TY - JOUR
T1 - Asymptotic behavior of the isoperimetric deficit for expanding convex plane curves
AU - Lin, Yu Chu
AU - Tsai, Dong Ho
PY - 2014/1/1
Y1 - 2014/1/1
N2 - We consider the expansion of a convex closed plane curve C0 along its outward normal direction with speed G(1/k), where k is the curvature and (Formula presented.) is a strictly increasing function. We show that if (Formula presented.), then the isoperimetric deficit (Formula presented.) of the flow converges to zero. On the other hand, if (Formula presented.), then for any number d ≥ 0 and (Formula presented.), one can choose an initial curve C0 so that its isoperimetric deficit D(t) satisfies (Formula presented.) for all t ∈[0, ∞). Hence, without rescaling, the expanding curve Ct will not become circular. It is close to some expanding curve Pt, where each Pt is parallel to P0. The asymptotic speed of Pt is given by the constant λ.
AB - We consider the expansion of a convex closed plane curve C0 along its outward normal direction with speed G(1/k), where k is the curvature and (Formula presented.) is a strictly increasing function. We show that if (Formula presented.), then the isoperimetric deficit (Formula presented.) of the flow converges to zero. On the other hand, if (Formula presented.), then for any number d ≥ 0 and (Formula presented.), one can choose an initial curve C0 so that its isoperimetric deficit D(t) satisfies (Formula presented.) for all t ∈[0, ∞). Hence, without rescaling, the expanding curve Ct will not become circular. It is close to some expanding curve Pt, where each Pt is parallel to P0. The asymptotic speed of Pt is given by the constant λ.
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U2 - 10.1007/s00028-014-0238-2
DO - 10.1007/s00028-014-0238-2
M3 - Article
AN - SCOPUS:84939885686
VL - 14
SP - 779
EP - 794
JO - Journal of Evolution Equations
JF - Journal of Evolution Equations
SN - 1424-3199
IS - 4-5
ER -