Time-smoothing for parabolic variational problems in metric measure spaces

Abstract

In 2013, Masson and Siljander determined a method to prove that the minimal p-weak upper gradient $g_{f_\varepsilon }$ for the time mollification $f_\varepsilon$, $\varepsilon >0$, of a parabolic Newton–Sobolev function $f\in L^p_\mathrm {loc}(0,\tau ;N^{1,p}_\mathrm {loc}(\Omega ))$, with $\tau >0$ and $\Omega$ open domain in a doubling metric measure space $(\mathbb {X},d,\mu )$ supporting a weak (1, p)-Poincaré inequality, $p\in (1,\infty )$, is such that $g_{f-f_\varepsilon }\rightarrow 0$ as $\varepsilon \rightarrow 0$ in $L^p_\mathrm {loc}(\Omega _\tau )$, $\Omega _\tau$ being the parabolic cylinder $\Omega _\tau :=\Omega \times (0,\tau )$. Their approach involved the use of Cheeger’s differential structure, and therefore exhibited some limitations; here, we shall see that the definition and the formal properties of the parabolic Sobolev spaces themselves allow to find a more direct method to show such convergence, which relies on p-weak upper gradients only and which is valid regardless of structural assumptions on the ambient space, also in the limiting case when $p=1$.