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

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

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:

Keywords: Distributional gradient, Non-local approximation, Sobolev spaces, Functions of bounded variation

Haim Brezis ^{1};
Petru Mironescu ^{2}

@article{CML_2023__15__27_0, author = {Haim Brezis and Petru Mironescu}, title = {Non-local approximations of the gradient}, journal = {Confluentes Mathematici}, pages = {27--44}, publisher = {Institut Camille Jordan}, volume = {15}, year = {2023}, doi = {10.5802/cml.91}, language = {en}, url = {https://cml.centre-mersenne.org/articles/10.5802/cml.91/} }

TY - JOUR AU - Haim Brezis AU - Petru Mironescu TI - Non-local approximations of the gradient JO - Confluentes Mathematici PY - 2023 SP - 27 EP - 44 VL - 15 PB - Institut Camille Jordan UR - https://cml.centre-mersenne.org/articles/10.5802/cml.91/ DO - 10.5802/cml.91 LA - en ID - CML_2023__15__27_0 ER -

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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