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

We define rigorously a solution to the fourth-order total variation flow equation in R n . If n ≥ 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 ≤ 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 ≠ 2. If n ≠ 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. We define rigorously a solution to the fourth-order total variation flow equation in . If , it can be understood as a gradient flow of the total variation energy in , the dual space of , which is the completion of the space of compactly supported smooth functions in the Dirichlet norm. However, in the low dimensional case , the space 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 . If , all annuli are calibrable, while in the case , 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. [ABSTRACT FROM AUTHOR]