Existence of parabolic minimizers to the total variation flow on metric measure spaces

We give an existence proof for variational solutions u associated to the total variation flow. Here, the functions being considered are defined on a metric measure space (X, d, μ) satisfying a doubling condition and supporting a Poincaré inequality. For such parabolic minimizers that coincide with a time-independent Cauchy–Dirichlet datum u on the parabolic boundary of a space-time-cylinder Ω × (0 , T) with Ω ⊂ X an open set and T> 0 , we prove existence in the weak parabolic function space Lw1(0,T;BV(Ω)). In this paper, we generalize results from a previous work by Bögelein, Duzaar and Marcellini by introducing a more abstract notion for BV -valued parabolic function spaces. We argue completely on a variational level.