Qiaofang XING

Abstract In this paper the author devotes to studying a logarithmic type nonlocal plane curve flow.Along this flow,the convexity of evolving curve is preserved,the perimeter decreases,while the enclosed area expands.The flow is proved to exist globally and converge to a finite circle in the C∞metric as time goes to infinity.

Keywords Convex curve,Nonlocal flow,Logarithmic type,Parallel curve

1 Introduction

The curve evolution problems have received much attention during the last few decades(see[1–4,17,24]).In particular,the popular curve shortening flow in the plane studied by Gage [8–10],Gage and Hamilton [11]and Grayson[12],etc.,is a very interesting and subtle case which,except for its own significance,provides a background for understanding the mean curvature flow [26]and other geometric evolution problems in higher dimensions.The book by Chou(Tso) and Zhu [6]provides an excellent and unified account of many results related to flowing curves by curvature.

Recently,there has been some interest in the nonlocal flow of convex closed plane curves,for example,the area-preserving flows are studied by Gage [11],Ma-Cheng [18]and Mao-Pan-Wang [20],the perimeter-preserving flows are investigated by Ma-Zhu [19]and Pan-Yang [23].In [16],Lin and Tsai summarized previous nonlocal curve flows to the following general form:

whereX0(u) ⊂R2is a given smooth closed strictly convex curve,parametrized byu∈S1,andX(u,t):S1×[0,T) →R2is a family of closed planar curves moving along its inward normal direction Nin(u,t) with given speed functionF(k(u,t))-λ(t).F(k) is a function of curvaturek(u,t) of the evolution curveX(u,t) satisfying the parabolic conditionF′(z)>0 for allzin its domain,andλ(t) is a function of time which depends on certain global quantities ofX(u,t),say its lengthL(t),enclosed areaA(t),or other possible global quantities of the integral of curvature over the entire curve in certain ways.In such a case the flow has nonlocal character.In the paper [25],Tsai-Wang gave a systematical study of area-preserving flows and perimeter-preserving flows ifF(k)=kα,α>0.

Different from the previous models,the purpose of this paper is to investigate a logarithmic type nonlocal curve flow which takes the form:

whereX0(u) is a smooth closed strictly convex curve in the plane,L(t) is the perimeter of the evolution curve at timetandkis its relative curvature.Most previous papers in the topic of nonlocal curve flow considered the flows with velocity ofkα+λ(t) (α∈R,α0),λ(t) being a constraint term.The velocity in the flow (1.1) has a different form,that is,a logarithmic form.The nonlocal flow (1.1) is not suitable for non-convex simple closed curves.The circle will be stable under this flow.If the initial curve is not circle,the points on the part where curvature is large will move inwards and the points on the part where curvature is small will move outwards.

The main result of the present paper is as follows.

Main TheoremA closed convex planar curveX0(u) which evolves according to (1.1)remains convex,decreases its length but enlarges its enclosed area,becomes more and more circular during the evolution process,and finally converges to a finite circle in theC∞metric astgoes to infinity.

To obtain the main theorem,we follow the classical proof steps for the study of curve flows.The first step is to show that the convexity of evolving curve is preserved along the flow via maximum principle.The second step is to obtain the local existence of flow by converting the problem to an equivalent parabolic type PDE.Then by showing that the curvature of evolving curve does not blow up(surely,a time-independent upper bound estimate on curvature,but not necessary,implies this),we can obtain the long time existence of the flow.Combining with the structure of the flow,we can employ Bonnesen inequality to prove that a globally existing flow converges to a circle in Hausdorffmetric as time goes to infinity.Furthermore,by establishing a time-independent upper bound on curvature and its derivatives,we can finally show the smooth convergence of the flow.

Though the proof steps are standard,we have to overcome some difficulties which are caused by the logarithmic form velocity.To this end,we establish two new type integral inequalities(see Theorem 2.1),in order to show the monotonicity of perimeter and enclosed area of the evolving curve.The upper bound of curvature is obtained via Tso’s support function method(see [5]).The gradient estimate of curvature is obtained via energy method.After a delicate combination of various new and classical techniques,the main theorem is proved finally.

This paper is organized as follows.In Section 2,two integral inequalities are proved to help us understand the final shape of the evolving curve under the flow (1.1).Section 3 is devoted to prove that if the evolving curve does not develop singularities then it converges to a circle in the Hausdorffmetric astgoes to infinity.In Section 4,the long time existence of the evolving curve is proved.In Section 5,it is shown that the limiting curve of the flow (1.1) is a finite circle in theC∞metric.

2 Two Integral Inequalities

In this section,if a convex curve evolving under (1.1) remains convex,we will prove two integral inequalities to help us obtain the monotonicity of the perimeterL(t)and the areaA(t)bounded by the evolving curve.

Lemma 2.1(Entropy Inequality for Curvature,see[13,Theorem 0.2]) For aC2closed and strictly convex curveγ,letLandAbe the perimeter ofγand the area it encloses respectively.One gets that

The equality in (2.1) holds if and only ifγis a circle.

