Calculus of Variations and Geometric Measure Theory
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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


Published Paper

Inserted: 21 apr 2020
Last Updated: 4 jan 2022

Journal: Manuscripta Math.
Year: 2022
Doi: 10.1007/s00229-021-01350-2

ArXiv: 2004.09243 PDF


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 Poincaré 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 Bögelein, Duzaar and Marcellini by introducing a more abstract notion for $\mathrm{BV}$-valued parabolic function spaces. We argue completely on a variational level.

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