The fractional nonlocal Ornstein-Uhlenbeck equation, Gaussian symmetrization and regularity

Feo, Filomena; Stinga, P. R.; Volzone, Bruno

doi:10.3934/dcds.2018142

Cited by 9 publications

(5 citation statements)

References 38 publications

(61 reference statements)

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“…This is an immediate consequence of the third formula for U in (20) and the results of Theorem 7, see [90,91]. For such explicit statement for negative powers L −s in other contexts like manifolds and discrete settings, see [34,39,45,70].…”

Section: Theorem 7 (Extension Problem For Positive Powersmentioning

confidence: 90%

“…It turns out these are quite concrete and useful ways of defining and understanding fractional operators. Indeed, when a heat kernel is available for the semigroup e −tL , then pointwise formulas for both positive and negative powers of L can be obtained, see [13,14,29,32,33,34,38,39,45,70,76,90,91,92,93,94]. For degenerate cases like the usual derivative or discrete derivatives see [1,11].…”

Section: The Methods Of Semigroupsmentioning

confidence: 99%

“…These were new even for the case of the fractional Laplacian. As it could be expected, these general extension problems found many applications such as free boundary problems [2,4], fractional derivatives [11], master equations [14,38,92], fractional elliptic PDEs [29,95,100], fractional Laplacians on manifolds [7,28,31,39,46] and in infinite dimensions [74], symmetrization [40,45], nonlocal Monge-Ampère equations [70], numerical analysis [73], biology [94] and inverse problems [48,51].…”

Section: The Methods Of Semigroupsmentioning

confidence: 99%

See 2 more Smart Citations

User’s guide to the fractional Laplacian and the method of semigroups

Stinga¹

2019

Fractional Differential Equations

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The method of semigroups is a unifying, widely applicable, general technique to formulate and analyze fundamental aspects of fractional powers of operators L and their regularity properties in related functional spaces. The approach was introduced by the author and José L. Torrea in 2009 (arXiv:0910.2569v1). The aim of this chapter is to show how the method works in the particular case of the fractional Laplacian L s = (−∆) s , 0 < s < 1. The starting point is the semigroup formula for the fractional Laplacian. From here, the classical heat kernel permits us to obtain the pointwise formula for (−∆) s u(x). One of the key advantages is that our technique relies on the use of heat kernels, which allows for applications in settings where the Fourier transform is not the most suitable tool. In addition, it provides explicit constants that are key to prove, under minimal conditions on u, the validity of the pointwise limitsThe formula for the solution to the Poisson problem (−∆) s u = f is found through the semigroup approach as the inverse of the fractional Laplacian u(x) = (−∆) −s f (x) (fundamental solution). We then present the Caffarelli-Silvestre extension problem, whose explicit solution is given by the semigroup formulas that were first discovered by the author and Torrea. With the extension technique, an interior Harnack inequality and derivative estimates for fractional harmonic functions can be obtained. The classical Hölder and Schauder estimates (−∆) ±s : C α → C α∓2s are proved with the method of semigroups in a rather quick, elegant way. The crucial point for this will be the characterization of Hölder and Zygmund spaces with heat semigroups.

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Section: Theorem 7 (Extension Problem For Positive Powersmentioning

confidence: 90%

Section: The Methods Of Semigroupsmentioning

confidence: 99%

Section: The Methods Of Semigroupsmentioning

confidence: 99%

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User’s guide to the fractional Laplacian and the method of semigroups

Stinga¹

2019

Fractional Differential Equations

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“…For an equivalent definition of (−∆ γ ) s and for other qualitative properties involving the fractional Ornstein-Uhlenbeck operator we refer to [20].…”

Section: The Extension Technique and The Gaussian Fractional Perimetermentioning

confidence: 99%

A quantitative dimension free isoperimetric inequality for the fractional Gaussian perimeter

Carbotti¹,

Cito²,

Manna³

et al. 2020

Preprint

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“…Actually, the effect of symmetrization on fractional elliptic problems like (1.1) has already been exploited in [24] and then in [54], [55], [45], [56], [28]. In those papers a symmetrization result in terms of mass concentration (i.e., an integral comparison, as in the parabolic case) is obtained in a somewhat indirect way.…”

Section: Introductionmentioning

confidence: 99%

Symmetrization for fractional elliptic problems: a direct approach

Ferone,

Volzone

2020

Preprint

Self Cite

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We provide new direct methods to establish symmetrization results in the form of mass concentration (i.e. integral) comparison for fractional elliptic equations of the type (−∆) s u = f (0 < s < 1) in a bounded domain Ω, equipped with homogeneous Dirichlet boundary conditions. The classical pointwise Talenti rearrangement inequality in [47] is recovered in the limit s → 1. Finally, explicit counterexamples constructed for all s ∈ (0, 1) highlight that the same pointwise estimate cannot hold in a nonlocal setting, thus showing the optimality of our results.

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The fractional nonlocal Ornstein-Uhlenbeck equation, Gaussian symmetrization and regularity

Cited by 9 publications

References 38 publications

User’s guide to the fractional Laplacian and the method of semigroups

User’s guide to the fractional Laplacian and the method of semigroups

A quantitative dimension free isoperimetric inequality for the fractional Gaussian perimeter

Symmetrization for fractional elliptic problems: a direct approach

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