Extension properties and boundary estimates for a fractional heat operator

Nyström, Kaj; Sande, Olow

doi:10.1016/j.na.2016.02.027

Cited by 54 publications

(55 citation statements)

References 19 publications

(1 reference statement)

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“…This is the so-called extension operator for the fractional powers (∂ t − x ) s , 0 < s < 1, of the heat operator. It was recently introduced independently by Nyström-Sande in [24], and Stinga-Torrea in [27]. These authors proved that, if for a given u ∈ S (R n+1 ), the function U solves the problem…”

Section: Notations and Preliminariesmentioning

confidence: 99%

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The structure of the singular set in the thin obstacle problem for degenerate parabolic equations

et al. 2021

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We study the singular set in the thin obstacle problem for degenerate parabolic equations with weight $$|y|^a$$ | y | a for $$a \in (-1,1)$$ a ∈ ( - 1 , 1 ) . Such problem arises as the local extension of the obstacle problem for the fractional heat operator $$(\partial _t - \Delta _x)^s$$ ( ∂ t - Δ x ) s for $$s \in (0,1)$$ s ∈ ( 0 , 1 ) . Our main result establishes the complete structure and regularity of the singular set of the free boundary. To achieve it, we prove Almgren-Poon, Weiss, and Monneau type monotonicity formulas which generalize those for the case of the heat equation ($$a=0$$ a = 0 ).

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Section: Notations and Preliminariesmentioning

confidence: 99%

“…The passage from (1.2) to (1.1) rests on the extension procedure for the operator (∂ t − x ) s , developed independently by Nyström and Sande in [24] and by Stinga and Torrea in [27]. Such result represents the parabolic counterpart of the famous Caffarelli and Silvestre's extension work [9].…”

Section: Introductionmentioning

confidence: 99%

The structure of the singular set in the thin obstacle problem for degenerate parabolic equations

et al. 2021

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“…It was recently introduced independently by Nyström-Sande in [46], and Stinga-Torrea in [57]. These authors proved that if for a given ϕ ∈ S (R n+1 ), the function u solves the problem…”

Section: A Sharp Harnack Inequality For the Parabolic Extension Problemmentioning

confidence: 99%

“…In more recent years Stinga and Torrea have generalized the extension procedure to different classes of operators, including uniformly elliptic operators in divergence form L = div(A(x)∇), see [57], or the heat operator H = ∂ ∂t − ∆ x , see [58]. This latter result was also established simultaneously and independently by Nyström and Sande in [46].…”

mentioning

confidence: 91%

“…Here, we have kept with the notations introduced before (1.5) above. We mention that, TWO CLASSICAL PROPERTIES OF THE BESSEL QUOTIENT I ν+1 /I ν AND THEIR IMPLICATIONS IN PDE' S 5 similarly to its elliptic predecessor (1.5), the operator (1.8) is the extension operator for the fractional powers (∂ t − ∆) s , 0 < s < 1, of the heat operator, see [46] and [58].…”

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confidence: 99%

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Two classical properties of the Bessel quotient 𝐼_{𝜈+1}/𝐼_{𝜈} and their implications in pde’s

Garofalo¹

2020

Contemporary Mathematics

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Two elementary and classical results about the Bessel quotient yν = I ν+1 Iνstate that on the half-line (0, ∞) one has for ν ≥ −1/2: (i) 0 < yν < 1; (ii) yν is strictly increasing. In this paper we show that (i) and (ii) have some nontrivial and interesting applications to pde's. As a consequence of them, we establish some sharp new results for a class of degenerate partial differential equations of parabolic type in R n+1 + × (0, ∞) which arise in connection with the analysis of the fractional heat operator (∂t − ∆) s in R n × (0, ∞), see Theorems 1.2, 1.4, 1.5 and 1.7 below.

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The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion

Vázquez

2017

Lecture Notes in Mathematics

102

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We describe the mathematical theory of diffusion and heat transport with a view to including some of the main directions of recent research. The linear heat equation is the basic mathematical model that has been thoroughly studied in the last two centuries. It was followed by the theory of parabolic equations of different types. In a parallel development, the theory of stochastic differential equations gives a foundation to the probabilistic study of diffusion.Nonlinear diffusion equations have played an important role not only in theory but also in physics and engineering, and we focus on a relevant aspect, the existence and propagation of free boundaries. Due to our research experience, we use the porous medium and fast diffusion equations as case examples.A large part of the paper is devoted to diffusion driven by fractional Laplacian operators and other nonlocal integro-differential operators representing nonlocal, long-range diffusion effects. Three main models are examined (one linear, two nonlinear), and we report on recent progress in which the author is involved.

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Extension properties and boundary estimates for a fractional heat operator

Cited by 54 publications

References 19 publications

The structure of the singular set in the thin obstacle problem for degenerate parabolic equations

The structure of the singular set in the thin obstacle problem for degenerate parabolic equations

Two classical properties of the Bessel quotient 𝐼_{𝜈+1}/𝐼_{𝜈} and their implications in pde’s

The Mathematical Theories of Diffusion: Nonlinear and Fractional Diffusion

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