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

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 Γ0 and Γ1 denote planar closed curves and T be a positive constant. Minimize the space–time functional over family of planar closed curves, which change from Γ0 at time t=0 into Γ1 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.