On an area-preserving inverse curvature flow of convex closed plane curves
Introduction
Let be a constant and let be a parametrization of a smooth convex closed curve1 . In this paper, we shall consider the following -type nonlocal flow for convex curves: where is the curvature of the evolving curve , the length of , the inward unit normal of and s the arc length parameter. Under the flow (1), the area bounded by the evolving curve is preserved (see Lemma 2.1), so this flow is called an area-preserving flow.
Since the initial curve has positive curvature for all and the function is strictly increasing on its domain , the flow (1) is parabolic. By parabolic theory or the explanation in some related nonlocal flow papers (see [11], [12], [18], [21], [22]), the flow (1) does have a unique smooth solution defined on for some time , and each evolving curve , , remains a smooth convex closed curve. For simplicity, we shall call such a convex solution of the flow (1) on .
The purpose of this paper is to study the asymptotic behavior of the flow (1). Similar to what happens in the -type length-preserving flow [10], singularity (see [2], i.e., curvature blow-up to +∞) may occur under the flow (1) for some initial convex closed curves. When , by a comparison argument, we can prove the existence of a convex closed curve such that under the flow (1) its curvature κ blows up to +∞ in finite time. See Section 3.
In view of the above explanation, the optimal result we can obtain is the following, which is the main result in this paper:
Theorem 1.1 Assume and , is a smooth convex closed curve. Consider the area-preserving flow (1) and assume that the curvature κ will not blow up to +∞ in any finite time during the evolution. Then the flow exists for all time and is area-preserving. Each remains smooth, convex, and it converges to a fixed round circle with radius in norm as .
Our work on the flow (1) is a supplement to previous studies in [15], [14], [17], [21], [9], [10]. For a brief survey on other related nonlocal flows of convex closed curves, please see the introduction section in [9], [21].
Before we start on the proof of Theorem 1.1, we point out again that there are lots of initial convex closed curves such that under (1) each flow solution is smoothly defined on . There are also lots of initial convex closed curves such that under (1) each flow solution is smoothly defined only on for some finite , which is due to the curvature blow-up as . Similar to what we have done in [10], numerical blow-up examples can be constructed.
Section snippets
Proof of Theorem 1.1
We shall decompose the proof of Theorem 1.1 into several lemmas. The major estimate is to show that if the curvature κ will not blow up to +∞ in finite time, then it will not drop down to 0 in finite time either. We shall decompose the proof into two cases: the case and the case . The case has been studied by [15]. Hence we can omit it.
A comparison with the length-preserving flow
In this section, we explore a relationship between the area-preserving flow (1) and the length-preserving flow in the paper [10], which has the form
Acknowledgements
Laiyuan Gao is supported by National Natural Science Foundation of China (No. 11801230). Shengliang Pan is supported by National Natural Science Foundation of China (No. 11671298). Dong-Ho Tsai is supported by MOST of Taiwan (No. 108-2115-M-007-013). The authors are grateful for the referee's helpful comments and positive opinions on the paper.
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