Existence of parabolic minimizers on metric measure spaces

Abstract

The objects of our studies are vector valued parabolic minimizers u associated to a convex Carathéodory integrand f obeying a p-growth assumption from below and a certain monotonicity condition in the gradient variable. Here, the functions being considered are defined on a metric measure space (X,d,μ). For such parabolic minimizers that coincide with a time-independent Cauchy–Dirichlet datum u0 on the parabolic boundary of a space–time-cylinder Ω×(0,T) with an open subset Ω⊂X and T>0, we prove existence in the parabolic Newtonian space Lp(0,T;N1,p(Ω;RN)). In this paper we generalize results from Bögelein et al. (2014,2015) to the metric setting and argue completely on a variational level.