Non-local approximations of the gradient
Confluentes Mathematici, Volume 15 (2023), pp. 27-44.

We revisit the proofs of a few basic results concerning non-local approximations of the gradient. A typical such result asserts that, if (ρ ε ) is a radial approximation to the identity in N and u belongs to a homogeneous Sobolev space W ˙ 1,p , then

V ε (x):=N N u(x+h)-u(x) |h|h |h|ρ ε (h)dh,x N ,

converges in L p to the distributional gradient u as ε0.

We highlight the crucial role played by the representation formula V ε =(u)*F ε , where F ε is an approximation to the identity defined via ρ ε . This formula allows to unify the proofs of a significant number of results in the literature, by reducing them to standard properties of the approximations to the identity.

We also highlight the effectiveness of a symmetric non-local integration by parts formula.

Relaxations of the assumptions on u and ρ ε , allowing, e.g., heavy tails kernels or a distributional definition of V ε , are also discussed. In particular, we show that heavy tails kernels may be treated as perturbations of approximations to the identity.

Revised after acceptance:
Published online:
DOI: 10.5802/cml.91
Classification: 46E35, 26A45
Keywords: Distributional gradient, Non-local approximation, Sobolev spaces, Functions of bounded variation
Haim Brezis 1; Petru Mironescu 2

1 Rutgers University, Department of Mathematics, Hill Center, Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA
2 Universite Claude Bernard Lyon 1, ICJ UMR5208, CNRS, Ecole Centrale de Lyon, INSA Lyon, Université Jean Monnet, 69622 Villeurbanne, France
License: CC-BY-NC-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
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     title = {Non-local approximations of the gradient},
     journal = {Confluentes Mathematici},
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Haim Brezis; Petru Mironescu. Non-local approximations of the gradient. Confluentes Mathematici, Volume 15 (2023), pp. 27-44. doi : 10.5802/cml.91.

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