Data for: Non-local, non-convex functionals converging to Sobolev norms

We study the pointwise convergence and the $\Gamma$-convergence of a family of non-local, non-convex functionals $\Lambda_\delta$ in $L^p(\Omega)$ for $p>1$. We show that the limits are multiples of $\int_{\Omega} |\nabla u|^p$. This is a continuation of our previous work where the case $p=1$ was considered.

我们研究了p>1时，L^p(Ω)空间中一族非局部非凸泛函Λ_δ的逐点收敛与Γ收敛（Γ-convergence）性质。我们证明了该泛函族的极限为∫_Ω |∇u|^p的常数倍。本研究是我们先前针对p=1情形开展的工作的延续。