Resumen de Existence of variational solutions to a Cauchy–Dirichlet problem with time-dependent boundary data on metric measure spaces

Michael Collins

• The objective of this work is an existence proof for variational solutions u to parabolic minimizing problems. Here, the functions being considered are defined on a metric measure space (X,d,μ). For such parabolic minimizers that coincide with Cauchy-Dirichlet data η 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)). In this paper we generalize results from Collins and Herán (Nonlinear Anal 176:56–83, 2018) where only time-independent Cauchy–Dirichlet data have been considered. We argue completely on a variational level.