On an area-preserving inverse curvature flow of convex closed plane curves

https://doi.org/10.1016/j.jfa.2021.108931Get rights and content

Abstract

This paper deals with the 1/κα-type area-preserving nonlocal flow of smooth convex closed plane curves for all constant α>0. Under this flow, the convexity of the evolving curve is preserved. Due to the existence of finite time curvature blow-up examples, it is shown that, if the curvature κ will not blow up in finite time, the evolving curve will converge smoothly to a circle as t.

Introduction

Let α>0 be a constant and let X0(φ):S1R2 be a parametrization of a smooth convex closed curve1 γ0R2. In this paper, we shall consider the following 1/κα-type nonlocal flow for convex curves:{Xt(φ,t)=(1L(t)X(,t)κα(,t)dsκα(φ,t))Nin(φ,t),t>0X(φ,0)=X0(φ),φS1, where κ(φ,t) is the curvature of the evolving curve X(φ,t), L(t) the length of X(,t), Nin(φ,t) the inward unit normal of X(φ,t) and s the arc length parameter. Under the flow (1), the area A(t) bounded by the evolving curve is preserved (see Lemma 2.1), so this flow is called an area-preserving flow.

Since the initial curve γ0 has positive curvature κ0(φ) for all φS1 and the function F(κ):=κα is strictly increasing on its domain κ(0,), 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 S1×[0,T) for some time T>0, and each evolving curve X(,t), t[0,T), remains a smooth convex closed curve. For simplicity, we shall call such X(φ,t) a convex solution of the flow (1) on S1×[0,T).

The purpose of this paper is to study the asymptotic behavior of the flow (1). Similar to what happens in the 1/κα-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 α=1, by a comparison argument, we can prove the existence of a convex closed curve γ0R2 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 α>0 and X0(φ),φS1, 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 t[0,) and is area-preserving. Each X(,t) remains smooth, convex, and it converges to a fixed round circle with radius A(0)/π in C norm as t.

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 X(φ,t) is smoothly defined on S1×[0,). There are also lots of initial convex closed curves such that under (1) each flow solution X(φ,t) is smoothly defined only on S1×[0,T) for some finite T>0, which is due to the curvature blow-up as tT. 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 0<α<1 and the case α>1. The case α=1 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{X(φ,t)t=(12πX(,t)κ1α(,t)dsκα(φ,t))Nin(φ,t),t>0,α>0X(φ,0)=X0(φ),φS1.

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.

References (24)

  • M. Green et al.

    Steiner polynomials, Wulff flows, and some new isoperimetric inequalities for convex plane curves

    Asian J. Math.

    (1999)
  • L.-Y. Gao et al.

    Nonlocal flow driven by the radius of curvature with fixed curvature integral

    J. Geom. Anal.

    (2020)
  • Cited by (16)

    • An eternal curve flow in centro-affine geometry

      2023, Journal of Functional Analysis
    • On an area-preserving locally constrained inverse curvature flow of convex curves

      2023, Nonlinear Analysis, Theory, Methods and Applications
    • The evolution of area-preserving and length-preserving inverse curvature flows for immersed locally convex closed plane curves

      2023, Journal of Functional Analysis
      Citation Excerpt :

      Unlike the flows studied previously in [11,17,25], the singularity could appear, i.e., the curvature blows up, during the evolution of the flow (1). This phenomenon has been illustrated by concrete examples in [13] and [14]. Moreover, if the flow is supposed to exist for all time, or equivalently, the curvature does not blow up at finite time, then it is shown that under both of the AP flow and the LP flow the evolving curve converges smoothly to a round circle as time goes to infinity.

    View all citing articles on Scopus
    View full text