Harnack's Inequality for Degenerate and Singular Parabolic Equations

By E. DiBenedetto, Ugo Gianazza, V. Vespri. 2011

355 citations recorded by the source corpus.

Subfields: Applied Mathematics, Computational Theory and Mathematics, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/7b1e0b6b45bf3f8a6628622b0aa8ee7f9f4ff2e3

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In Applied Mathematics

In Computational Theory and Mathematics

In Mathematical Physics

Citations in the corpus

Local Regularity for Parabolic Nonlocal Operators (2012, neutral): “[11], [17]): Let I = (T1, T2), Q = I × Ω.”
A Priori Estimates for Fractional Nonlinear Degenerate Diffusion Equations on Bounded Domains (2013, methodology): “Harnack inequalities of elliptic type of Theorem 8.2: for all t≥ t∗ sup x∈BR(x0) u(t,x) ≤ H2 inf x∈BR(x0) u(t,x). (1.7) The latter inequalities imply the more standard forward Harnack inequalities of [24, 25], and to our knowledge have not been observed before, even in the case s= 1. All the above c”
A variational approach to parabolic equations under general and p,q-growth conditions (2020, result): “About regularity we could quote much more, included the more recent book, published in 2012 by DiBenedetto–Gianazza– Vespri [32].”
Infinite Speed of Propagation and Regularity of Solutions to the Fractional Porous Medium Equation in General Domains (2015, background): “4 The Classical Case s D 1 The classical case has been intensively studied since the 1980s by many authors [1–3, 16, 17, 21, 22, 28, 29, 34, 35, 37, 42]; see also the books [30, 36, 38, 65], as already mentioned in the introduction.”
On the regularity theory for mixed local and nonlocal quasilinear elliptic equations (2021, methodology): “MIXED LOCAL AND NONLOCAL QUASILINEAR PARABOLIC EQUATIONS 3 DiBenedetto [19] and DiBenedetto-Gianazza-Vespri [20].”
Optimal Existence and Uniqueness Theory for the Fractional Heat Equation (2016, background): “onger result may seem surprising, but it is known in nonlinear diﬀusions, where it is typical to have “reversed in time” Harnack inequalities, especially in the regime of Fast Diﬀusion Equations, cf. [12, 19]. Proof. Formula (8.13) follows by the global bounds (8.2), simply by recalling the heat ker”
Harnack inequality for kinetic Fokker-Planck equations with rough coefficients and application to the Landau equation (2016, methodology): “The method has been extended to degenerate cases, like the p-Laplacian, first in the elliptic case by Ladyzhenskaya and Uralt’seva [48], and then, degenerate parabolic cases were covered by DiBenedetto [24] (see also DiBenedetto, Gianazza and Vespri [26, 25, 27]).”
The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion (2017, background): “Obtaining valid versions of the Harnack inequality was challenging in this range [48, 49], see also the monograph [110].”
On the H\"older regularity of signed solutions to a doubly nonlinear equation (2020, methodology): “In order to use the known regularity theory for the parabolic p-Laplacian (see [5, 8] for an account of the theory), it turns out to be more convenient to consider the equation satisfied by w := v, i.”
Intrinsic scaling properties for nonlocal operators (2013, methodology): “ons of second order with bounded measurable coeﬃcents. Nowadays they form a standard tool for the study of various questions of nonlinear partial diﬀerential equations of second order, cf. [CC95] and [DGV12]. Note that the property formulated in Lemma 1.1 is also referred to as expansion of positivi”