TY - JOUR

T1 - The variational problem for a certain space-time functional defined on planar closed curves

AU - Okabe, Shinya

N1 - Funding Information:
The author would like to express his deepest gratitude to Professor Y. Tonegawa of Hokkaido University who suggested Problem 1.1 to him and gave him many useful suggestions and encouragement. I also thank the referee for giving me helpful comments. This research was partly supported by Grant-in-Aid for Young Scientists (B) (No. 21740110).

PY - 2012/5/15

Y1 - 2012/5/15

N2 - 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.

AB - 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.

KW - Initial final value problem

KW - Space-time functional

KW - Variational problem

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U2 - 10.1016/j.jde.2012.01.020

DO - 10.1016/j.jde.2012.01.020

M3 - Article

AN - SCOPUS:84858283828

SN - 0022-0396

VL - 252

SP - 5155

EP - 5184

JO - Journal of Differential Equations

JF - Journal of Differential Equations

IS - 10

ER -