The variational problem for a certain space-time functional defined on planar closed curves

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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 languageEnglish
Pages (from-to)5155-5184
Number of pages30
JournalJournal of Differential Equations
Volume252
Issue number10
DOIs
Publication statusPublished - 2012 May 15

Keywords

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

ASJC Scopus subject areas

  • Analysis
  • Applied Mathematics

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