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The anisotropic polyharmonic curve flow for closed plane curves

Journal Article


Abstract


  • We study the curve diffusion flow for closed curves immersed in the Minkowski plane M, which is equivalent to the Euclidean plane endowed with a closed, symmetric, convex curve called an indicatrix that scales the length of a vector in M depending on its length. The indiactrix ∂U (where U⊂ R 2 is a convex, centrally symmetric domain) induces a second convex body, the isoperimetrix I~. This set is the unique convex set that miniminises the isoperimetric ratio (modulo homothetic rescaling) in the Minkowski plane. We prove that under the flow, closed curves that are initially close to a homothetic rescaling of the isoperimetrix in an averaged L 2 sense exists for all time and converge exponentially fast to a homothetic rescaling of the isoperimetrix that has enclosed area equal to the enclosed area of the initial immersion.

Publication Date


  • 2019

Citation


  • Parkins, S. & Wheeler, G. (2019). The anisotropic polyharmonic curve flow for closed plane curves. Calculus of Variations and Partial Differential Equations, 58 (2), 70-1-70-35.

Scopus Eid


  • 2-s2.0-85063292465

Start Page


  • 70-1

End Page


  • 70-35

Volume


  • 58

Issue


  • 2

Place Of Publication


  • Germany

Abstract


  • We study the curve diffusion flow for closed curves immersed in the Minkowski plane M, which is equivalent to the Euclidean plane endowed with a closed, symmetric, convex curve called an indicatrix that scales the length of a vector in M depending on its length. The indiactrix ∂U (where U⊂ R 2 is a convex, centrally symmetric domain) induces a second convex body, the isoperimetrix I~. This set is the unique convex set that miniminises the isoperimetric ratio (modulo homothetic rescaling) in the Minkowski plane. We prove that under the flow, closed curves that are initially close to a homothetic rescaling of the isoperimetrix in an averaged L 2 sense exists for all time and converge exponentially fast to a homothetic rescaling of the isoperimetrix that has enclosed area equal to the enclosed area of the initial immersion.

Publication Date


  • 2019

Citation


  • Parkins, S. & Wheeler, G. (2019). The anisotropic polyharmonic curve flow for closed plane curves. Calculus of Variations and Partial Differential Equations, 58 (2), 70-1-70-35.

Scopus Eid


  • 2-s2.0-85063292465

Start Page


  • 70-1

End Page


  • 70-35

Volume


  • 58

Issue


  • 2

Place Of Publication


  • Germany