Theorem 2.1For aC2closed and strictly convex curveγ,one gets

and moreover,the equality in (2.2) or (2.3) holds if and only if the curve is a circle.

ProofFrom the classical isoperimetric inequalityL2≥4πA,we have

So (2.2)is proved and the equality holds if and only if the curve is a circle.It remains to prove(2.3).

Inspired by the method of the paper [15,Lemma 18],we use the monotonicity formula for convex parallel curves to prove (2.3).Suppose thatγβis a convex curve outer parallel to the curveγwith distancer >0.Their perimeter,enclosed area,and curvature are related by (see[7,p.47])

wheresis an arc length parameter ofγ.As a consequence,we have the infinitesimal identities

LetThen from (2.3)–(2.5),we have

From (2.4),we also note that

So we haveH(r)≤0 for allr≥0.Specially,whenr=0,

The equality holds if and only ifγis a circle.

3 The Final Shape of the Evolving Curve

LetX0(u):S1→R2be a convex closed curve in the plane andX(u,t)=(x(u,t),y(u,t)):S1×[0,T)→R2be a family of closed planar curves which evolve under the flow (1.1).

Letg(u,t) = |Xu| = (x2u+y2u)denote the metric along the curve.Then the arc-length element is given by ds=g(u,t)du,or formally

The tangent T,normal N,tangent angleθ,curvaturekand perimeterLof the curve and areaAit bounds are defined in the standard way:

Since changing the tangential components of the velocity vectorXtaffects only the parametrization,not the geometric shape of the evolving curve (see [6,11,22]),we can choose a suitable tangential componentwhich makesθindependent oft,such that the geometric analysis of the evolving curve can be simplified,i.e.,we may consider the following evolution problem which is equivalent to flow (1.1):

Just as Gage,Hamilton and others have done,we can derive the evolution equations of the perimeterL(t),the enclosed areaA(t) and the curvaturekof the evolving curve as follows.

Lemma 3.1Under the flow (3.1) the geometric quantities of the evolving curve evolve as

In the following,suppose that the flow (3.1) exists on time interval [0,∞).It is first proved that the evolving curve is convex,i.e.,the curvaturek=1ρ(ρis the curvature radius of the evolution curve) is always positive.Then the Hausdorffconvergence of the evolving curve is proved by showing that the isoperimetric ratioconverges to 4πast→∞.

Theorem 3.1The flow (3.1) keeps the convexity of the curve during the evolution process.

ProofFrom the evolution equation (3.4),it follows that

namely,

LetThe maximum principle implies that

and

Thus the curvature

We get that the evolving curve is convex during the evolution process.

Theorem 3.2Under the flow (3.1) with an initial convex curve,the perimeterL(t) of the evolving curve decreases and its enclosed areaA(t) increases.Furthermore,the isoperimetric ratiois decreasing to 4πwhich means that the evolving curve becomes more and more circular.

ProofFrom (2.2)–(2.3) of Theorem 2.2,we can see thatLt≤0 andAt≥0.So the perimeterL(t) decreases and the enclosed areaA(t) increases during the evolution process.SinceL≤L0andA≥A0,together with the classical isoperimetric inequality,we can get

So we can conclude that there exists a positive numberRsuch that

By the evolution equations of (3.2)–(3.3),and the inequalitythe isoperimetric ratioof the evolving curve evolves according to

Ast→∞we have the decay of the isoperimetric ratio

Employing Theorem 3.2 and the Bonnesen inequality (see [21]),we can get the following Theorem 3.3.We omit the details of the proof(which are similar to that of [11,Corollary 2.5])here.

Theorem 3.3If an evolving curve under the flow (3.1) does not develop singularities,then it converges to a circle in the Hausdorffmetric.

4 The Long Time Existence

Following the standard method of curve flows (see [11,14,23]),we study the long time behavior of the flow (1.1) which is equivalent to a Cauchy problem of curvature.

Lemma 4.1The nonlocal flow (1.1) is equivalent to the following Cauchy problem:

where (θ,t)∈[0,2π]×[0,T).

The local existence of the flow (1.1) is an application of the classical theory of parabolic equations.In this section,we will use the method in the paper[25]to obtain a time-independent upper bound of the curvature.Then the flow (1.1) can be extended on time interval [0,+∞).

Theorem 4.1Under the flow (1.1),the curvaturekhas a uniform upper bound for all(θ,t)∈[0,2π]×[0,∞).

ProofGiven a convex curve in the plane,the Bonnesen inequality (see [21]) says

for allrin≤r≤routwhererin,routare the inradius and the outradius of the given convex curve.So

Since the isoperimetric ratioI(t) ofX(θ,t) is decreasing as well as the isoperimetric deficit under the flow(1.1),we have 1 ≤I(t)≤I(0)for allt∈[0,ϖ).Letrin(t)androut(t)denote the inradius and the outradius ofX(θ,t) respectively.LetTherefore

Sincewe get

Both the inradius and the outradius have time-independent positive bounds.

