The total variation flow with time dependent boundary values (Q502265)

In this paper, the authors consider the nonlinear parabolic equation \[ u_t = \operatorname{div}\left(\frac{Du}{|Du|}\right) \] defined in the domain \(\Omega \times [0,T]\), where \(\Omega\) is Lipschitz. The initial data are given by an \(L^2\) function and the lateral boundary values \(g\) are in \(L^1_{w*}((0, T); BV(\Omega))\) with \(g_t \in L^2\). In this interesting paper, they construct an existence theory for the solution to such Cauchy-Dirichlet problem following an approach due do Lichnewsky-Teman and full developed by Bögelein, Duzaar and Marcellini.