Time-smoothing for parabolic variational problems in metric measure spaces

Abstract

In a work of 2013, Masson and Siljander proved that the time mollification f ε , ε > 0, of a parabolic Newton-Sobolev function f ∈ L p loc (0, τ ; N 1,p loc (Ω)), with τ > 0 and Ω open domain in a doubling metric measure space (X, d, µ) supporting a weak (1, p)-Poincaré inequality, p ∈ (1, ∞), is such that the minimal p-weak upper gradient g f −fε → 0 as ε → 0 in L p loc (Ω τ ), Ω τ being the parabolic cylinder Ω τ := Ω × (0, τ ). Their original version of this deep result 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 actually allow to infer that such convergence for the time mollifications can be shown in a much simpler way, regardless of structural assumptions on the ambient space, and also in the limiting case when p = 1.