FLATϕCURVATURE FLOW OF CONVEX SETS

For an arbitrary initial compact and convex subsetK 0of ℝ n , and for an arbitrary normϕon ℝ n , we construct a flatϕcurvature flowK(t) such thatK(t) is compact and convex throughout the evolution. Previously and using similar methods, R. McCann had shown that flatϕcurvature flow in the plane preser...

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Bibliographic Details
Published inTaiwanese journal of mathematics Vol. 16; no. 1; pp. 1 - 12
Main Author Caraballo, David G.
Format Journal Article
LanguageEnglish
Published Mathematical Society of the Republic of China 01.02.2012
Subjects
Convexity
Curvature
Existence theorems
Integers
Interfacial tension
Kinetics
Surface energy
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Summary:For an arbitrary initial compact and convex subsetK 0of ℝ n , and for an arbitrary normϕon ℝ n , we construct a flatϕcurvature flowK(t) such thatK(t) is compact and convex throughout the evolution. Previously and using similar methods, R. McCann had shown that flatϕcurvature flow in the plane preserves convex, balanced sets. More recently, G. Bellettini, V. Caselles, A. Chambolle, and M. Novaga showed that flatϕcurvature flow in ℝ n preserves compact, convex sets. We also establish a new Hölder continuity estimate for the flow. Flatϕcurvature flows, introduced by F. Almgren, J. Taylor, and L. Wang, model motion byϕ-weighted mean curvature. Under certain regularity assumptions, they coincide with smoothϕ-weighted mean curvature flows given by partial differential equations as long as the smooth flows exist. 2000Mathematics Subject Classification: 53C44, 49N60, 49Q20, 52A20. Key words and phrases: Flat flow, Curvature flow, Convex, Mean curvature, Anisotropic, Mobility, Crystal growth, Hölder continuity.
ISSN:1027-5487
2224-6851
DOI:10.11650/twjm/1500406525