# CONFLUENTES MATHEMATICI

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 $\left({\rho }_{\epsilon }\right)$ is a radial approximation to the identity in ${ℝ}^{N}$ and $u$ belongs to a homogeneous Sobolev space ${\stackrel{˙}{W}}^{1,p}$, then

 ${V}_{\epsilon }\left(x\right):=N\underset{{ℝ}^{N}}{\int }\frac{u\left(x+h\right)-u\left(x\right)}{|h|}\frac{h}{|h|}{\rho }_{\epsilon }\left(h\right)\phantom{\rule{0.166667em}{0ex}}dh,\phantom{\rule{4pt}{0ex}}x\in {ℝ}^{N},$

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

We highlight the crucial role played by the representation formula ${V}_{\epsilon }=\left(\nabla u\right)*{F}_{\epsilon }$, where ${F}_{\epsilon }$ is an approximation to the identity defined via ${\rho }_{\epsilon }$. 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 ${\rho }_{\epsilon }$, allowing, e.g., heavy tails kernels or a distributional definition of ${V}_{\epsilon }$, are also discussed. In particular, we show that heavy tails kernels may be treated as perturbations of approximations to the identity.

Accepted:
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
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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. https://cml.centre-mersenne.org/articles/10.5802/cml.91/

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