anisotropic flows for convex plane curves - Project Euclid

Vol. 97, No. 3

DUKE MATHEMATICAL JOURNAL

© 1999

ANISOTROPIC FLOWS FOR CONVEX PLANE CURVES KAI-SENG CHOU and XI-PING ZHU

Introduction. Modeling the dynamics of melting solids or similar phenomena has been a topic of study for a long time. For sharp interfaces, the motion of the interface, that is, the boundary of the solid, is usually related to its curvature in a certain way. One of the early models was proposed by Mullins [20] for grain boundaries. This two-dimensional model is given by the equation V = k,

(1)

where V and k are, respectively, the normal velocity and curvature of the interface. Equation (1) is sometimes called the “curve shortening problem” because it is the negative L2 -gradient flow of the length of the interface. By drawing pictures, one can be easily convinced that a simple closed curve stays simple and smooth along (1) and shrinks to a point in finite time, with the limiting shape of a circle. However, a mathematical treatment of (1) turned out to be rather delicate. In fact, a rigorous study did not exist until the early 1980s, when differential geometers considered (1) as a tool in the search for simple closed geodesics on surfaces. It was also regarded as a model case for more general curvature flows, which are believed to be important in the topological classification of low-dimensional manifolds. As a first attempt, Gage [10] proved that the isoperimetric ratio decreases along convex curves. Then in Gage and Hamilton [12], it was shown that a convex curve stays convex and shrinks to a point in finite time. Moreover, if one normalizes the flow by dilating it so that the enclosing area is constant, the normalized flow converges smoothly to a circle. Finally, Grayson [14] completed this line of investigation by showing that a simple curve evolves into a convex one before shrinks to a point. For an alternative approach to this result, one may consult Hamilton [18]. Recently Mullins’s theory was generalized by Gurtin [15], [16] and by Angenent and Gurtin [4], [5] (see also the monograph of Gurtin [17]) to include anisotropy and the possibility of a difference in bulk energies between phases. Anisotropy is indispensable in dealing with crystalline materials. For perfect conductors, the temperatures in both phases are constant. The equation becomes β(θ)V = g(θ)k + F,

(2)

Received 23 April 1996. 1991 Mathematics Subject Classification. Primary 35K57; Secondary 58E15. Authors’ research supported by an Earmarked Grant of Hong Kong. Zhu partially supported by the Foundation for Outstanding Young Scholars and the National Science Foundation of China. 579

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where θ is the tangent angle of the interface, β > 0 is a kinetic coefficient, g is of the form g = d 2 f/dθ 2 + f , f > 0 is the interfacial energy, and the constant F is the energy of the solid relative to its surrounding. The presence of θ reflects anisotropy, and the particular form in which f appears in g is a consequence of thermodynamics. It was also noted that β may depend on the normal velocity as well. However, we do not elaborate on this point in this work. In this paper, we consider regular interfaces and always assume that the interfacial energy is strictly stable; that is, g is positive. This means that (2) is essentially an equation of parabolic type. We should nevertheless point out that other cases are also important. For a detailed discussion on their physical relevance, one is referred to [4] or [17]. Basic properties concerning the flow (2) (as well as other curvature flows for immersed curves) can be found in Angenent [2], [3]. A general result therein ensures the existence of a maximal solution for (2) for rather general initial interfaces. With the presence of F , apart from shrinking the flow may expand depending on the “size” of the initial interfaces as well. Denote by L(t) and A(t) the length of, and the area enclosed by, the interface at time t, respectively. It was proved in [4] that when F ≥ 0, A(t) tends to zero in finite time; when F < 0, A(t) tends to zero in finite time, provided the length of the initial interface is sufficiently small, and it tends to infinity provided the initial interface contains a sufficiently large disk. Both the expanding and the collapsing cases have been taken up by several authors. Soner [22] considered (2) as well as its higher-dimensional generalization by means of the concept of viscosity solutions. Among other things, he proved that in the expanding case, the solution is asymptotic to a dilation of the Wulff region of 1/β, answering a conjecture raised in [4] . This result was further strengthened in [5], where, in particular, the assumption required in [22], which states that the polar diagram of β is convex, was removed. Less is known for the collapsing case. When F vanishes, the equation becomes β(θ)V = g(θ)k.

(3)

Equation (3) arises from geometry. When g/β(θ) = g/β(θ + π), it has a natural interpretation as the curve-shortening problem for a Minkowski geometry. Using this point of view, Gage [11] showed that a convex curve shrinks to a point as in the isotropic case. Moreover, the normalized curve converges smoothly to a self-similar solution of (3), which is unique up to dilations and translations. Without assuming any symmetry on g/β, Gage and Li [13] showed that the normalized curve “subconverges” to self-similar solutions of (3), of which uniqueness is not known. One can also observe that when F is negative, (2) has a stationary solution of which curvature satisfies g(θ)k + F = 0.

(4)

The specified form g = d 2 f/dθ 2 + f guarantees that this equation is uniquely solvable (up to translations) for a convex curve. Linearized stability of this stationary

ANISOTROPIC FLOWS FOR CONVEX PLANE CURVES

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solution was studied and, in fact, employed, to prove the above-mentioned asymptotic result in the expanding case in [5]. It was shown that the stationary point is a hyperbolic fixed point for the associated linearized semiflow. The unstable manifold is of dimension 1 and consists of two trajectories v ± (θ, t); both decay to zero as t → −∞. v + is positive and corresponds to the expanding case; v − is negative and corresponds to the collapsing case (for convex curves, since the stationary solution is uniformly convex). Angenent and Gurtin studied the expanding case and posed the asymptotic behavior of the collapsing case as an open problem. (See also [17].) In this context, the results of [11] and [13] may be regarded as a partial answer to their question. In this work, we study (2) for convex interfaces under the strict stability assumption. Rather complete results are obtained. We summarize them in the following theorem. Main Theorem. Consider (2) where β, g are positive, smooth 2π-periodic functions, and F is a constant. For any given smooth and convex γ0 , there exists a negative F ∗ such that the maximal solution γ (·, t) of (2) starting at γ0 is uniformly convex for t ∈ (0, tmax ), and the following hold: (a) When F > F ∗ , tmax is finite and γ (·, t) shrinks to a point as t → tmax . Furthermore, if we normalize γ so that its enclosing area is constant, the normalized flow subconverges smoothly to a self-similar solution of the flow (3). (b) When F = F ∗ , tmax = ∞. If, in addition, it is assumed that g=

d 2f + f, dθ 2

(5)

for some f , then the flow converges to a stationary solution (4) smoothly. (c) When F < F ∗ , tmax = ∞, and the interface γ (·, t) expands to infinity as t approaches infinity. If, in addition, it is assumed that the polar diagram of β is uniformly convex, that is, β −1 + d 2 β −1 /dθ 2 > 0, for all θ, γ (·, t)/t converges smoothly to the boundary of the Wulff region of −F /β. Some remarks are in order. First, in the shrinking case (a), we complete the answer to a question raised in [5] (see also [17]). In fact, we prove that the curvature of the normalized flow has a uniform (positive) lower and an upper bound in [ε, tmax ) for ε > 0 small. By the construction of a Liapunov function, it can be shown that any limit point of γ (·, t) must be a self-similar solution of (3). In the case when self-similar solutions are unique (up to translations), as, for instance, when g/β(θ) = g/β(θ +π), we can show that the normalized flow actually converges smoothly to a self-similar solution. Since the uniqueness of self-similar solutions of (3) is in general not known, we can only make a weaker assertion: The normalized flow “subconverges” to a selfsimilar solution. More precisely, any sequence γ˜ (·, tj ), tj → tmax , where γ˜ is the normalized flow, contains a subsequence that converges to a self-similar solution of (3).

