Optimal C∞ -approximation of functions with exponentially or sub-exponentially integrable derivative

We discuss Meyers-Serrin’s type results for smooth approximations of functions b= b(t, x) : R× Rn→ Rm, with convergence of an energy of the form ∫R∫Rnw(t,x)φ(|Db(t,x)|)dxdt,where w> 0 is a suitable weight function, and φ: [0 , ∞) → [0 , ∞) is a convex function with φ(0) = 0 having exponential or subexponential growth.

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 \[ \int_{\mathbb{R}}\int_{\mathbb{R}^n} w(t,x) \varphi\left(|Db(t,x)|\right)\mathrm{d} x \mathrm{d} t\,, \] where $w>0$ is a suitable weight function, and $\varphi:[0,\infty)\to [0,\infty)$ is a convex function with $\varphi(0)=0$ having exponential or sub-exponential growth.