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The total variation flow with time dependent boundary values

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Abstract

In this paper we consider the Cauchy–Dirichlet problem for the total variation flow on a space-time cylinder \(\Omega _T=\Omega \times (0,T)\) with a bounded Lipschitz domain \(\Omega \) in \(\mathbb {R}^n\), when the initial datum \(u_o\) is in \(L^2(\Omega )\) and the time dependent lateral boundary values g are given by a function in \(L^{1}_{w*} (0,T;\mathrm{BV}(\Omega ))\) with time derivative \(\partial _tg\in L^2(\Omega _T)\).

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Acknowledgments

V. Bögelein has been supported by the DFG-Project BO3598/1-1 “Evolutionsgleichungen mit pq-Wachstum”.

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Correspondence to Frank Duzaar.

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Communicated by F. H. Lin.

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Bögelein, V., Duzaar, F. & Scheven, C. The total variation flow with time dependent boundary values. Calc. Var. 55, 108 (2016). https://doi.org/10.1007/s00526-016-1041-4

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  • DOI: https://doi.org/10.1007/s00526-016-1041-4

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