Optimal $C^{\infty }$-approximation of functions with exponentially or sub-exponentially integrable derivative

Abstract

We discuss Meyers-Serrin's type results for smooth approximations of functions $b=b(t,x):\mathbb{R}\times {\mathbb{R}}^n\to {\mathbb{R}}^m$, with convergence of an energy of the form \[ \int_{\mathbb{R}}\int_{\mathbb{R}^n} w(t,x) φ\left(|Db(t,x)|\right)\mathrm{d} x \mathrm{d} t\,, \] where $w>0$ is a suitable weight function, and $\phi :[0,\infty )\to [0,\infty )$ is a convex function with $\phi (0)=0$ having exponential or sub-exponential growth.

18 pages; remark 25 extended