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 $$p=1$$ .