Non-local approximations of the gradient

Haim Brezis; Petru Mironescu

doi:10.5802/cml.91

Non-local approximations of the gradient
[English translation of your title]

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

Confluentes Mathematici, Tome 15 (2023), pp. 27-44.

Résumé

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 $ℝ^{N}$ and $u$ belongs to a homogeneous Sobolev space ${\dot{W}}^{1, p}$ , then

V_{ε} (x) : = N \int_{ℝ^{N}} \frac{u (x + h) - u (x)}{| h |} \frac{h}{| h |} ρ_{ε} (h) d h, x \in ℝ^{N},

converges in $L^{p}$ to the distributional gradient $\nabla u$ as $ε \to 0$ .

We highlight the crucial role played by the representation formula $V_{ε} = (\nabla u) * F_{ε}$ , 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.

Reçu le : 2023-04-04
Accepté le : 2023-11-30
Accepté après révision le : 2023-12-11
Publié le : 2024-04-08

DOI : 10.5802/cml.91

Classification : 46E35, 26A45
Mots clés : Distributional gradient, Non-local approximation, Sobolev spaces, Functions of bounded variation

Affiliations des auteurs :

Haim Brezis ¹ ; Petru Mironescu ²

Licence :

CC-BY-NC-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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Haim Brezis; Petru Mironescu. Non-local approximations of the gradient. Confluentes Mathematici, Tome 15 (2023), pp. 27-44. doi : 10.5802/cml.91. https://cml.centre-mersenne.org/articles/10.5802/cml.91/

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