Almost classical solutions to the total variation flow

The paper examines one-dimensional total variation flow equation with Dirichlet boundary conditions. Thanks to a new concept of "almost classical" solutions we are able to determine evolution of facets -- flat regions of solutions. A key element of our approach is the natural regularity determined by nonlinear elliptic operator, for which $x^2$ is an irregular function. Such a point of view allows us to construct solutions. We apply this idea to implement our approach to numerical simulations for typical initial data. Due to the nature of Dirichlet data any monotone function is an equilibrium. We prove that each solution reaches such steady state in a finite time.Comment: 3 figure

A duality based approach to the minimizing total variation flow in the space H−sH^{-s}

We consider a gradient flow of the total variation in a negative Sobolev space $H^{-s}$ $(0\leq s \leq 1)$ under the periodic boundary condition. If $s=0$, the flow is nothing but the classical total variation flow. If $s=1$, this is the fourth order total variation flow. We consider a convex variational problem which gives an implicit-time discrete scheme for the flow. By a duality based method, we give a simple numerical scheme to calculate this minimizing problem numerically and discuss convergence of a forward-backward splitting scheme. Several numerical experiments are given