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Next, when F = F ∗ , we show that the curvature of γ (·, t) admits uniform (positive) lower and upper bounds in [1, ∞). Without assuming (5), the following is valid: Let c be the unique point in the region c = (c1 , c2 ) : c = |c|(cos θ, sin θ) and 0 ≤ |c| < −F /β(θ) satisfying

2π 0

geiθ dθ = 0, β(c1 cos θ + c2 sin θ) + F

and let γˆ = γ − ct. Then γˆ (·, t) converges to a convex curve of which curvature satisfies g(θ )k + β(c1 cos θ + c2 sin θ) + F = 0.

(6)

Thus, when (5) is violated, the flow runs to infinity along the direction c/|c| at constant speed, with shape asymptotic to a solution of (6). Finally, in the expanding case, we improve previous works by establishing smooth rather than uniform convergence of the corresponding support functions when the polar diagram of β is uniformly convex. This paper is organized as follows. In Section 1, we collect some basic facts on the model (2). In Section 2, we derive a gradient estimate for the curvature of the interface. Using this estimate, we prove the “cusp theorem,” which asserts that the curvature becomes unbounded if and only if the interface shrinks to a point. We study the asymptotic behavior of the normalized flow, in the next two sections. First we derive a bound on the entropy of the normalized flow, which is crucial in controlling the normalized curvature form above. Then we apply it to obtain estimates of all orders for the normalized curvature in Section 4. Once this is done, subconvergence to self-similar solutions can be established rather easily. In Section 5, we study the convergence of the modified flow γˆ to a stationary solution when F = F ∗ . We conclude this paper with a discussion on the expanding case in Section 6. Our main theorem is contained in Propositions 4.3, 5.8, and 6.1. When F is nonzero, it is easy to see that some flows of (2), which start at closed simple curves, develop self-intersections after some time. However, when F vanishes, the flow preserves simple curves. In a companion paper [9], we will show that the flow always evolves into a convex one before it shrinks to a point. 1. The anisotropic flow. In the interfacial motion studied in [4], [5], [15], [16], the interface is described as a simple closed curve γ (·, t) in the plane, of which evolution is governed by the equation ∂γ = (θ)k + (θ) N, ∂t

(1.1)


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where γ is oriented in counterclockwise direction, θ is the angle between the tangent of γ and the x-axis, and N and k are, respectively, the unit inward normal and curvature of γ . The functions and are smooth 2π-periodic functions of θ. Throughout this paper, we assume that is positive. The specified forms of and , such as = (d 2 f/dθ 2 + f )/β for some function f and = F /β for some constant F , arising from physical consideration, are not considered until Section 5. Let L(t) and A(t) be the length of, and the area enclosed by, γ (·, t), respectively. By direct computations (see (A.6) and (A.7) in the appendix), we have dL = − k(k + )ds (1.2) dt γ and dA =− dt

γ

(k + )ds.

(1.3)

It is clear that both A(t) and L(t) decrease along the flow when γ (·, t) is convex and is nonnegative. However, it is not clear what happens otherwise. From the appendix, we know that the curvature of γ satisfies the following parabolic equation: ∂2 ∂k = 2 (k + ) + k 2 (k + ), ∂t ∂s

(1.4)

where ∂/∂s is the derivative with respect to the arc-length of γ . By representing γ (·, t) as graphs over γ0 , one can show that (1.1) admits a solution satisfying γ (·, 0) = γ0 for small t > 0. In fact, it was proved in [2] that for very general initial γ0 , (1.1) has a maximal solution γ (·, t) defined in [0, tmax ), tmax ≤ ∞, where tmax is finite if and only if the curvature of γ (·, t) becomes unbounded as t approaches tmax . For the sake of completeness, we give a proof of this fact when γ0 is convex. There are two special parametrizations of γ (·, t) that we use alternatively. The first one is the representation of γ (·, t) as a polar graph. We fix a point inside γ0 as the origin and introduce the polar coordinate. For small t > 0, we can write γ (·, t) = r(α, t)(cos α, sin α),

α ∈ [0, 2π ],

(1.5)

where the polar angle α = α(·, t) is invertible for small t > 0. The tangent and normal vectors of γ are given by ∂r ∂r cos α − r sin α, sin α + r cos α D, T = ∂α ∂α ∂r ∂r N =− sin α + r cos α, − cos α + r sin α D, ∂α ∂α

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where

D=

∂r ∂α

2

1/2 +r

2

,

respectively. By differentiating (1.5), we have ∂α ∂r ∂γ (·, t) = DT + (α, t)(cos α, sin α). ∂t ∂t ∂t

(1.6)

Equating the coefficients of T and N in (1.1) and (1.6) yields ∂α ∂r/∂t 1 ∂r ∂r =− (cos α, sin α) · T = − 2 ∂t D D ∂t ∂α

(1.7)

and k + =

∂r r ∂r (cos α, sin α) · N = − . ∂t D ∂t

(1.8)

Recall that in the polar coordinates, the curvature is given by k=

2 1 ∂ 2r ∂r 2 − r . + 2 + r ∂α D3 ∂α 2

(1.8) becomes

2 ∂r D(θ) (θ ) ∂ 2 r ∂r 2 r = , −2 −r − ∂t ∂α r rD 2 ∂α 2

(1.9)

which is a quasilinear parabolic equation for r. Notice that ∂r sin θ = sin α + r cos α D. ∂α By a local existence result for quasilinear parabolic equations, we immediately conclude that for any smooth γ0 that admits a polar representation, there exists a unique smooth solution r(α, t) to (1.9) satisfying r(α, 0) = |γ0 |(α) for small t > 0. Regard (1.7) as an ordinary differential equation of the form dα = F (α, t), dt α(0; u) given, where u is a parameter. As α(0; u) is invertible, by the continuous dependence on parameters, we know that α(t; u) is invertible for small t. In conclusion, we have found a parametrization u such that (1.1) holds for small t.


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Lemma 1.1. Let γ0 be a smooth plane curve that is strictly star-shaped with respect to some point. Then there exists a unique solution of (1.1) satisfying γ (·, 0) = γ0 , which is also strictly star-shaped with respect to the same point, for small t > 0. Next we show that for a convex γ0 , γ (·, t) becomes uniformly convex (i.e., k(·, t) is positive) instantly. We have ∂k ∂k ∂α ∂k (·, t) = (α, t) + (α, t) ∂t ∂t ∂α ∂t ∂k (k + ) ∂r ∂k = (α, t) + ∂t rD ∂α ∂α by (1.7). Using ds = Ddα and dθ = kds in (1.4), we obtain ∂k ∂k (k + ) ∂r ∂k (α, t) = (·, t) − ∂t ∂t rD ∂α ∂α 1 ∂ 1 ∂ (k + ) ∂r ∂k = (k + ) + k 2 (k + ) − D ∂α D ∂α rD ∂α ∂α

∂ 1 ∂k 3kθ θ k + ∂r ∂k = + + − D ∂α D ∂α D D rD ∂α ∂α

2 + k (θ θ + ) k + (θθ + ) .

