### Abstract

We consider the evolution equations F_{t} = -(H_{-1})^{α}ν, where 0 < α < 1, ν is the unit outer normal vector and H_{-1} is the harmonic mean curvature defined by H_{-1} = ((κ_{1}
^{-1} + κ_{2}
^{-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 language | English |
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Pages (from-to) | 109-116 |

Number of pages | 8 |

Journal | Calculus of Variations and Partial Differential Equations |

Volume | 12 |

Issue number | 2 |

Publication status | Published - 2001 Mar |

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### ASJC Scopus subject areas

- Mathematics(all)
- Analysis
- Applied Mathematics

### Cite this

**Contraction of surfaces by harmonic mean curvature flows and nonuniqueness of their self similar solutions.** / Anada, Koichi.

Research output: Contribution to journal › Article

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TY - JOUR

T1 - Contraction of surfaces by harmonic mean curvature flows and nonuniqueness of their self similar solutions

AU - Anada, Koichi

PY - 2001/3

Y1 - 2001/3

N2 - 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.

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.

UR - http://www.scopus.com/inward/record.url?scp=0005868621&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0005868621&partnerID=8YFLogxK

M3 - Article

VL - 12

SP - 109

EP - 116

JO - Calculus of Variations and Partial Differential Equations

JF - Calculus of Variations and Partial Differential Equations

SN - 0944-2669

IS - 2

ER -