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Contraction of convex surfaces by nonsmooth functions of curvature

Journal Article


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Abstract


  • We consider the motion of convex surfaces with normal speed given by arbitrary strictly monotone, homogeneous degree one functions of the principal curvatures (with no further smoothness assumptions). We prove that such processes deform arbitrary uniformly convex initial surfaces to points in finite time, with spherical limiting shape. This result was known previously only for smooth speeds. The crucial new ingredient in the argument, used to prove convergence of the rescaled surfaces to a sphere without requiring smoothness of the speed, is a surprising hidden divergence form structure in the evolution of certain curvature quantities.

Publication Date


  • 2016

Citation


  • Andrews, B. & McCoy, J. (2016). Contraction of convex surfaces by nonsmooth functions of curvature. Communications in Partial Differential Equations, 41 (7), 1089-1107.

Scopus Eid


  • 2-s2.0-84979536730

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=6846&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/5817

Number Of Pages


  • 18

Start Page


  • 1089

End Page


  • 1107

Volume


  • 41

Issue


  • 7

Place Of Publication


  • United States

Abstract


  • We consider the motion of convex surfaces with normal speed given by arbitrary strictly monotone, homogeneous degree one functions of the principal curvatures (with no further smoothness assumptions). We prove that such processes deform arbitrary uniformly convex initial surfaces to points in finite time, with spherical limiting shape. This result was known previously only for smooth speeds. The crucial new ingredient in the argument, used to prove convergence of the rescaled surfaces to a sphere without requiring smoothness of the speed, is a surprising hidden divergence form structure in the evolution of certain curvature quantities.

Publication Date


  • 2016

Citation


  • Andrews, B. & McCoy, J. (2016). Contraction of convex surfaces by nonsmooth functions of curvature. Communications in Partial Differential Equations, 41 (7), 1089-1107.

Scopus Eid


  • 2-s2.0-84979536730

Ro Full-text Url


  • http://ro.uow.edu.au/cgi/viewcontent.cgi?article=6846&context=eispapers

Ro Metadata Url


  • http://ro.uow.edu.au/eispapers/5817

Number Of Pages


  • 18

Start Page


  • 1089

End Page


  • 1107

Volume


  • 41

Issue


  • 7

Place Of Publication


  • United States