicit counterexamples are provided which fail to satisfy the Harnack inequality in any one of the forward, backward, or elliptic form.…”

Section: Novelty and Significancementioning

confidence: 99%

Intrinsic Harnack inequalities for quasi-linear singular parabolic partial differential equations

DiBenedetto¹,

Gianazza²,

Vespri³

2007

Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur.

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ABSTRACT. -Intrinsic Harnack estimates for non-negative solutions of singular, quasi-linear, parabolic equations are established, including the prototype p-Laplacian equation (1.4) below. For p in the supercritical range 2N/(N + 1) < p < 2, the Harnack inequality is shown to hold in a parabolic form, both forward and backward in time, and in an elliptic form at fixed time. These estimates fail for the heat equation (p → 2). It is shown by counterexamples that they fail for p in the subcritical range 1 < p ≤ 2N/(N + 1). Thus the indicated supercritical range is optimal for a Harnack estimate to hold. The novel proofs are based on measure-theoretical arguments, as opposed to comparison principles, and are sufficiently flexible to hold for a large class of singular parabolic equations including the porous medium equation and its quasi-linear versions.

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“…when p > 2; the corresponding approach to the Hölder continuity for the singular case, namely when 1 < p < 2, will be dealt with in [6].…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

A new proof of the Hölder continuity of solutions to p-Laplace type parabolic equations

Gianazza¹,

Surnachev²,

Vespri³

2010

Advances in Calculus of Variations

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It is a well-known fact that solutions to nonlinear parabolic partial differential equations of p-laplacian type are Hölder continuous. One of the main features of the proof, as originally given by DiBenedetto and DiBenedetto-Chen, consists in studying separately two cases, according to the size of the solution. Here we present a new proof of the Hölder continuity of solutions, which is based on the ideas used in the proof of the Harnack inequality for the same kind of equations recently given by E. DiBenedetto, U. Gianazza and V. Vespri. Our method does not rely on any sort of alternative, and has a strong geometric character.

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Harnack type estimates and Hölder continuity for non-negative solutions to certain sub-critically singular parabolic partial differential equations

Cited by 38 publications

References 8 publications

A boundary Harnack inequality for singular equations of 𝑝-parabolic type

A boundary Harnack inequality for singular equations of 𝑝-parabolic type

Intrinsic Harnack inequalities for quasi-linear singular parabolic partial differential equations

A new proof of the Hölder continuity of solutions to p-Laplace type parabolic equations

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