On the Isoperimetric Inequality and Surface Diffusion Flow for Multiply Winding Curves
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On the Isoperimetric Inequality and Surface Diffusion Flow for Multiply Winding Curves Tatsuya Miura
& Shinya Okabe
Communicated by I. Fonseca
Abstract In this paper we establish a general form of the isoperimetric inequality for immersed closed curves (possibly non-convex) in the plane under rotational symmetry. As an application, we obtain a global existence result for the surface diffusion flow, providing that an initial curve is H 2 -close to a multiply covered circle and is sufficiently rotationally symmetric.
1. Introduction It is well known that the behavior of the isoperimetric ratio plays an important role in surface diffusion flow, which is a kind of higher order geometric flow. In this paper we first establish a general form of the isoperimetric inequality for rotationally symmetric immersed closed curves in the plane, which are possibly non-convex, and then apply it to obtain a global existence result for the surface diffusion flow for curves, which we call the curve diffusion flow (CDF), for short. 1.1. Isoperimetric Inequality For a planar closed Lipschitz curve γ , let L(γ ) and A(γ ) denote the length and the signed area, respectively, where we choose the area of a counterclockwise circle to be positive (see Sect. 2 for details). We define the isoperimetric ratio of γ as ⎧ 2 ⎨ L(γ ) (A(γ ) > 0), (1.1) I (γ ) := 4π A(γ ) ⎩ ∞ (A(γ ) ≤ 0). The classical isoperimetric inequality asserts that inf I (γ ) = 1 in a certain class, and the infimum is attained if and only if γ is a round circle, cf. [39].
T. Miura, S. Okabe
Our first purpose is to obtain a generalized isoperimetric inequality that extracts the information of rotation number; namely, we try to find a class X n of immersed closed curves such that inf X n I (γ ) = n, where n ≥ 2, so that the infimum is attained by an n-times covered circle. This is however not easily done by restricting admissible curves to n-times rotating curves. Indeed, even in such a class the isoperimetric ratio can be arbitrarily close to 1 due to an example of a large circle with small (n − 1)-loops; this example leads us to seek an appropriate “global” assumption on the admissible class. In this paper we focus on rotationally symmetric curves. For an integer n ∈ Z and a positive integer m ∈ Z>0 , we define the class An,m to consist of all immersed curves in W 2,1 (S1 ; R2 ) of rotation number n and of m-th rotational symmetry, where we choose the counterclockwise rotation to be positive (see Definitions 2.1 and 2.2 for details). We are now in a position to state our first main theorem, which gives a fully general version of the isoperimetric inequality for rotationally symmetric curves. Theorem 1.1. Let n ∈ Z and m ∈ Z>0 . Then inf
γ ∈An,m
I (γ ) = i n,m := n + m − m
n m
.
(1.2)
The index i n,m is nothing but a unique element in (n + mZ) ∩ {1, . . . , m}, and i n,m = n holds if and only if 1 ≤ n ≤ m. The infimum in (1.2) is attained if and only if i n,m = n and γ is a counterclockwise n-times covered round circle. Theorem 1.1 covers general n and m, althoug
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