Harnack's Inequality for Degenerate and Singular Parabolic Equations

DiBenedetto, Emmanuele; Gianazza, Ugo; Vespri, Vincenzo

doi:10.1007/978-1-4614-1584-8

Cited by 250 publications

(311 citation statements)

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“…As remarked in [16], this result would follow from Lemma 3.3, if we could control the dependency of the constants with respect to the amount of positivity, i.e. the constant δ in Lemma 3.3.…”

Section: Proof Of Lemma 33mentioning

confidence: 74%

“…The weighted version (with Muckenhoupt weights) was investigated by Chiarenza and Serapioni [9]. In the case 2 ≤ p < ∞, the corresponding Harnack inequality was proved by DiBenedetto, Gianazza and Vespri in [15], see also [16], and by Kuusi using a different approach in [35].…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

“…Our proofs of Theorem 1.2 and Theorem 1.3 are loosely based on the strategy developed in [35], but also rely on the extension of certain arguments introduced in [13] and [16]. Among our contributions, we single out exactly which assumptions are needed on the underlying geometry for the the results to hold.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

“…Our proof here is different to the proof developed in [35] which is based on a covering type lemma together with a delicate analysis. The usage of Lemma 2.7, as an alternative to a covering type lemma, was first mentioned in [16]. (2) Cold: As it turns out, (1.11) implies that the Sobolev norm of a supersolution is small and that its average in space does not change to much in time.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

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Harnack estimates for degenerate parabolic equations modeled on the subelliptic p -Laplacian

Avelin

Capogna

Citti

et al. 2014

Advances in Mathematics

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Abstract. We establish a Harnack inequality for a class of quasi-linear PDE modeled on the prototypewhere p ≥ 2, X = (X 1 , . . . , Xm) is a system of Lipschitz vector fields defined on a smooth manifold M endowed with a Borel measure µ, and X * i denotes the adjoint of X i with respect to µ. Our estimates are derived assuming that (i) the control distance d generated by X induces the same topology on M; (ii) a doubling condition for the µ-measure of d−metric balls and (iii) the validity of a Poincaré inequality involving X and µ. Our results extend the recent work in [15], [35], to a more general setting including the model cases of (1) metrics generated by Hörmander vector fields and Lebesgue measure; (2) Riemannian manifolds with non-negative Ricci curvature and Riemannian volume forms; and (3) metrics generated by non-smooth Baouendi-Grushin type vector fields and Lebesgue measure. In all cases the Harnack inequality continues to hold when the Lebesgue measure is substituted by any smooth volume form or by measures with densities corresponding to Muckenhoupt type weights.2000 Mathematics Subject Classification.

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Section: Proof Of Lemma 33mentioning

confidence: 74%

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

See 2 more Smart Citations

Harnack estimates for degenerate parabolic equations modeled on the subelliptic p -Laplacian

Avelin

Capogna

Citti

et al. 2014

Advances in Mathematics

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“…Because u ∈ C ∞ (Q + ) and also, u ∈ C ∞ (Q 1 ), where Q + and Q 1 is mentioned as before, many authors (see [8,15,16,18,20,22,26]) discussed the Harnark inequalities in Q + and the smoothness of free boundary…”

Section: ) C = O(t γ ) For Large T Andmentioning

confidence: 98%

Regularity and geometric character of solution of a degenerate parabolic equation

Pan

2016

Bull. Math. Sci.

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This work studies the regularity and the geometric significance of solution of the Cauchy problem for a degenerate parabolic equation u t = u m . Our main objective is to improve the Hölder estimate obtained by pioneers and then, to show the geometric characteristic of free boundary of degenerate parabolic equation. To be exact, for the weak solution u(x, t), the present work will show that:1. The function φ = (u(x, t)) β ∈ C 1 (R n ) for given t > 0 if β is large sufficiently; 2. The surface φ = φ(x, t) is tangent to R n at the boundary of the positivity set of u(x, t); 3. The function φ(x, t)is a classical solution to another degenerate parabolic equation.Moreover, some explicit derivative estimates and expressions about the speed of propagation of u(x, t) and the continuous dependence on the nonlinearity of the equation are obtained.

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Infinite Speed of Propagation and Regularity of Solutions to the Fractional Porous Medium Equation in General Domains

Bonforte

Figalli

Ros‐Oton

2016

Comm Pure Appl Math

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We study the positivity and regularity of solutions to the fractional porous medium equations× Ω, for m > 1 and s ∈ (0, 1) and with Dirichlet boundary data u = 0 in (0, ∞) × (R N \ Ω), and nonnegative initial condition u(0, ·) = u 0 ≥ 0. Our first result is a quantitative lower bound for solutions which holds for all positive times t > 0. As a consequence, we find a global Harnack principle stating that for any t > 0 solutions are comparable to d s/m , where d is the distance to ∂Ω. This is in sharp contrast with the local case s = 1, where the equation has finite speed of propagation.After this, we study the regularity of solutions. We prove that solutions are classical in the interior (C ∞ in x and C 1,α in t) and establish a sharp C s/m x regularity estimate up to the boundary. Our methods are quite general, and can be applied to wider classes of nonlocal parabolic equations of the form u t + LF (u) = 0 in Ω, both in bounded or unbounded domains.

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Harnack's Inequality for Degenerate and Singular Parabolic Equations

Cited by 250 publications

References 0 publications

Harnack estimates for degenerate parabolic equations modeled on the subelliptic p -Laplacian

Harnack estimates for degenerate parabolic equations modeled on the subelliptic p -Laplacian

Regularity and geometric character of solution of a degenerate parabolic equation

Infinite Speed of Propagation and Regularity of Solutions to the Fractional Porous Medium Equation in General Domains

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