LetE(0) be the inscribed circle ofX(θ,0) with radiusrin(0).LetE(0) shrink according towhereL(0) is the length ofX(θ,0).ThenE(t) is enclosed byX(θ,t) for allt∈[0,min{ϖ,t1}) by the maximum principle,whereThe radiusr(t) ofE(t) is given by

Letp(θ,t) = -〈X(θ,t),Nin(θ,t)〉 be the support function ofX(θ,t).If the center ofE(0) is chosen to be the originO,thenp(θ,t) (with respective toO) ofX(θ,t) is defined on [0,2π]×[0,min{ϖ,t1}) and

Define

The first and the second derivations ofQwith respect toθare

Under the nonlocal flow (1.1),the evolution of the support function is

We compute the evolution ofQonS1×[0,T0) to get

Apparently,there is a large enough constantQ∗independent ofθandt,such that whenQ>Q∗,we have

LetQmax(t) = max{Q(θ,t) |θ∈[0,2π]}.ThenQmax(t) is decreasing ifQ > Q∗.By the maximum principle,there exists a positive constant Φ=max{Qmax(0),Q∗}such thatQ(θ,t)≤Φ for (θ,t) ∈[0,2π]×[0,T0).Combining with (4.4),we can get that the curvaturekhas a uniform upper boundk(θ,t) ≤eΦ(C−β)fort∈[0,T0).Using the above method,we can prove thatk(θ,t) ≤holds on time intervals [T0,2T0],[3T0,4T0],···,whereCis a constant depending on the initial curveX0.So the proof is done.

Corollary 4.1The solution to the flow (1.1) exists fort∈[0,∞).

ProofBy Theorem 3.1,the evolving curve is convex under the flow(1.1).(4.1)is uniformly parabolic in any finite time interval.Theorem 4.1 tells us that the curvature has a uniform upper bound for allt≥0.The regularity theory of parabolic equations ensures that all derivatives ofkare bounded.So the flow can be extended to the time interval [0,∞).

5 C∞Convergence

It is shown that the evolving curve converges to a circle in theC0sense (Hausdorffconvergence).AndC∞convergence means all the derivatives of curvaturek(or curvature radiusρ)converge to 0 ast→∞.In this section,we investigate convergence of the flow (1.1) and finish the proof of Main Theorem by showing that the evolving curve converges to a circle in theC∞metric.

Lemma 5.1Under the flow (1.1),we have

whereM1is a positive constant independent ofθandt.

Proofwhereαis a constant to be chosen later.It follows from (4.1)that

and

By Theorem 4.1,khas a uniform upper bound,and the lengthL(t) has an upper boundL0which is independent of time.So there exists a constantindependent of time such thatNow we choose

Sincek(θ,t)is uniformly bounded andis also uniformly bounded,k(θ,t)is equicontinuous.So for any sequencek(θ,ti),we can choose a subsequencek(θ,tin) converging uniformly to a functionk∞(θ).We know that the curve converges to a circle in the Hausdorffsense,sok∞(θ)=1R,whereRis a positive constant mentioned in(3.5).Since every subsequence converges to1R,we get thatk(θ,t)converges to1R.We can obtain the followingC∞convergence.

Theorem 5.1Under the flow (1.1),we have

ProofEvidently,the curvature equation is uniformly parabolic and the regularity theory ensures that all the space and time derivatives ofkare bounded by constants depending only on the order of differentiations.By induction and the Ascoli-Arzela theorem,we obtain (5.2).

Next,we can imitate the method in[8,Subsections 5.7.6–5.7.14]to estimate the convergence of derivatives of the radius curvature.

By (4.1),the evolution equation of the curvature radiusρ(θ,t) is

Lemma 5.2Letρ(i)denote the ith derivative ofρ(θ,t) with respect toθ.We have

ProofIn fact,this is just a special version of the Wirtinger inequality.

Lemma 5.3There exists a positive constantC1depending on the initial curve such that

holds for sufficiently larget.

ProofBy direct computation

So integrating with respect totyields

Lemma 5.4For any 0<µ<1,we can find a constantC2,such that

ProofCompute that

By the above estimate ofρ(1)and sinceast→∞,we have

By [11,Lemma 5.7.6],we complete the proof.

Corollary 5.1Under the flow (1.1),for any

ProofIn one-dimensional case

and we apply this toρ(1).

Next,we can follow the routine of [11]and use a similar method as in the proof of Lemma 5.4 and Corollary 5.1 to obtain the estimates of higher order derivatives ofρ.

Corollary 5.2Under the flow(1.1),for any 0<µ<1,

Sinceρis exponentially convergent to a constant,bothL(t) andkalso exponentially converge.Integrating (1.1) shows that the evolving curveX(·,t) converges to a fixed limitX∞.Therefore combining with Theorems 3.1–3.3,Corollaries 4.1 and 5.2 and Theorem 5.1,we can get the proof of Main Theorem.

AcknowledgementThe author is grateful to the anonymous referee for his or her careful reading the original manuscript of this paper,proposing some corrections and giving the invaluable comments and suggestions.