The Method of Intrinsic Scaling

Urbano, José Miguel

doi:10.1007/978-3-540-75932-4

Cited by 64 publications

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“…The main ideas underlying the original proof by DiBenedetto, namely the so-called intrisic scaling method, are discussed in [7]. A thorough presentation of this same set of techniques is given in the recent monograph [9].…”

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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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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“…We turn our attention to the regularity of solutions of ( 3.1 ). The proof is based on the method of intrinsic scaling introduced by DiBenedetto for the porous medium equation [ DiB79 , Urb08 ]. It is also similar in spirit to the proof in [ KZ18 ] where regularity was proved for a degenerate diffusion equation posed on with a potentially singular drift term.…”

Section: Existence and Regularity Of Solutionsmentioning

confidence: 99%

“…There are two a priori estimates that play a key role in the proof of Hölder regularity: a Cacciopoli-type energy estimate and a logarithmic estimate. The proof of the energy estimate is essentially the same as [ Urb08 , Proposition 2.4] and we state it without proof.…”

Section: Proof Of Hölder Regularitymentioning

confidence: 99%

Phase Transitions for Nonlinear Nonlocal Aggregation-Diffusion Equations

Carrillo

Gvalani

2021

Commun. Math. Phys.

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We are interested in studying the stationary solutions and phase transitions of aggregation equations with degenerate diffusion of porous medium-type, with exponent $$1< m < \infty $$ 1 < m < ∞ . We first prove the existence of possibly infinitely many bifurcations from the spatially homogeneous steady state. We then focus our attention on the associated free energy, proving existence of minimisers and even uniqueness for sufficiently weak interactions. In the absence of uniqueness, we show that the system exhibits phase transitions: we classify values of m and interaction potentials W for which these phase transitions are continuous or discontinuous. Finally, we comment on the limit $$m \rightarrow \infty $$ m → ∞ and the influence that the presence of a phase transition has on this limit.

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“…Such scalings, originally introduced by DiBenedetto-see for instance [19,43] for a general introduction to the methods-have become a standard tool in order to deal with the regularity of parabolic problems with p-growth; see for instance their use in the context of higher integrability in the work of Kinnunen and Lewis [29,30], Bögelein [4,6] and Acerbi and Mingione [1], and in the study of the porous medium equation in the work of Kinnunen and Lindqvist [31]. Apart from the typical difficulties involved in the use of the arguments related to intrinsic geometry, here we find two new ones, one of which turns out to reveal additional, subtle features of the intrinsic geometry approach.…”

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confidence: 99%

Regularity in parabolic systems with continuous coefficients

2011

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Under the only assumption of continuous coefficients, we prove a partial Hölder continuity result for solutions to parabolic systems with polynomial growth. A key component throughout the argument is the use of DiBenedetto's intrinsic geometry (Degenerate Parabolic Equations. Universitext. Springer, New York, 1993) to accommodate the inhomogeneity in the system. A main technical point is that, although we are proving the Hölder continuity of solutions, we employ the intrinsic geometry using cylinder stretched according to the size of the (spatial) gradient, in an iteration scheme that does not necessarily imply the boundedness of the gradient itself.

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The Method of Intrinsic Scaling

Abstract: The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.

Cited by 64 publications

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

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

Phase Transitions for Nonlinear Nonlocal Aggregation-Diffusion Equations

Regularity in parabolic systems with continuous coefficients

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