平面封閉曲線之非局部流

科技部補助專題研究計畫成果報告 期末報告

平面封閉曲線之非局部流

計 畫 類 別 ： 個別型計畫

計 畫 編 號 ： MOST 105-2115-M-006-002- 執 行 期 間 ： 105年08月01日至106年07月31日 執 行 單 位 ： 國立成功大學數學系暨應用數學所

計 畫 主 持 人 ： 林育竹

計畫參與人員： 此計畫無其他參與人員

中　華　民　國　106　年　10　月　31　日

中 文 摘 要 ： 非局部曲率流衍生於許多應用領域, 諸如介面問題、影像處理等。

近年來，相關的研究主要探討平面上簡單封閉曲線在非局部曲率流 下的演化行為。在這個研究專題報告中,我們研究探討的主題是關於 平面上非簡單封閉曲線 (non-simple closed curves) 在

k^{alpha}-type 非局部保長的曲率流的驅動下之漸進行為. 我們證 明非簡單凸的封閉曲線,在這種型態保長的曲率流曲下 ,假設曲率有 均勻上界的情況下,演化曲線會存留至時間為無窮大,而且會均勻地 漸進收斂至m-fold circle.

中 文 關 鍵 詞 ： 非局部流、簡單封閉曲線、非簡單封閉曲線、有號面積、拋物型方 程、漸進行為、 保長流、凸的

英 文 摘 要 ： The nonlocal curvature flow, arising in many application fields, such as phase transitions, image processing, etc., has received much attention in recent years. Among them, the initial curves are assumed to be simple closed and convex, and thus the global solution will converge to a round circle smoothly. In this project, we focus on the asymptotic behavior of the k^{α}-type length-preserving nonlocal flow for nonsimple convex closed curves. Under the assumption that the curvature has a uniform upper bound, the envolving curve will exist and remain convex in infinite time, and finally converge to a m-fold circle uniformly as the time t tends to infinity.

英 文 關 鍵 詞 ： nonlocal flow, simple closed plane curves, non-simple closed curves, signed area (enclosed algebraic area),

parabolic equation, asymptotic behavior, length-preserving, convex

Study on Length-Preserving Nonlocal Flow of Non-simple Convex Closed Plane Curves

Yu-Chu Lin October 31, 2017

1 Introduction

In this project we are concerned with the asymptotic behavior of non-simple convex closed plane curves under certain nonlocal ‡ows for which the speed functions depend on some global quantities, such as the length, the enclosed algebraic area, the mean of the curvature. The corresponding evolution equation of curvature is generally a parabolic equation. The nonlocal curvature ‡ow, arising in many application

…elds, such as phase transitions, image processing([7], [9], [22]), has received much attention in recent years.

Especially, there has been some interest in the nonlocal ‡ow of convex simple closed plane curves recently.

See the papers by Gage [11], Chao-Lin-Wang [6], Jiang-Pan [14], Pan-Yang [20], Ma-Cheng [17], Ma-Zhu [18], and Lin-Tsai [16], Tsai-Wang [23]. All of these papers deal with the evolution of a given convex simple closed plane curves driven by certain parabolic curvature ‡ows with speed function F (k) (t) : As for the speed function, F (k) is a given functions of curvature satisfying the parabolic condition for all z in its domain and (t) is a function of time, which may be dependent on certain global quantities of evolving curves, say length L (t) , enclosed area A (t), the mean of curvature, or the mixture. Their geometric features are the area-preserving ‡ow, the length-preserving ‡ow, or the gradient ‡ow of the isoperimetric di¤erence/ratio. Especially, the convergence of the area-preserving and length preserving ‡ow has been completely clari…ed by Tsai and Wang, their results including all papers mentioned above. Among them, the initial curves are assumed to be simple closed and convex, and thus the global solution will converge to a round circle smoothly.