(1.10)

It follows from the strong maximum principle for parabolic equations that the following lemma is valid. Lemma 1.2. γ (·, t) is uniformly convex for t > 0 if γ0 is convex. Moreover, observe that (1.10) is of the form ∂ 2k ∂k ∂k = A 2 +B + Ck, ∂t ∂α ∂α where A, B, and C are bounded as long as k is bounded. Also, the boundedness of k yields a positive lower bound for A. Hence k satisfies a uniformly parabolic equation with bounded coefficients as long as k is bounded. By a result in Krylov and Safonov [19], one can control the Hölder norm of k by max |k|. Using a standard argument, we have the following lemma. Lemma 1.3. There exists a maximal solution γ (·, t) to (1.1) satisfying γ (·, 0) = γ0 , where γ0 is convex, in [0, tmax ). If tmax is finite, then k(·, t) becomes unbounded as t approaches tmax . In view of Lemma 1.2, we may always assume that γ (·, t) is uniformly convex for t ∈ [0, tmax ). In particular, the tangent angle θ can be used to parametrize γ . Let h(θ, t) = γ (θ, t), (sin θ, − cos θ), θ ∈ [0, 2π], be the support function of γ (θ, t).

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From (1.1), we have

∂γ ∂γ ∂u ∂h = + , (sin θ, − cos θ) = −(k + ) ∂t ∂t ∂u ∂θ

since ∂γ /∂u is tangential to γ . Using the formula

k=

(1.11)

−1

∂ 2h +h ∂θ 2

,

we obtain from (1.11) the following equation for k(θ, t): ∂k ∂2 = k 2 2 (k + ) + k 2 (k + ). ∂t ∂θ

(1.12)

Both (1.11) and (1.12) are basic in our study. To conclude this section, we give a preliminary estimate on the blow-up rate of k. Lemma 1.4. Let kmax (t) = maxθ k(θ, t). If tmax is finite, then kmax (t) ≥ C(tmax − t)−1/2 , for some constant C depending only on and . Proof. From (1.12), dkmax ≤ max (θθ + ) k 3 + max (θθ + ) k 2 θ θ dt 3 ≤ C1 (1 + kmax ) . Hence

−2 1 + kmax (t) ≤ 2C1 (tmax − t)

after an integration from t to tmax . 2. A gradient estimate. In this section, we adapt the method in [3] to derive a gradient estimate for the curvature of the flow, which is used to estimate the entropy of the normalized flow in the next section. As an application of this estimate, we also show that the curve must collapse to a point when tmax is finite. Setting v(θ, t) = (θ )k(θ, t) + (θ), it follows from (1.12) that 2 −1 ∂v 2 ∂ v = (v − ) +v , (θ, t) ∈ R × [0, tmax ). (2.1) ∂t ∂θ 2 Lemma 2.1. Assume k(θ, 0) > 0 and 2 ∂v 2 (θ, 0) ≤ M 2 v + ∂θ


587

for all θ . Then for any (θ, t), either ∂ 2v v + 2 (θ, t) ≥ 0 ∂θ or

∂v v2 + ∂θ

2

(θ, t) ≤ max M 2 , 2L∞ .

Proof. For fixed (θ0 , t0 ), t0 > 0, we set B = [v 2 + (∂v/∂θ)2 ]1/2 (θ0 , t0 ). Suppose that B 2 > max{M 2 , 2L∞ }. We claim that [v + (∂ 2 v/∂θ 2 )](θ0 , t0 ) ≥ 0. To prove this claim, we write v(θ0 , t0 ) = B cos ξ, ∂v (θ0 , t0 ) = B sin ξ ∂θ for ξ ∈ (−π, π ). Notice that from v > −L∞ > −B, ξ is uniquely determined. Letting v ∗ (θ ) = B cos(θ0 − θ + ξ ),

θ ∈ (θ0 + ξ − π, θ0 + ξ + π)

be a stationary solution of (2.1), we consider w = v −v ∗ in (θ0 +ξ −π, θ0 +ξ +π)× [0, tmax ). By our assumption, w is positive along (θ0 +ξ ±π, t), and w(θ, 0) vanishes somewhere in the interval (θ0 + ξ − π, θ0 + ξ + π). We claim that w(θ, 0) has exactly two zeros in this interval. In fact, at each zero θ¯ of w(θ, 0) in (θ0 + ξ − π, θ0 + ξ ), we have ¯ = B cos(θ0 − θ¯ + ξ ), v(θ¯ , 0) = v ∗ (θ) and ∂v ∂w (θ¯ , 0) = (θ¯ , 0) − B sin θ0 − θ¯ + ξ ∂θ ∂θ ¯ 0) 1/2 ≤ −B sin θ0 − θ¯ + ξ + M 2 − v 2 (θ,

1/2 < −B sin θ0 − θ¯ + ξ + B 2 − B 2 cos2 (θ0 − θ¯ + ξ ) = 0. And, similarly, at each zero θ in (θ0 + ξ, θ0 + ξ + π), we have ∂w θ, 0 > 0. ∂θ So there are exactly two zeros of w(θ, 0) in (θ0 + ξ − π, θ0 + ξ + π).

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By the Sturmian theory [1], the number of zeros of w(·, t) is nonincreasing in time. Now, the function v ∗ has been chosen so that w(θ0 , t0 ) has a zero of multiplicity 2. Therefore, w(θ0 , t0 ) is positive for θ = θ0 , and we must have ∂ 2 w/∂θ 2 (θ0 , t0 ) ≥ 0. As a consequence of Lemma 2.1, we can bound the gradient of the curvature by its L1 -norm. Lemma 2.2. For t ∈ [0, tmax ), 2 ∂(k) ∂v 2 v + (θ, 0) + (1 + 2π)L∞ ∂θ (·, t) ∞ ≤ max θ ∂θ L 2π + θ L∞ + (θ)k(θ, t)dθ. 0

Proof. For a fixed (θ1 , t1 ) such that ∂v (θ1 , t1 ) ≥ M + L∞ ∂θ for M 2 = maxθ (v 2 + (∂v/∂θ )2 )(θ, 0), we choose θ2 to be the first θ > θ1 such that either ∂v/∂θ (θ, t1 ) = 0 or [v 2 + (∂v/∂θ)2 ](θ, t1 ) ≤ max{M 2 , 2L∞ }. By the above lemma, we have ∂ 2v v + 2 (θ, t1 ) ≥ 0 ∂θ for all θ ∈ (θ1 , θ2 ). Therefore, θ2 2 ∂v ∂v ∂ v (θ1 , t1 ) = (θ2 , t1 ) − (θ, t1 )dθ 2 ∂θ ∂θ ∂θ θ1 θ2 ∂v ≤ (θ2 , t1 ) + v(θ, t1 )dθ. ∂θ θ1 From this gradient estimate, we deduce a pointwise estimate on k as follows. Let (k)max (t) = maxθ (θ )k(θ, t)=(θ0 )k(θ0 , t) for some θ0 . We have 2π ∂(k) 1 (k)max (t) − (·, t) (θ)k(θ, t)dθ ≤ 2π ∞. 2π 0 ∂θ L By Lemma 2.2, we have 2π 1 (k)max (t) ≤ 2π + (θ)k(θ, t)dθ 2π 0 2

