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We consider the expansion of a convex closed plane curve C 0 along its outward normal direction with speed G (1/ k ), where k is the curvature and $${G \left(z \right) :\left(0, \infty \right) \rightarrow \left( 0, \infty \right)}$$ G z : 0 , ∞ → 0 , ∞ is a strictly increasing function. We show that if $${{\rm lim}_{z \rightarrow \infty} G \left(z \right) = \infty}$$ lim z → ∞ G z = ∞ , then the isoperimetric deficit $${D \left(t \right) : = L^{2}\left(t \right) -4 \pi A \left(t \right)}$$ D t : = L 2 t - 4 π A t of the flow converges to zero. On the other hand, if $${{\rm lim}_{z \rightarrow \infty}G \left(z \right) = \lambda \in (0,\infty)}$$ lim z → ∞ G z = λ ∈ ( 0 , ∞ ) , then for any number d ≥ 0 and $${\varepsilon > 0}$$ ε > 0 , one can choose an initial curve C 0 so that its isoperimetric deficit $${D \left(t \right)}$$ D t satisfies $${\left \vert D \left(t \right) -d \right \vert < \varepsilon}$$ D t - d < ε for all $${t \in (0, \infty)}$$ t ∈ ( 0 , ∞ ) . Hence, without rescaling, the expanding curve C t will not become circular. It is close to some expanding curve P t , where each P t is parallel to P 0 . The asymptotic speed of P t is given by the constant $${\lambda}$$ λ .
Journal of Evolution Equations – Springer Journals
Published: Dec 1, 2014
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