Contrary to the nonlocal ‡ow of simple closed curves, there was little research of the nonlocal ‡ow for non-simple closed curves. Recently, relevant studies are included in these papers by Dziuk-Kuwert- Schätzle [8], Escher-Ito [10], Kong-Wang [15]. Therefore, in this project we will study the convergence of non-simple closed plane curves under certain nonlocal ‡ows for which the speed function is still of the form where is involved certain global quantities of the evolving curves. A non-simple closed plane curves has self-intersections. The curves in those papers mentioned above are all simple closed and convex, and the convergence to a round circle is almost proved via modifying the argument in [12]. Especially, almost all of their proofs depend crucially on the Bonnesen inequality. However, it is unknown whether the Bonnesen inequality holds for a general non-simple closed plane curves. Therefore, it is to be expected that the problem will become more di¢ cult to deal with in the non-simple cases.

For an immersed curve ; the enclosed algebraic area (or the signed area) is de…ned by:

A = 1 2

Z

xdy ydx = ZZ

R²

(x; y; ) dxdy; (1)

here (x; y; ) is the winding number of relative to the point (x; y) : In general, for an immersed curve, the enclosed algebraic area may be positive, zero or negative. Consider a family of time-dependent immersed curves

parametrized by the maps X (u; t) : S

[0; T ) ! R

: Let L (t), t 2 [0; t); be the length of ( ; t) : S

! R

and let A (t) be its enclosed algebraic area . We have the evolution of the macroscopic, geometric quantities associated the curve, length and area as below.

Lemma 1.1 If the family of closed immersed curves f

g parametrized by smooth maps X (u; t) : S

[0; T ) ! R

; satis…es the equation

@X

@t = W;

then dL dt (t) =

Z

t

hW; kNi ds; (2)

and dA

dt (t) = Z

t

hW; Ni ds; (3)

where h ; i is the standard inner product in R

, N is the unit inward normal and k is the signed curvature at the point X (u; t) :

In this project, we focus on the k -type nonlocal ‡ow for convex immersed curves driven by the following

equation (

@t

(u; t) = (F (k) (t)) N

(u; t) ; X (u; 0) = X

(u) ; u 2 S

;

(4) with the speed function

F (k) (t) = k 1 2m

Z

k

ds; > 0 (LP ): (5)

Here s is the arclength parameter of X (u; t) : S

[0; T ) ! R

and the constant > 0 is arbitrary;

X (u; t) evolves along its inward normal direction N

(u; t), m 2 is the winding number of X ( ; t), k is the curvature at the point X (u; t) ; and X

(u) R

is a given smooth convex immersed curve, parametrized by u 2 S

: In view of Lemma 1.1, we shall readily see that under the speed function (5) ; the

‡ow is length-preserving. On the other hand, an isoperimetric inequality of Osserman [19] tells that for any closed immersed curve,

L

4 X

jm

j A

; (6)

where m

and A

are the winding number and the area (in the usual sense) of jth component of the curve respectively. Under the convex con…nement, the enclosed algebraic area A (t) 0 whenever the ‡ow exists. Since the immersed curve has self-intersections, the curvature of the envolving curve may blow up in …nite time if the initial curve has a tiny loop. Here, we assume the curvature has a uniform upper bound throughout Section 2 and thus prove that under this assumption the ‡ow converges to a m-fold circle uniformly.

2 The k -type length-preserving ‡ow with > 0

As the nonlocal ‡ow (5) is parabolic and X

(u) is smooth, similar to [11] (or [5]), there is a unique smooth solution X (u; t) to the ‡ow de…ned on S

[0; T ) for some short time T > 0: Since X

(u) is convex, by continuity, we may assume that each X ( ; t) is also convex for t 2 [0; T ): We shall show shortly that the convexity is preserved as long as the solution to the ‡ow exists. Under the convex assumption, we can use the outward normal angle 2 S

= [0; 2m ] to parameterize X ( ; t). By X ( ; t) we mean the unique point on X ( ; t) at which its outward normal N

is given by (cos ; sin ) : Therefore, all evolution equations of the ‡ow can be expressed in ( ; t) coordinates and the evolution of the curvature k ( ; t) of X ( ; t) is given

by (

k

( ; t) = k

( ; t) [(k ) ( ; t) + k ( ; t) (t)] ; k( ; 0) = k

( ) > 0; ( ; t) 2 S

[0; T );

