The fourth-order total variation flow in ${\mathbb{R}}^n$

We define rigorously a solution to the fourth-order total variation flow equation in ${\mathbb{R}}^n$. If $n\ge 3$, it can be understood as a gradient flow of the total variation energy in $D^{-1}$, the dual space of $D_0^1$, which is the completion of the space of compactly supported smooth functions in the Dirichlet norm. However, in the low dimensional case $n\le 2$, the space $D^{-1}$ does not contain characteristic functions of sets of positive measure, so we extend the notion of solution to a larger space. We characterize the solution in terms of what is called the Cahn-Hoffman vector field, based on a duality argument. This argument relies on an approximation lemma which itself is interesting. We introduce a notion of calibrability of a set in our fourth-order setting. This notion is related to whether a characteristic function preserves its form throughout the evolution. It turns out that all balls are calibrable. However, unlike in the second-order total variation flow, the outside of a ball is calibrable if and only if $n\ne 2$. If $n\ne 2$, all annuli are calibrable, while in the case $n=2$, if an annulus is too thick, it is not calibrable. We compute explicitly the solution emanating from the characteristic function of a ball. We also provide a description of the solution emanating from any piecewise constant, radially symmetric datum in terms of a system of ODEs.