Non-local approximations of the gradient

Brezis, Haim; Mironescu, Petru

Confluentes Mathematici, Tome 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 ${\dot{W}}^{1,p}$, then

$$V_{\epsilon }\left(x\right):=N\underset{{ℝ}^N}{\int }\frac{u(x+h)-u\left(x\right)}{\left|h\right|}\frac{h}{\left|h\right|}{\rho }_{\epsilon }\left(h\right)dh,x\in {ℝ}^N,$$

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.

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

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