∂v 2 ∞ + 2π max v + (θ, 0) + (1 + 2π)L + θ L∞ . θ ∂θ (2.2) Furthermore, for θ ∈ [θ0 − 1/4π, θ0 + 1/4π],


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∂(k) 1 (k)max (t) − (θ)k(θ, t) ≤ (·, t) ∞. 4π ∂θ L Combining this with Lemma 2.2 yields (k)max (t) ≤ 2(θ )k(θ, t) 2

∂v 1 2 max v + (θ, 0) + (1 + 2π)L∞ + θ L∞ + θ 2π ∂θ (2.3) for all θ , |θ − θ0 | ≤ 1/4π . Next, we show that the curvature becomes unbounded only when the enclosed area of the flow becomes zero. This fact was first established in [12] and Chou [6] for the curve-shortening flow. It can also be found in [3] when (θ) = (θ + π) and (θ ) = −(θ + π ) and in [13] when ≡ 0. Our proof basically follows [12]. Lemma 2.3. Let 0 < T < ∞. The curvature k(θ, t) is bounded as long as A(t) is bounded away from zero in [0, T ). Proof. Let β ∗ (t) = sup{β : (k)(θ, t) ≥ β on an interval of length equal to π}. For any small ε > 0, there exists an interval of π such that k(θ, t) ≥ β ∗ (t) − ε. Then γ (·, t) lies between two parallel lines whose distance d is given by d=

a+π

2L∞ sin(θ − a) dθ ≤ ∗ . k(θ, t) β (t) − ε

a

On the other hand, we have d

L(t) ≥ A(t). 2

Hence β ∗ (t) − ε ≤ L∞

L(t) . A(t)

Letting ε ↓ 0, we obtain β ∗ (t) ≤ L∞ L(t)/A(t). So, in view of (1.2), β ∗ (t) is bounded as long as A(t) is bounded away from zero in [0, T ). We estimate k by β ∗ . First of all, we examine the integral 0

2π

log(k)dθ.

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We have d 2π (θ ) log (θ )k(θ, t)dθ dt 0 2 2π ∂(k) (k)2 − + k (θθ + ) dθ = ∂θ 0 ∂(k) 2 ∂(k) 2 2 2 = (k) − dθ + (k) − dθ ∂θ ∂θ Ii J i 2 2π ∂ + k + dθ, ∂θ 2 0 where Ii , i ∈N are mutually disjoint open intervals satisfying ∪i Ii = {θ : (k)(θ, t) > β ∗ }, and J is the complement of the union of Ii ’s. By the definition of β ∗ , the length of each Ii does not exceed π . Applying Wirtinger’s inequality b 2 b df 2 f dθ ≤ dθ, dθ a a which holds for f (a) = f (b) = 0, 0 ≤ b − a ≤ π, to the function k − β ∗ on each Ii yields 2 ∂(k) 2 ∗ (k) − dθ ≤ 2β (t) kdθ. ∂θ Ii Ii On the other hand,

∂(k) (k) − ∂θ

2

J

dθ ≤

2

J

(k)2 dθ

≤ 2πβ ∗ (t)2 . Therefore, d dt

0

2π

log(k)dθ

∗

2π

(k)dθ + 2πβ ∗ (t)2 2π dL ∗ − = 2β (t) + max + θθ dθ + 2πβ ∗ (t)2 , − θ dt 0 ≤ 2β (t) + max + θθ θ

which implies 2π (θ ) log (θ )k(θ, t)dθ ≤ 0

0

2π

0

(θ) log (θ)k(θ, 0)dθ + C L(0) + T , (2.4)


591

after an integration. Letting kmin (t) = minθ k(θ, t), from (1.12), one has dkmin 3 2 ≥ min(θθ + )kmin + min(θθ + )kmin θ θ dt with kmin (0) > 0. Therefore, kmin (t) cannot decrease to zero faster than exponentially. So k has a positive lower bound in [0, T ). From (2.4), there exists a constant C1 such that (θ )k(θ, t) ≤ C1 except on intervals of length less than 1/4π. However, by (2.3), k is also bounded on these intervals. Finally, we have the following result. Proposition 2.4. The maximal solution of (1.1) with convex initial data either exists for all time or shrinks to a point in finite time. Proof. First notice that L(t) is uniformly bounded when tmax is finite. For p > 1, we compute d Lp Lp−1 dA dL = − L pA dt A dt dt A2 2π 2π 2π p−1 L = kdθ − pA dθ + L dθ + L ds . −pA A2 0 0 0 γ Using Gage’s inequality [10], πL ≤ A

2π

kdθ,

0

which holds for any closed convex curve; we have 2π 2π Lp−1 d Lp ≤ kdθ − pA dθ −A dt A A2 0 0 2π Lp−1 ≤ dθ −πL − pA A2 0 π Lp ≤ C− A A for large p. Here the constant C depends only on , , p, and tmax . It follows that Lp (0) Lp (t) ≤ max C, . A(t) A(0) Hence L(t) must tend to zero as A(t) tends to zero.

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3. An entropy estimate for the normalized flow. In this section, we study the asymptotic behavior of (1.1) in the shrinking case. By the result of the previous sections, the flow shrinks if tmax is finite. In fact, the converse is also true. If tmax = ∞ and γ (·, t) shrinks to a point eventually, then for some large t , γ (·, t ) is contained in a very small circle. By the containment principle (see [5]; see also Lemma 5.3 in Section 5), this implies that tmax is finite, contradicting our assumption. For simplicity, we use ω instead of tmax to denote the extinguishing time. From (1.3), we have ω A(t) 1 2π 1 = (θ)dθ + (θ)ds dt. 2(ω − t) 2 0 2(ω − t) t γ Hence

A(t) 1 2π −→ (θ)dθ as t ↑ ω. 2(ω − t) 2 0 Without loss of generality, we may assume that the flow shrinks at the origin. We dilate γ (·, t) by setting γ˜ (·, t) = (2ω − 2t)−1/2 γ (·, t). The curvature and the enclosed area of γ˜ are given, respectively, by ˜ t) = (2ω − 2t)1/2 k(·, t) k(·, and ˜ = (2ω − 2t)−1 A(t). A(t) Notice that, according to our normalization, A˜ is not necessarily a constant, but 1 2π ˜ (θ)dθ. lim A(t) = t↑ω 2 0 We also rescale the time according to

ω−t 1 . τ = − log 2 ω

˜ τ ) is Then the equation for k˜ = k(θ, 2 (k) ˜ √ ∂ ∂ k˜ ˜ − k˜ + 2ωe−τ k˜ 2 (θθ + ) . = k˜ 2 + k ∂τ ∂θ 2

(3.1)

To study the asymptotic behavior of γ˜ as τ → ∞, we examine the entropy of the normalized flow 2π ˜ (τ ) = ˜ τ )dθ. Ᏹ (θ) log (θ)k(θ, 0


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The concept of entropy for curvature flows has been used in Hamilton [18] and Gage and Li [13]. The rate of change of Ᏹ˜ (τ ) is

