A new approach to Sobolev spaces and connections to $\mathbf\Gamma$-convergence

Abstract.

This is a follow-up of a paper of Bourgain, Brezis and Mironescu [2]. We study how the existence of the limit

$$\int_\Omega \! \int_\Omega \omega\left( \frac{|f(x)-f(y)|}{|x-y|} \right) \rho_\varepsilon(x-y) \, dx \, dy \quad \text{as $\varepsilon \downarrow 0$}, $$

for $\omega : [0,\infty) \to [0,\infty)$ continuous and $(\rho_\varepsilon) \subset L^1({\mathbb R}^N)$ converging to $\delta_0$, is related to the weak regularity of $f \in L^1_{\rm loc}(\Omega)$. This approach gives an alternative way of defining the Sobolev spaces W ^1,p. We also briefly discuss the $\Gamma$-convergence of (1) with respect to the $L^1(\Omega)$-topology.