Existence of parabolic minimizers on metric measure spaces

The objects of our studies are vector valued parabolic minimizers u associated to a convex Caratheodory 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, mu). For such parabolic minimizers that coincide with a time-independent Cauchy-Dirichlet datum uo on the parabolic boundary of a space-time-cylinder Omega x (0, T) with an open subset Omega subset of X and T > 0, we prove existence in the parabolic Newtonian space L-P (0, T; N-1,N-p (Omega; R-N)). In this paper we generalize results from Biigelein et al. (2014,2015) to the metric setting and argue completely on a variational level. (C) 2018 Elsevier Ltd. All rights reserved.