2024-03-29T07:00:27Zhttps://eprints.lib.hokudai.ac.jp/dspace-oai/requestoai:eprints.lib.hokudai.ac.jp:2115/853872022-11-17T02:08:08Zhdl_2115_45007hdl_2115_116The fourth-order total variation flow in R^n1000070144110Giga, Yoshikazu1000080635657Kuroda, HirotoshiŁasica, Michałopen accessfourth-ordertotal variation flowcalibrabilitysubdifferentialradial solution410We 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.Department of Mathematics, Hokkaido University2022-05-18engdepartmental bulletin paperVoRhttps://doi.org/10.14943/103229http://hdl.handle.net/2115/8538710.14943/103229Hokkaido University Preprint Series in Mathematics1145138https://eprints.lib.hokudai.ac.jp/dspace/bitstream/2115/85387/1/The%20fourth-order%20total%20variation%20flow%20in%20Rn.pdfapplication/pdf609.22 KB2022-05-18