Γ-convergence, Sobolev norms, and BV functions

Abstract

We prove that the family of functionals $(I_{\delta })$ defined by ${\displaystyle I_{\delta }(g)=\underset{|g(x)-g(y)|>\delta }{\underset{{\mathbb{R}}^N\times {\mathbb{R}}^N}{\int \int }}\frac{{\delta }^p}{|x-y{|}^{N+p}}\mathrm{dx}\mathrm{dy},\hspace{1em}\forall g\in L^p({\mathbb{R}}^N),}$ for $p\ge 1$ and $\delta >0$, $\Gamma$-converges in $L^p({\mathbb{R}}^N)$, as $\delta$ goes to zero, when $p\ge 1$. Hereafter $\left| \right|$ denotes the Euclidean norm of ${\mathbb{R}}^N$. We also introduce a characterization for bounded variation (BV) functions which has some advantages in comparison with the classic one based on the notion of essential variation on almost every line.