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The fourth-order total variation flow in R^n

Giga, Yoshikazu Kuroda, Hirotoshi Łasica, Michał 北海道大学

2022.05.18

概要

We define rigorously a solution to the fourth-order total variation flow equation in Rn. If n ≥ 3, it can be understood as a gradient flow of the total variation energy in D-1 ,the dual space of D10, 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.

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