(θ ) ∂ k˜ (θ, τ )dθ ˜ τ ) ∂τ k(θ, 0 2π 2 ˜ √ ∂ (k) ˜ − + 2ωe−τ k˜ (θθ + ) dθ (3.2) = k˜ + k ∂θ 2 0 2π √ 2π ˜ dθ, u(θ, τ )dθ + 2ωe−τ k˜ (θθ + ) + 23e−τ k =

d ˜ Ᏹ(τ ) = dτ

2π

0

0

2 (k) ˜ ∂ ˜ u = k˜ + k˜ − − 23e−τ k, ∂θ 2

where

with 3 > 0 to be chosen later. We compute d dτ

0

2π

2π ˜ 2 2 d ∂(k) ˜ dθ k˜ − − − 23e−τ k dτ 0 ∂θ ˜ ˜ 2π ∂ kτ ∂ k 2 k˜ k˜τ − 2 = ∂θ ∂θ 0 ˜ − 23e−τ k˜τ dθ + 23e−τ k

udθ =

∂ 2 k˜ −τ ˜ −τ ˜ ˜ ˜ + k kτ + 3e k − 3e kτ dθ =2 ∂θ 2 0 2 ˜ 2π 2 ˜ ∂ k 2 ∂ k ˜ ˜ ˜ =2 + k k + k − k˜ ∂θ 2 ∂θ 2 0

√ −τ ˜ 2 + 2ωe k θθ + dθ

2π

2π

+

23e 0

2π

=2 0

−τ

˜ kdθ − 23

2π 0

e

−τ

2 ˜ ∂ k 2 k˜ + k˜ − k˜ ∂θ 2

√ + 2ωe−τ k˜ 2 θθ + dθ

2 ˜ 2 2 ˜ ∂ k ∂ k 1 ˜ ˜ ˜ ˜ k + k − k + k ∂θ 2 ∂θ 2 2 ˜ √ −τ ˜ 2 ∂ k + 2ωe k + k˜ θθ + dθ ∂θ 2

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CHOU AND ZHU

2π

+ 23

e

−τ

˜ kdθ − 23

0

2π

e

−τ

0

k˜ 2

∂ 2 k˜ ˜ + k dθ ∂θ 2

2π √ ˜ − 23 e−τ kdθ 2ωe−2τ k˜ 2 (θθ + ) dθ 0 0 2 ˜ 2 2 ˜ 2π ∂ ∂ k k 1 2 ˜ − 2k˜ ˜ =2 k˜ + k + k ∂θ 2 ∂θ 2 0 2 ˜ ∂ k ˜ + k˜ + k ∂θ 2 2 ∂ 2 k˜ −τ ˜ + k − 43e k˜ ∂θ 2 2 ∂ 2 k˜ −τ ˜ + 43e k˜ + k ∂θ 2 ˜ 2 − 43e−τ k ˜ 2 + 2 − 2 + 43e−τ k 2 2 2 −2τ ˜ 2 −2τ ˜ + 43 e k − 43 e k dθ 2π

+ 23

∂ 2 k˜ ˜ 2ωe + k θθ + dθ +2 2 ∂θ 0 2 ˜ 2π 2π −τ ˜ −τ ˜ 2 ∂ k ˜ + 23 e kdθ − 23 e k + k dθ ∂θ 2 0 0 2π √ 2π −2τ 2 −τ ˜ + 23 e kdθ − 23 2ω e k˜ θθ + dθ

2π

√

−τ

k˜ 2

0

2π

1 2 u dθ + 2

2π

0

0

k˜

∂2

0

k˜ ˜ =2 + k dθ ∂θ 2 0 0 2 ˜ 2π 2π −τ ˜ 2 ∂ k ˜ ˜ e k + k dθ − 83 e−τ kdθ + 83 2 ∂θ 0 0 2π 2π −2 dθ − 832 e−2τ k˜ 2 dθ

∂ 2 k˜ ˜ θθ + dθ +2 2ωe k + k 2 ∂θ 0 2 ˜ 2π 2π −τ ˜ −τ ˜ 2 ∂ k ˜ dθ e kdθ − 23 e k + k + 43 ∂θ 2 0 0 √ 2π −2τ 2 − 23 2ω e k˜ θθ + dθ

2π

√

−τ

0

˜2

595 2 ˜ 2π 2π 2π 1 2 −τ ˜ 2 ∂ k ˜ =2 u dθ + 2 udθ + 63 e k + k dθ ∂θ 2 0 0 0 2π − 832 e−2τ k˜ 2 dθ ANISOTROPIC FLOWS FOR CONVEX PLANE CURVES

0

√ + 2 2ω

2π

e

−τ

˜2

k

0

√

− 23 2ω 0

2π

∂ 2 k˜ ˜ θθ + dθ + k 2 ∂θ

e−2τ k˜ 2 θθ + dθ

2π 1 2 u dθ + 2 udθ =2 0 0 2 ˜

2π ∂ k −τ ˜ −τ ˜ ˜ ˜ + 43 e k k + k − − 23e k dθ ∂θ 2 0 2π 2π ˜ + 832 + 43 e−τ kdθ e−2τ k˜ 2 dθ

2π

0 2π

0

2 ˜ √ −τ ˜ 2 ∂ k ˜ 3 + 2ω θθ + dθ +2 e k + k ∂θ 2 0 2π √ 2π −2τ 2 2 −2τ ˜ 2 − 83 e k dθ − 23 2ω e k˜ θθ + dθ

0

0

2π 1 2 −τ ˜ ˜ =2 u dθ + 2 e−τ kdθ 1 + 23e k udθ + 43 0 0 0 2 ˜ 2π √ −τ ˜ 2 ∂ k ˜ +2 e k + k 3 + 2ω + dθ θθ ∂θ 2 0 √ 2π −2τ 2 − 23 2ω e k˜ θθ + dθ. 2π

2π

2π

0

Using

2π

2

23e

−τ

˜ kudθ =2

0

˜ −1/2 udθ 23e−τ 1/2 k

0

≥−

0

2π

1 2 u dθ − 4

2π

32 e−2τ k˜ 2 dθ,

0

we get 2π 2π 2π 2π 1 2 d 2 u dθ + udθ ≥ 2udθ − 43 e−2τ k˜ 2 dθ dτ 0 0 0 0 2 ˜ 2π 2π −τ ˜ −τ ˜ 2 ∂ k ˜ e kdθ + 2 e k + k + 43 ∂θ 2 0 0

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CHOU AND ZHU

√ √ × 3 + 2ω θθ + dθ − 23 2ω Now we choose 3=

√

2π 0

e−2τ k˜ 2 θθ + dθ.

2ω max |θθ + | θ

to get d dτ

0

2π

2π 2π 1 2 2 u dθ + udθ ≥ 2udθ − 63 e−2τ k˜ 2 dθ 0 0 0 2π ˜ e−τ kdθ + 43 0 2π 2 ˜ √ −τ ˜ 2 ∂ k ˜ e k + k 3 + 2ω + dθ. +2 θθ ∂θ 2 0

2π

The last term in the right-hand side of this inequality can be estimated as follows. 2 ˜ 2π √ −τ ˜ 2 ∂ k ˜ 3 + 2ω θθ + dθ e k + k 2 ∂θ 2 0 2π √ ˜ 3 + 2ω θθ + dθ e−τ ku =2 0

+2

2π

0 2π

√ e−τ k˜ + 23e−τ k˜ 3 + 2ω θθ + dθ

√ u √ 2e−τ k˜ 3 + 2ω θθ + dθ √ 2 0 2π √ 2 1 2π 1 2 ≥− u dθ − 2 e−2τ k˜ 2 3 + 2ω θθ + dθ 2 0 0 2π 2π 1 2 1 u dθ − 832 e−2τ k˜ 2 dθ. ≥− 2 0 0 ≥2

Finally, we arrive at 2π 2π d 1 2π 1 2 u dθ + 2 udθ ≥ udθ dτ 0 2 0 0 2π 2 −2τ ˜ 2 e k dθ + 43 − 143 0

2π

0

e

˜ kdθ.