(7) where k

( ) > 0 is the curvature of the initial curve X

( ) and

(t) = 1 2m

Z

X( ; t)

k

ds = 1 2m

Z

k ( ; t) d (LP). (8)

Since X

( ) is a closed curve, k

( ) satis…es the integral condition Z

0

cos k

( ) d =

Z

sin

k

( ) d = 0: (9)

In view of (7), (9) is preserved under the ‡ow.

Following the argument in Theorem 4.1.4 of [12], one can verify that the nonlocal ‡ow (5) is equivalent to the equation (7) with (t) given by (8). Since the smooth initial data k

( ) > 0 satis…es (9), we can just focus on the curvature equation (7) from now on, and study its long time behavior. Furthermore, set v = k and thus we sometimes consider the following evolution equation instead of (7):

( v

( ; t) = v

( ; t) [v ( ; t) + v ( ; t) (t)] ; v( ; 0) = v

( ) > 0; ( ; t) 2 S

[0; T );

with p = 1 +

; and

(t) = 1 2m

Z

v ( ; t) d :

In addition, we further assume that the curvature will not blow up in …nite time throughout this article; in other words, v

(t) C (T ) for all t 2 [0; T ): Under this assumption, our main result is stated as follows.

Theorem 2.1 Let

be a smooth convex closed curve with the winding number m 2: Assume that under the LP ‡ow (5), the curvature k has a uniform upper bound during the envolution. Then ( ; t) exists and remains convex on [0; 1): Moreover, it will converge uniformly to a m-fold circle as t ! 1:

2.1 Convexity

We …rst give the gradient estimate for the curvature, which proof is analogous to that in Lemma I1.12 in [3].

Lemma 2.2 Under the LP ‡ow (5) on S

[0; T ) with > 0; there holds the estimate max

S_m¹ [0;t]

max max

S¹_m [0;t]

v

; max

S¹_m f0g

; 8 t 2 [0; T ); (10)

where v = k and = v

+ v

: In particular, we have

jv ( ; t)j C (T ) ; 8 ( ; t) 2 S

[0; T ); (11) whenever v

(t) C (T ) for all t 2 [0; T ):

Proof. Direct computation shows that

@

= 2 v

(v + v (t)) + [ v

2 v

v (v + v)] + pv

v 2 pv

v

(t) v

+ pv

v 2 v

v (v + v) + 2 v

(v + v) ; p > 0 (12) On the time interval [0; t] ; let be the constant max

v

: Then whenever

(s) = (

; s) >

at any time s 2 [0; t] ; we have

(

; s) = 2v (

; s) (v (

; s) + v (

; s)) = 0; (

; s) 0:

Since (

; s) > ; we must have v (

; s) 6= 0 and so (v (

; s) + v (

; s)) = 0: Due to the evolution inequality (12) we get @

(

; s) 0 at (

; s) : By the maximum principle, we have estimate (10). Finally, the estimate (11) follows from (10) and the assumption.

Next, we will prove a time-dependent positive lower bound of the curvature. To prove it, we consider

the cases 0 < < 1 and 1 separately.

2.1.1 The case 0 < < 1.

Motivated by the idea of Andrews [3], we obtain the following:

Lemma 2.3 Assume that the maximum curvature k

(t) C (T ) under the LP ‡ow on S

[0; T ) with 0 < < 1: Then there exists a constant c (T ) > 0 such that

k ( ; t) c (T ) > 0; 8 ( ; t) 2 S

[0; T ): (13) Proof. By (11) and the mean value theorem, we have

jk (

; t) k (

; t)j C (T )

for all (

; t) ; (

; t) 2 S

[0; T ): By the identity @k =@ = ( 1)

