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The fourthorder total variation flow in R^n
Title:  The fourthorder 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:  fourthorder  total variation flow  calibrability  subdifferential  radial solution 
Issue Date:  18May2022 
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 fourthorder total variation flow equation in Rn. If n ≥ 3, it can be understood as a gradient flow of the total variation energy in D1 ,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 D1 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 CahnHoffman 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 fourthorder 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 secondorder 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

Submitter: 数学紀要登録作業用
