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 $(mathcal{X}, d, mu)$ satisfying a doubling condition and supporting a Poincare inequality. For such parabolic minimizers that coincide with a time-independent Cauchy-Dirichlet datum $u_0$ on the parabolic boundary of a space-time-cylinder $Omega times (0, T)$ with $Omega subset mathcal{X}$ an open set and $T > 0$, we prove existence in the weak parabolic function space $L^1_w(0, T; mathrm{BV}(Omega))$. In this paper, we generalize results from a previous work by Bogelein, Duzaar and Marcellini by introducing a more abstract notion for $mathrm{BV}$-valued parabolic function spaces. We argue completely on a variational level.