Contraction of surfaces by harmonic mean curvature flows and nonuniqueness of their self similar solutions

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Abstract

We consider the evolution equations Ft = -(H-1)αν, where 0 < α < 1, ν is the unit outer normal vector and H-1 is the harmonic mean curvature defined by H-1 = ((κ1 -1 + κ2 -1)/2)-1. In this paper, we prove the nonuniqueness of their strictly convex self similar solutions for some 0 < α < 1. This result implies that there are non-spherical self similar solutions.

Original languageEnglish
Pages (from-to)109-116
Number of pages8
JournalCalculus of Variations and Partial Differential Equations
Volume12
Issue number2
Publication statusPublished - 2001 Mar

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Harmonic mean
Mean Curvature Flow
Self-similar Solutions
Nonuniqueness
Contraction
Normal vector
Strictly Convex
Mean Curvature
Evolution Equation
Imply
Unit

ASJC Scopus subject areas

  • Mathematics(all)
  • Analysis
  • Applied Mathematics

Cite this

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abstract = "We consider the evolution equations Ft = -(H-1)αν, where 0 < α < 1, ν is the unit outer normal vector and H-1 is the harmonic mean curvature defined by H-1 = ((κ1 -1 + κ2 -1)/2)-1. In this paper, we prove the nonuniqueness of their strictly convex self similar solutions for some 0 < α < 1. This result implies that there are non-spherical self similar solutions.",
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AB - We consider the evolution equations Ft = -(H-1)αν, where 0 < α < 1, ν is the unit outer normal vector and H-1 is the harmonic mean curvature defined by H-1 = ((κ1 -1 + κ2 -1)/2)-1. In this paper, we prove the nonuniqueness of their strictly convex self similar solutions for some 0 < α < 1. This result implies that there are non-spherical self similar solutions.

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