| Abstrakt: |
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 0 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 L w 1 (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. [ABSTRACT FROM AUTHOR] |
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