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Hokkaido University Preprint Series in Mathematics >
The fourth-order total variation flow in R^n
| Title: | The fourth-order total variation flow in R^n |
| Authors: | Giga, Yoshikazu Browse this author →KAKEN DB | | Kuroda, Hirotoshi Browse this author →KAKEN DB | | Łasica, Michał Browse this author |
| Keywords: | fourth-order | | total variation flow | | calibrability | | subdifferential | | radial solution |
| Issue Date: | 18-May-2022 |
| Publisher: | Department of Mathematics, Hokkaido University |
| Journal Title: | Hokkaido University Preprint Series in Mathematics |
| Volume: | 1145 |
| Start Page: | 1 |
| End Page: | 38 |
| Abstract: | 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. |
| Type: | bulletin (article) |
| URI: | http://hdl.handle.net/2115/85387 |
| Appears in Collections: | 理学院・理学研究院 (Graduate School of Science / Faculty of Science) > Hokkaido University Preprint Series in Mathematics
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Submitter: 数学紀要登録作業用
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