## V. Buffa - M. Collins - C. Pacchiano Camacho

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

created by buffa on 21 Apr 2020

modified on 04 Jan 2022

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BibTeX]

*Published Paper*

**Inserted:** 21 apr 2020

**Last Updated:** 4 jan 2022

**Journal:** Manuscripta Math.

**Year:** 2022

**Doi:** 10.1007/s00229-021-01350-2

**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 PoincarĂ© 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 BĂ¶gelein, Duzaar and Marcellini by introducing a more abstract notion for $\mathrm{BV}$-valued parabolic function spaces. We argue completely on a variational level.