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

Abstract

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 Poincar\'e 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 B\"ogelein, Duzaar and Marcellini by introducing a more abstract notion for $\mathrm{BV}$-valued parabolic function spaces. We argue completely on a variational level.