Time-smoothing for parabolic variational problems in metric measure spaces

Abstract

In 2013, Masson and Siljander determined a method to prove that the $p$-minimal 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 $(1,p)$-Poincar\'e inequality, $p\in(1,\infty)$, is such that $g_{f-f_\varepsilon}\to0$ as $\varepsilon\to0$ 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 $p=1$.