@k

=@s; where s is the arc length parameter of X ( ; t) ; and the property that the length of X ( ; t) is decreasing in both ‡ow, we also obtain

k

(s

; t) k

(s

; t) C (T ) js

s

j C (T ) ; 0 < < 1 (14) for all t 2 [0; T ) and all s

; s

on the curve X ( ; t) :

On the other hand, for each t 2 [0; T ); there exists some (t) 2 S

such that L (t) =

Z

1

k ( ; t) d = 2m

k ( (t) ; t) = L (0) :

This implies, at each time t 2 [0; T ); the existence of some value of s (t) such that 0 < k

(s (t) ; t) (L (0) =2m )

; where 0 < < 1: This, together with (14), implies the existence of a constant c (T ) >

0 such that k (s; t) c (T ) for all t 2 [0; T ) and all s on X ( ; t) : 2.1.2 The case 1:

Lemma 2.4 Assume that the maximum curvature k

(t) C (T ) under the LP ‡ow (5) on S

[0; T ) with 1: Then there exists a constant c (T ) > 0 such that

k ( ; t) c (T ) > 0; 8 ( ; t) 2 S

[0; T ): (15) Proof. Applying the similar argument as in [18] (which is for the case = 1), consider the quantity

( ; t) = 1 k ( ; t)

L (t) 2m

1 2m

Z

Z

k ( ; ) d d ; ( ; t) 2 S

[0; T ); (16) where max

m

( ; 0) 0; and compute

t

= (k ) k + (t) (t) 1

2m Z

0

k d 1

2m Z

0

k ( ; t) d

= k

( ; t) ( ; t) ( + 1)k

( ; t)

( ; t) k ( ; t) k

: (17) By the maximum principle,

1

k ( ; t) max

2Sm¹

1

k

( ) + L (t) L (0)

2m + 1

2m Z

0

Z

k ( ; ) d d (18)

for all ( ; t) 2 S

[0; T ): By the assumption k

(t) C (T ) and the fact that L (t) = L (0), we get

estimate (15).

2.2 The long time behavior

Throughout this section, we assume that the curvature has a uniform upper bound. In view of Lemm 2.3 and 2.4, the solution to (4) exists on [0; 1): To go further we recall the following inequality in Andrews [3]:

Lemma 2.5 Let M be a compact Riemannian manifold with a volume form d ; and let be a continuous function on M: Then for any decreasing continuous function F : R ! R; we have

Z

M

d Z

M

F ( ) d Z

M

d Z

M

F ( ) d : (19)

If F is strictly decreasing, then equality holds if and only if is a constant function on M:

Remark 2.6 If F is increasing, then we have in (19).

Applying Andrew’s inequality to a convex closed immersed curve and to the m-fold circle S

with F (z) = z ; z > 0, we have

Z

( ) k ( ) d

Z

F ( ( ))

k ( ) d L Z

0

( ) F ( ( )) k ( ) d : Choosing ( ) = k ( ) gives

Z

k

d L

2m Z

0

k d ; > 0:

This, together with (3), implies that dA

dt =

Z

k

d L (t) 2m

Z

k d 0: (20)

It means that under the LP ‡ow, the enclosed algebraic area is increasing for all > 0.

Next, we prove the following:

Lemma 2.7 Let g (t) 0 be a di¤ erentiable function on [0; 1) with R

0

g (t) dt < 1: If there exists a constant C < 0 such that g

(t) C on [0; 1) or there exists a constant C > 0 such that g

(t) C on [0; 1), then we must have

g (t) ! 0 as t ! 1: (21)

Lemma 2.8 Assume that under the LP ‡ow (5) with > 0; the curvature has a uniform upper bound on [0; 1). Then we have

dA

dt (t) ! 0 as t ! 1 (22)

Proof. We …rst consider the case 1: Let g (t) = dA=dt and we know that g (t) 0 on [0; 1) with R

0

g (t) dt = A (1) A (0) < 1 due to (6) and (20). For the LP ‡ow, compute g

(t) =

Z

( 1) v (v + v (t)) d + L(0) 2m

Z

v

(v + v (t)) d := I (t) + II (t) ; where (t) = (2 m)

