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 $g_{f_{\epsilon }}$ for the time mollification $f_{\epsilon }$, $\epsilon >0$, of a parabolic Newton-Sobolev function $f\in L_{\mathrm{loc}}^p(0,\ddot{I}„;N_{\mathrm{loc}}^{1,p}(\hat{I}\text{c◯}))$, with $\ddot{I}„>0$ and $\hat{I}\text{c◯}$ open domain in a doubling metric measure space $(\mathbb{X},d,\hat{I}¼)$ supporting a weak $(1,p)$-Poincaré inequality, $p\in (1,\infty )$, is such that $g_{f-f_{\epsilon }}\to 0$ as $\epsilon \to 0$ in $L_{\mathrm{loc}}^p(\hat{I}{\text{c◯}}_{\ddot{I}}„)$, $\hat{I}{\text{c◯}}_{\ddot{I}}„$ being the parabolic cylinder $\hat{I}{\text{c◯}}_{\ddot{I}}„:=\hat{I}\text{c◯}\times (0,\ddot{I}„)$. 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 $p=1$.

Accepted, to appear in Annali dell'Università di Ferrara