The total variation flow with time dependent boundary values

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 \(\partial _tg\in L^2(\Omega _T)\).