R

0

vd :By (11), we have jI (t)j =

Z

( 1) v (v + v (t)) d

=

Z

( 1) v

+ v

v (t) d C

;

t 2 [0; 1); for some constant C

> 0 independent of time. Also we have jII (t)j = L(0)

2

Z

pv

v

+ v

(v (t)) d C

; 8 t 2 [0; 1); p = 1 + 1

> 1: (23) for another constant C

independent of time. Therefore we obtain g

(t) C

; t 2 [0; 1); for some constant C

independent of time. (22) follows due to Lemma 2.7.

For the case 0 < < 1; since the curvature has a uniformly upper bound, the curvature has a uniform positive lower bound due to Lemmas 2.3. Therefore, it is easy to see the above discussion is also valid for 0 < < 1:

Finally, we will prove the envolving curve will converge to a m-fold circle as t ! 1 in the following lemma and thus we complete the proof of Theorem 2.1.

Lemma 2.9 Under the LP ‡ow (5) on S

[0; 1) with > 0; we have

t!1

lim k ( ; t) 2m

L (0)

= 0: (24)

Proof. Assume not. Then there exists a sequence of times ft

g

going to in…nity such that k ( ; t

) 2m

L (0)

" > 0; 8 i (25)

for some " > 0: As v and v are both uniformly bounded, there is a subsequence of ft

g

; still denoted as ft

g

; such that v ( ; t

) = k ( ; t

) converges uniformly on S

to a Lipschitz continuous function w( ) 0 as i ! 1: In particular, since > 0; both k ( ; t

) and k

( ; t

) converge uniformly to w

( ) and w

( ) respectively. At this moment, we can not guarantee that w

( ) 0 is strictly positive for the case > 1. Since the curvature has a uniformly upper bound, the curvature has a uniform positive lower bound for the case 0 < < 1; due to Lemmas 2.3. The following argument is obviously valid for the case 0 < < 1: Hence we focus on the case 1 from now on.

In the case 1, we cannot conclude that the integral of 1=k ( ; t

) converges to the integral of w

( ) : One needs to do more. By (22), we have

0 = lim

i!1

dA dt (t

) =

Z

w

( ) d + L(0) 2

Z

w ( ) d ; 1 1

0: (26)

Also note that

L (0) = Z

0

1

k ( ; t

) d ; 8 i and by Fatou’s lemma, we have

0

Z

1

w

( ) d L (0) : (27)

Thus the Lebesgue integral R

0

w

( ) d converges, i.e., w

( ) > 0 almost everywhere on S

. Now by (26) and (27), we have

2 Z

0

w

( ) d

Z

1 w

( ) d

Z

w ( ) d : (28)

On the other hand, by inequality (19), we also have (take F ( ) = 1=

+ " ; 0; " > 0) Z

0

w ( ) d Z

0

1

w

( ) + " d 2m Z

0

w ( )

w

( ) + " d ; 8 " > 0:

Letting the constant " ! 0

; we obtain Z

0

w ( ) d Z

0

1

w

( ) d 2m Z

0

w

( ) d : (29)

Combine (28) and (29) to get 2m

Z

w

( ) d = Z

0

1 w

( ) d

Z

w ( ) d ; i.e., the following iterated integral vanishes

1 2

Z

Z

1 w

(x)

1

w

(y) (w (x) w (y)) dxdy = 0: (30) As the integrand in (30) is nonpositive almost everywhere on S

S

and w ( ) 0 is a continuous function, we must have w ( ) = C everywhere on S

; where C is the constant given by (2m =L (0)) due to (26). This implies that k ( ; t

) converges uniformly to the constant 2m =L (0) as i ! 1; contradicting to (25). The proof is done.

3 Future Work

For the nonsimple closed plane curves, we just …nished the discussion of the long time behavior of the length-preserving ‡ow for nonsimple convex closed plane curves under the assumption that the curvature has a uniform upper bound. In the future, we hope that we can give an explicit example in this case.