0

Lemma 3.1. We have √ ∞ 2π ˜ τ )dθdτ ≤ L(0) + ω 2ω (θ )e−τ k(θ, 0

(3.3) −τ

2π 0

|(θ)|dθ.


597

Proof. Integrate the equation dL dL dt = dτ dt dτ √ = − 2ω

2π

(θ)e

−τ

˜ τ )dθ − 2ωe−2τ k(θ,

0

2π

(θ)dθ.

0

Lemma 3.2. We have e−τ k˜max (τ ) → 0

as τ → ∞.

Proof. First we claim that vmax (t) = maxθ ((θ)k(θ, t) + (θ)) is nondecreasing for t close to ω. For, as kmax (t) → ∞, we can find t0 such that vmax (t) > max{M, L∞ }

for all t ∈ [t0 , ω),

where M is the constant given in Lemma 2.1. Let t1 and t2 satisfy t0 ≤ t1 < t2 < ω and choose θ1 such that v(θ1 , t1 ) = vmax (t1 ). To prove the claim, it suffices to show v(θ1 , t1 ) ≤ v(θ1 , t)

for t ∈ [t1 , t2 ].

Suppose that, on the contrary, there exists t ∈ (t1 , t2 ) such that v(θ1 , t1 ) > v(θ1 , t ). By Lemma 2.1, there is a first t > t1 such that v(θ1 , t1 ) = v(θ, t ) and v(θ1 , t1 ) > v(θ1 , t) for t ∈ (t , t + ε), ε > 0 small. But then v(θ1 , t) > max{M, L∞ } for all t ∈ [t1 , t ], and by Lemma 2.1 again, ∂v/∂t ≥ 0 for all t in [t , t + ε); the contradiction holds. Hence vmax (t) is nondecreasing for t ≥ t0 . From Lemma 3.1, we have ∞ 2π √ (θ)e−2τ 2ωk θ, ω − ωe−2τ dθdτ ≤ C. 0

0

By (2.2),

∞√

2ωe−2τ vmax (τ )dτ ≤ C .

0

However, since vmax is nondecreasing, there is τ0 > 0 such that vmax (τ )

∞√ τ

2ωe

−2τ

≤

∞ τ

√ e−2s 2ωvmax (s)ds,

τ ≥ τ0 .

Therefore, e

−τ

k˜

(τ ) ≤ 2 max

∞√ τ

2ωe−2s vmax (s) ds

√ + 2ωe−2τ max |(θ)| → 0 θ

as τ → ∞.

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CHOU AND ZHU

Now we apply Lemma 3.2 to (3.3) to get d dτ

0

2π

udθ ≥

1 2

u2 dθ + 2

2π

0

2π

+ 43 0

1

2π

udθ

0

˜ ˜ − 43e−τ k)dθ e−τ k(1

≥ 2π 2 0 dθ

2π

2 udθ

+2

0

2π

udθ,

τ ≥ τ0 ,

0

for large τ0 . It follows that

2π

u(θ, τ )dθ ≤ 0,

τ ≥ τ0 .

0

Therefore, d ˜ Ᏹ(τ ) ≤ 23 dτ

2π

e

−τ

√ ˜ kdθ + 2ω

0

0 2π

≤ 33

2π

e−τ k˜ (θθ + ) dθ

˜ e−τ kdθ.

0

By Lemma 3.1, Ᏹ˜ (τ ) is uniformly bounded for τ ≥ τ0 . We have proved the following result. Proposition 3.3. Suppose that the maximal solution of (1.1) shrinks to a point. Then the entropy of the normalized flow γ˜ is uniformly bounded in [0, ∞). 4. Convergence of the normalized flow. In this section, we show that the normalized flow subconverges to self-similar solutions of ∂γ /∂t = kN. To establish this, we first use the entropy estimate obtained in the last section to bound the normalized curvature from above. Next, by expressing the flow as polar graphs, we obtain a positive lower bound for the normalized curvature via the Harnack inequality of Krylov and Safonov. After obtaining the two-sided bounds on the normalized curvature, its higher-order estimates fall out from parabolic theory. Subconvergence of the normalized flow then follows easily by the construction of a Liapunov function. Lemma 4.1. Suppose the solution of (1.1) shrinks to a point. Then the curvature of the normalized flow is uniformly bounded with a bound depending only on , , and γ0 . Proof. According to Lemma 6.1 in [13], the minimum width w(τ ) of γ˜ (·, τ ) satisfies w(τ ) ≥ exp C − Ᏹ˜ (τ )/2πmin ,


599

where C depends on only. By Lemma 6.3 in [13] and Proposition 3.3, the inradius of γ˜ satisfies 1 r˜in (τ ) ≥ w(τ ) 3 ≥ r0 > 0,

τ ∈ [0, ∞).

˜ ) tends to a positive constant when τ approaches infinity, this estimate implies As A(τ that the diameter of γ˜ satisfies ˜ ) ≤ D0 < ∞, D(τ

τ ∈ [0, ∞).

For a fixed t, let θ0 satisfy (θ0 )k(θ0 , t) = (k)max (t). By (2.3), ˜ τ ) + Ce−τ , C constant, k˜ (τ ) ≤ 2(θ)k(θ, max

for θ, |θ − θ0 | ≤ 1/4π . By Lemma 1.4, k˜max (τ ) has a uniform positive lower bound. Therefore, we have

k˜

max

˜ τ ), (τ ) ≤ 4(θ)k(θ,

τ large, and |θ − θ0 | ≤

Now, from the entropy estimate, 2π log k˜ dθ C≥ 0 ˜ log k dθ + = |θ −θ0 |≤1/4π

|θ−θ0 |>1/4π

1 . 4π

log k˜ dθ

1 1 ˜ ) min log k˜ max (τ ) − e−1 L(τ 2π 4 1 1 ≥ min log k˜ max (τ ) − 4e−1 D0 . 2π 4

≥

Hence k˜ is uniformly bounded. Notice that we have used x log x ≥ −e−1 for x > 0. ˜ Let h(θ, ˜ t) = (2ω)−1/2 eτ h(θ, t) be Next we derive a positive lower bound for k. the support function for γ˜ . Then h˜ satisfies the equation √ ∂ h˜ ˜ = − k˜ + 2ωe−τ + h. ∂τ

(4.1)

Fix τ0 > 0. From our estimate on the inradius of γ˜ , we know that we can put a disk of radius r0 inside γ˜ (·, τ0 ). We take the center of this disk as the origin. Notice that γ˜ (·, τ0 ) is now contained in a concentric disk of radius D0 . By Lemma 4.1, we have −τ ∂e h˜ sup (θ, τ ) ≤ Ce−τ ∂τ θ

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CHOU AND ZHU

˜ Integrating for some constant C depending only on , , and an upper bound for k. this inequality gives −τ e 1 h(θ, ˜ τ1 ) − e−τ0 h(θ, ˜ τ0 ) ≤ C e−τ1 − e−τ0 . Therefore, we can find 5 > 0 independent of τ0 such that r0 ˜ τ ) ≤ 2D0 , ≤ h(θ, 2

τ ∈ [τ0 , τ0 + 5].

