@article{Giga2023, 
author = {Yoshikazu Giga and Hirotoshi Kuroda and Michał Łasica},
title = {The fourth-order total variation flow in              R        n},
year = {2023},
journal = {Mathematics in Engineering},
volume = {5},
number = {6},
pages = {1-45},
keywords = {fourth-order, total variation flow, calibrability, subdifferential, radial solution},
url = {https://www.sciopen.com/article/10.3934/mine.2023091},
doi = {10.3934/mine.2023091},
abstract = {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.}
}