Furthermore, we hope that we can deal with the k -type area-preserving ‡ow with > 0: With loss of the classical isoperimetric inequality and the Bonnesen inequality, it is to be expected that the problem will become more di¢ cult to deal with.

Finally, we remark note that the above nonlocal ‡ow (5) is not suitable for non-convex closed curves due to the curvature term k ; since it may not be de…ned for k < 0: To deal with the nonlocal ‡ow with = 2n 1, n 2 N, for simple nonconvex closed curves, we will compare the intrinsic and extrinsic distances of simple nonconvex closed curves in future work. Such a comparison is useful in certain places, for example in curvature ‡ows of plane curves. In Andrews-Bryan [1] and Huisken [13], they compared intrinsic and extrinsic distances of an evolving time-depending simple closed plane curve

R

driven by the curve shortening ‡ow (see Gage-Hamilton [12]) and derived long time convergent behavior of the evolving curve

: Motivated by [1] and [13], we want to study the comparison more thoroughly and hope to obtain interesting geometric results. Further, we can deal with curvature ‡ows at all.

References

[1] B. Andrews, P. Bryan, Curvature bound for curve shortening ‡ow via distance comparison and a direct proof of Grayson’s theorem. J. Reine Angew. Math. 653 (2011), 179–187.

[2] U. Abresch, J. Langer, The normalized curve shortening ‡ow and homothetic solutions, J. Di¤.

Geom., 23 (1986) 175-196.

[3] B. Andrews, Evolving convex curves, Cal. Var. & PDEs, 7 (1998) 315-371.

[4] B. Andrews, Classi…cation of limiting shapes for isotropic curve ‡ows, J. Amer. Math. Soc., 16 (2003) 443- 459.

[5] S. Angenent, On the formation of singularities in the curve shortening ‡ow, J. Di¤. Geom., 33 (1991) 601-633.

[6] X.-L. Chao, X.-R. Ling, X.-L Wang, On a planar area-preserving curvature ‡ow , Proc. of the AMS, 141 (2013), 1783-1789.

[7] S. Dai, On the shortening rate of collections of plane convex curves by the area-preserving mean curvature ‡ow, SIAM J. Math. Anal., 42 (2010), no. 1, 323–333.

[8] G. Dziuk, E. Kuwert, R. Schätzle, Evolution of elastic curves in R

: existence and computation,

SIAM J. Math. Anal., 33 (2002), 1228-1245.

[9] I. Capuzzo Dolcetta, S. Finzi Vita, R. March, Area-preserving curve-shortening ‡ows: from phase separation to image processing, Inter-faces Free Bound., 4 (2002) ,325-343.

[10] J. Escher, K. Ito, Some dynamic properties of volume preserving curvature driven ‡ows, Mathema- tische Annalen, 333 (2005) 213-230.

[11] M. Gage, On an area-preserving evolution equation for plane curves, Contemp. Math., 51 (1986), 51-62.

[12] M. Gage, R. Hamilton, The heat equation shrinking convex plane curves, J. Di¤. Geom., 23 (1986), 69–96.

[13] G. Huisken, A distance comparison principle for evolving curves, Asian J. of Math., 2 (1) (1998),127- 133.

[14] L. Jiang, S.-L. Pan, On a non-local curve evolution problem in the plane, Comm. Anal. Geom., 16 (2008), no. 1, 1–26.

[15] L.H. Kong, X.L. Wang, Area-preserving evolution of non-simple symmetric plane curves, J. Evol.

Equ., 14 (2014) 387-401.

[16] Y.C. Lin, D.H. Tsai, Application of Andrews and Green-Osher inequalities to nonlocal ‡ow of convex plane curves, J. Evol. Equ., 12 (2012) 833-854.

[17] L. Ma, L. Cheng, A non-local area preserving curve ‡ow, Geom. Dedicata, 171(2014), 231-247.

