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

Abstract

AbstractThe 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 $$({\mathcal {X}}, d, \mu )$$ ( X , d , μ ) . For such parabolic minimizers that coincide with Cauchy-Dirichlet data $$\eta $$ η on the parabolic boundary of a space-time-cylinder $$\varOmega \times (0, T)$$ Ω × ( 0 , T ) with an open subset $$\varOmega \subset {\mathcal {X}}$$ Ω ⊂ X and $$T > 0$$ T > 0 , we prove existence in the parabolic Newtonian space $$L^p(0, T; {\mathcal {N}}^{1,p}(\varOmega ))$$ L p ( 0 , T ; N 1 , 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.