(4.2)

Now consider γ˜ as polar graphs as in Section 1. The normalized curvature k˜ = √ 2ωe−τ k satisfies √ √ ∂ k˜ ∂ 1 ∂ k˜ 3θ k˜ ∂ r˜ /∂α(k˜ + 2ωe−τ ) 2ωe−τ θ ∂ k˜ − = + ∂τ ∂α D˜ ∂α D˜ ∂α D˜ D˜ r˜ D˜ √ ˜ + k˜ 2 (θ θ + ) k˜ + 2ωe−τ (θθ + ) − k, τ ∈ [τ0 , τ0 + 5], (4.3) where D˜ = [(∂ r˜ /∂α)2 + r˜ 2 ]1/2 . ˜ τ ) ≥ k0 > 0 for all (θ, τ ), where k0 only depends on , , Lemma 4.2. k(θ, and r0 . Proof. From (4.2), we know that ∂ r˜ ≤ constant. ∂α Also, since

−˜r ∂ 2 r˜ /∂α 2 + 2(∂ r˜ /∂α)2 + r˜ 2 k˜ = D˜ 3

and k˜ is bounded,

2 ∂ r˜ ∂α 2 ≤ constant.

Hence k˜ satisfies a uniformly parabolic equation with bounded coefficients in R×[τ0 , τ0 + 5]. We need the following version of Harnack’s inequality due to Krylov and Safonov [19]: Suppose u is a nonnegative solution of ∂ 2u ∂u ∂u = a 2 +b + cu, ∂τ ∂α ∂α

for all (α, τ ) ∈ (−1, 1) × (0, 2),

where a, b, c are measurable, and 0 < λ ≤ a(α, τ ), a(α, τ ), |b(α, τ )|, and |c(α, τ )| are bounded by M. Then there exists N = N(λ, M) > 0 such that 1 u(0, 1) ≤ u(α, 2), N

1 for |α| ≤ . 2


Now, we let

601

5 5 ˜ ˜ max k α, τ0 + = k α0 , τ 0 + . α 2 2

˜ Applying Harnack’s inequality to u(α, τ ) ≡ k(2πθ + θ0 , τ0 + (5/2)τ ) we conclude 5 1 ˜ ˜ τ0 + 5) k α0 , τ0 + ≤ k(α, N 2 for all α, |α − α0 | ≤ π . By Lemma 1.4, we have ˜ τ0 + 5) ≥ C k(α, N for all α ∈ [0, 2π]. Since τ0 is arbitrary and 5 is independent of τ0 , this means that k˜ has a positive uniform lower bound. We are now in position to show subconvergence. In the previous works [12] and [13], the functional H τ, (a, b) =

2π

(θ) log h˜ θ, τ ; (a, b) dθ,

0

˜ τ ; (a, b)) is the support function of γ˜ using (a, b) as the origin, is the where h(θ, Liapunov function. It is important to choose (a, b) such that h˜ is positive and H is bounded. In the case ≡ 0, Gage and Li [13] made the choice based on a monotonicity formula due to Firey. In our general case, this monotonicity formula is unlikely to be valid. With the two-sided bounds of k˜ at our disposal, we can construct another Liapunov function. Since k˜ is uniformly bounded away from zero, we may assume √ k˜ + 2ωe−τ ≥ k0 > 0

(4.4)

for all τ > 0. Then γ is nesting, and, if we fix the point at which γ shrinks as the origin, h is always positive. We claim that in fact, h˜ ≥ k0 /2 for all τ . For, if h˜ < k0 /2 ˜ 0 , τ0 + 1) < 0. But this is for some (θ0 , τ0 ), by (4.1) and (4.4) one must have h(θ impossible. Hence h˜ ≥ k0 /2 for all τ . Consider the functional

2π

J (τ ) = 0

˜ 2

∂h − h˜ 2 + 2 log h˜ + 23e−τ dθ, ∂θ

where 3 is a constant to be determined. We have

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CHOU AND ZHU

h˜ τ ∂ 2 h˜ ˜ ˜ −τ − 3e + h h + dθ τ ∂θ 2 h˜ 0 2π 2π 1 ˜ ˜ ˜ √ −τ ˜ k − h k + 2ωe − h − 2 3e−τ dθ = −2 k˜ h˜ 0 0 2π 2π 2π 1 ˜ ˜ 2 2ω −2τ 2 k − h + e dθ − 2 ≤− 3e−τ dθ ˜ ˜ ˜ ˜ k h k h 0 0 0 2π 2ω −τ 1 ˜ ˜ 2 k − h + 4π 2 e ||2max − 3)e−τ ≤− k0 k˜ h˜ 0

dJ =2 dτ

2π

−

≤ 0, if 3 = 2ω/k02 ||2max . Hence dJ ≤− dτ

2π 0

1 ˜ ˜ 2 k − h ≤ 0. ˜k h˜

(4.5)

From parabolic theory, we have a uniform Hölder estimate on k˜ (in θ and τ ). From the boundedness of J , we immediately deduce 0 = lim

τ →∞

dJ . dτ

˜ we conclude Furthermore, from the estimates for h˜ and k, 2π 2 lim k˜ − h˜ dθ = 0. τ →∞ 0

(4.6)

Proposition 4.3. Any sequence γ˜ (·, τj ), τj → ∞, contains a subsequence converging smoothly to a self-similar solution of the flow ∂γ = kN. ∂t

(4.7)

˜ τj ) are Proof. By Lemma 4.1 and Lemma 4.2, we know that γ˜ (·, τj ) and k(·, ˜ uniformly bounded. Also, k(·, τj ) has a positive uniform lower bound. By parabolic theory, we infer all higher-order bounds on γ˜ (·, τj ). Hence we may extract a convergent subsequence from γ˜ (·, τj ), which, by (4.6), must converge smoothly to a solution of the equation ˜ k˜ = h. In other words, its limit is a self-similar solution of (4.7). Remark 4.4. We would like to emphasize that the asymptotic results we developed and proved in this and the last three sections apply to the flow (1.1), where > 0 and are smooth 2π -periodic functions. In particular, the requirement that is of the form F /β, which implies that is either identically zero or of the same sign, is not needed.