[18] L. Ma, A.-Q. Zhu, On a length preserving curve ‡ow, Monatshefte Für Mathematik, 165 (2012), 57-78.

[19] R. Osserman, The isoperimetric inequality, Bull. Amer. Math. Soc., 84 (1978), 1182-1238.

[20] S.-L. Pan, J. Yang, On a non-local perimeter-reserving curve evolution problem for convex plane curves, Manuscripta Math., 127 (2008), no. 4, 469–484.

[21] S.-L. Pan, H. Zhang, On a curve expanding ‡ow with a nonlocal ter m, Comm. Contemp. Math., 12 (2010), no. 5, 815–829.

[22] G. Sapiro, A. Tannenbaum, Area and length preserving geometric invariant scale-spaces, Pattern Anal. & Machine Intel., IEEE Transactions, 17 (1995), 67-72.

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curves, Calculus of Variations & PDEs, Volume 54 (2015), Issue 4, 3603-3622.

105年度專題研究計畫成果彙整表

計畫主持人：林育竹 計畫編號：105-2115-M-006-002- 計畫名稱：平面封閉曲線之非局部流

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商標權 0

營業秘密 0

積體電路電路布局權 0

著作權 0

品種權 0

其他 0

技術移轉 件數 0 件

收入 0 千元

國 外

學術性論文

期刊論文 0

研討會論文 0 篇

專書 0 本

專書論文 0 章

技術報告 0 篇

其他 0 篇

智慧財產權 及成果

專利權 發明專利 申請中 0

件

已獲得 0

新型/設計專利 0

商標權 0

營業秘密 0

積體電路電路布局權 0

著作權 0

品種權 0

其他 0

技術移轉 件數 0 件

收入 0 千元

參 與 計 畫 人 力

本國籍

大專生 0

人次

碩士生 0

博士生 0

博士後研究員 0

專任助理 0

非本國籍

大專生 0

碩士生 0

博士生 0

博士後研究員 0

專任助理 0

其他成果

（無法以量化表達之成果如辦理學術活動

、獲得獎項、重要國際合作、研究成果國 際影響力及其他協助產業技術發展之具體 效益事項等，請以文字敘述填列。）　　

科技部補助專題研究計畫成果自評表

請就研究內容與原計畫相符程度、達成預期目標情況、研究成果之學術或應用價 值（簡要敘述成果所代表之意義、價值、影響或進一步發展之可能性）、是否適 合在學術期刊發表或申請專利、主要發現（簡要敘述成果是否具有政策應用參考 價值及具影響公共利益之重大發現）或其他有關價值等，作一綜合評估。

1. 請就研究內容與原計畫相符程度、達成預期目標情況作一綜合評估

■達成目標

□未達成目標（請說明，以100字為限）

　　□實驗失敗 　　□因故實驗中斷 　　□其他原因 說明：

2. 研究成果在學術期刊發表或申請專利等情形（請於其他欄註明專利及技轉之證 號、合約、申請及洽談等詳細資訊）

論文：□已發表　□未發表之文稿　■撰寫中　□無 專利：□已獲得　□申請中　■無

技轉：□已技轉　□洽談中　■無 其他：（以200字為限）

3. 請依學術成就、技術創新、社會影響等方面，評估研究成果之學術或應用價值

（簡要敘述成果所代表之意義、價值、影響或進一步發展之可能性，以500字 為限）

非局部流衍生於許多應用領域，諸如介面問題、影像處理等。近年來，相關的 研究主要探討平面上簡單封閉曲線的演化行為. 我們在這個專題中針對一般的 封閉曲線(這類的曲線通常會自我相交)的演化行為，進行探討。期待對於一般 封閉曲線在非曲部流下的收斂行為能有更全面地了解，同時提升應用層面。

4. 主要發現

本研究具有政策應用參考價值：■否　□是，建議提供機關

（勾選「是」者，請列舉建議可提供施政參考之業務主管機關）

本研究具影響公共利益之重大發現：□否　□是　

說明：（以150字為限）