Contraction of convex surfaces by nonsmooth functions of curvature. Issue 7 (2nd July 2016)

• Record Type: Journal Article
• Authors: Andrews, Ben; McCoy, James
• Is Part Of: Communications in partial differential equations. Volume 41:Issue 7(2016)
• Page Start: 1089
• Page End: 1107
• Publication Date: 2016-07-02
• Subjects: Curvature flow -- parabolic partial differential equation; 35K55 -- 53C44; Differential equations, Partial -- Periodicals; 515.353
• DOI: 10.1080/03605302.2016.1202266
• Languages: English
• ISSNs: 0360-5302

Abstract

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.