### Abstract

We consider a variational problem for a certain space-time functional defined on planar closed curves. The functional is related to the functional appeared in Bellettini and Mugnai (2008) [4]. The variational problem is stated as follows: "Let Γ _{0} and Γ _{1} denote planar closed curves and T be a positive constant. Minimize the space-time functional over family of planar closed curves, which change from Γ _{0} at time t=0 into Γ _{1} at time t=T". Concerning the variational problem, we prove the existence of minimizer in a radially symmetric class and determine all the minimizers for each initial final data. Moreover we show that there exists a unique non-radially symmetric critical point in a neighborhood of a certain radially symmetric minimizer.

Original language | English |
---|---|

Pages (from-to) | 5155-5184 |

Number of pages | 30 |

Journal | Journal of Differential Equations |

Volume | 252 |

Issue number | 10 |

DOIs | |

Publication status | Published - 2012 May 15 |

### Keywords

- Initial final value problem
- Space-time functional
- Variational problem

### ASJC Scopus subject areas

- Analysis
- Applied Mathematics