603

Remark 4.5. Under the symmetry condition (θ + π) = (θ), it was proved in [11] that a self-semilar solution to (4.7) is unique up to homothety and translations. It is not hard to show that in this case, in fact, γ˜ (·, τ ) converges smoothly to a self-similar solution of (4.7). One may use the idea in Simon [21] to show that this is also true in the general case. However, we do not explore this point here. 5. Convergence to a stationary solution. In this section, we write (1.1) in the form β(θ)

∂γ = g(θ)k + F N ∂t

(5.1)

as in [4]. We always assume β, g > 0 in C ∞ S 1 and F ∈ R. The equation is subject to the initial condition γ (·, t) = γ0 ,

(5.2)

in which, without loss of generality, γ0 is assumed to be uniformly convex. In this section, we show that to each γ0 , there corresponds a unique negative value F ∗ such that the solution of (5.1) and (5.2) shrinks or expands depending on whether F > F ∗ or F < F ∗ . Also, after a certain modification which does not change the shape of γ (·, t), the modified flow of (5.1)F ∗ and (5.2) converges to a limiting convex curve. We begin by stating a comparison principle that is a direct consequence of the maximum principle applied to the equation of the flow satisfied by its support function. In fact, it remains valid even in the nonconvex case. (See the “containment principle” in [4] and [5].) Lemma 5.1. (a) Let γ1 and γ2 be two solutions of (5.1), where γ1 (·, 0) is contained inside γ2 (·, 0). Then γ1 (·, t) is contained inside γ2 (·, t) for t > 0. (b) Let γ and γ be solutions of (5.1)F , (5.2) and (5.1)F , (5.2), respectively. Suppose that F ≤ F . Then γ (·, t) is contained inside γ (·, t) for t > 0. Lemma 5.2. Let γ be a solution of (5.1) and (5.2) with negative F . Denote the curvature of γ0 by k0 . (a) If g(θ )k0 (θ ) + F > 0 for all θ, then tmax is finite and γ (·, t) shrinks to a point. (b) If g(θ )k0 (θ ) + F < 0 for all θ, then tmax = ∞, and the domain bounded by γ (·, t) occupies the entire plane as t approaches infinity.

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CHOU AND ZHU

Proof. From (2.1), the function vmin (t) = minθ [(g(θ)k(θ, t) + F )/β(θ)] is nondecreasing. In particular, for negative F , g(θ)k(θ, t) > β(θ)v(θ, t) ≥ β(θ)vmin (0) > 0. So k(θ, t) has a uniform positive lower bound. Substituting this into (2.1) yields dvmin ≥ c0 vmin , dt

for some c0 > 0.

If tmax = ∞, vmin grows at least exponentially. From (2.1), we have dvmin 3 ≥ c1 vmin , dt

for some c1 > 0,

for large t. Hence vmin blows up in finite time, and the contradiction holds. So tmax must be finite. Similarly, by looking at (2.1), one can see that vmax (t) is nonincreasing. In particular, vmax (t) ≤ vmax (0) < 0. So k is uniformly bounded and tmax = ∞. Finally, from (1.11), we have ∂h (θ, t) = −v(θ, t) ≥ −vmax (0) > 0, ∂t that is, h(θ, t) ≥ −vmax (0)t + h(θ, 0). Thus γ (·, t) encloses any bounded set for sufficiently large t. By combining Lemmas 5.1 and 5.2, we have the following. Lemma 5.3. Let γ be the solution of (5.1) and (5.2) and F < 0. (a) If γ0 is contained inside a circle of radius less than minθ (−g(θ)/F ), then γ shrinks to a point. (b) If γ0 contains a disk of radius greater than maxθ (−g(θ)/F ), then γ expands to infinity. Define I1 = {F : γ is a solution of (5.1)F and (5.2) that shrinks to a point}, and I2 = {F : γ is a solution of (5.1)F and (5.2) that expands to infinity as t → ∞}.


605

By the continuous dependence of γ on F , both I1 and I2 are open. By Lemma 5.2, they are also nonempty. Using Lemma 5.1, we conclude that I1 = (F¯ , ∞) and I2 = (−∞, F ), where −∞ < F ≤ F¯ < 0. We can show that for each F ∈ [F , F¯ ], the ¯ ¯ ¯ flow (5.1)F and (5.2) exists for all time, with curvature uniformly pinched between two positive constants. First of all, we claim that there exists l0 > 0 such that L(t) ≥ l0 ,

∀t ∈ [0, ∞).

(5.3)

For, if L(t) is so small that γ (·, t) is contained inside a circle of radius less than minθ (−g(θ )/F ), then γ (·, t) shrinks to a point according to Lemma 5.3, contradicting the assumption F ∈ [F , F¯ ]. Hence (5.3) must hold. Next we claim that there exists ¯ L0 > 0 such that L(t) ≤ L0 ,

∀t ∈ [0, ∞).

(5.4)

We compute 2π 2π 2π d L2 L = 2 − 2A kdθ − 2A dθ + L dθ + L ds dt A A 0 0 0 γ L 2 ≤ 2 αA − βL A βL α L2 − = A β A for some α, β depending only on g, β, k0 , and l0 . Hence α L2 L2 (t) ≤ max , (0) . A β A

(5.5)

In the case when L(t) becomes very large, we can inscribe a large disk inside γ (·, t). However, by Lemma 5.3, γ (·, t) would expand to infinity, again contradicting the assumption F ∈ [F , F¯ ]. Hence (5.4) must hold. ¯ (5.3) and (5.5) together imply that the inradius of γ (·, t) satisfies rin (t) ≥ r0 ,

∀t ∈ [0, ∞),

(5.6)

for some positive r0 . Lemma 5.4. Let γ satisfy (5.1)F and (5.2) for F ∈ [F , F¯ ]. Then k(·, t) is uniformly ¯ bounded in [0, ∞). Proof. Recall the definition of the “support center” of a convex curve introduced in Chou [7]. Let h be the support function of a convex curve. The support center of the convex curve is defined to be 2π c= h(θ)xdθ, x = (cos θ, sin θ). 0

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It is easy to see that h(θ) − c · x ≥ ρrin for some absolute constant ρ > 0. Let c(t) be the support center of γ (·, t). We consider the auxilliary function w(θ, t) =

−∂h/∂t (θ, t) . h(θ, t) − c(t) · x − ρr0 /2

Notice by (5.6) that w is well defined. For any T > 0, suppose that the maximum of w over [0, 2π ] × [0, T ] is attained at (θ0 , t0 ), t0 > 0. At (θ0 , t0 ), we have

∂w −∂ 2 h/∂t∂θ ∂h/∂t ∂h/∂θ − c · (− sin θ, cos θ) + 0= = , ∂θ (hˆ − δ)2 hˆ − δ 0≤

∂w −∂ 2 h/∂t 2 ∂h/∂t (∂h/∂t − dc/dt · x) + = , ∂t (hˆ − δ)2 hˆ − δ

and 0≥

∂ 2 w −∂ 3 h/∂t∂θ 2 ∂h/∂t (∂ 2 h/∂θ 2 + c · x) + = , ∂θ 2 hˆ − δ (hˆ − δ)2

where hˆ = h − c · x and δ = ρr0 /2. Using (1.11), (∂h/∂t)2 ∂h/∂t dc/dt · x ∂ 2 h/∂t 2 ≥ + hˆ − δ (hˆ − δ)2 (hˆ − δ)2 ∂h/∂t dc/dt · x k 2 (∂ 3 h/∂t∂θ 2 + ∂h/∂t) ≥ + hˆ − δ (hˆ − δ)2 2ˆ 2 k ∂ h ˆ 1 ∂h ∂h dc ≥ + ·x +h−δ . 2 2 ∂t ∂t dt (hˆ − δ)2 (hˆ − δ) ∂θ In other words, w2 +

k 2 δ dc/dt · x kw ≥ w+ (−w). ˆ ˆ (h − δ) (h − δ) (hˆ − δ)

Using ρr0 ≤ hˆ ≤ L0 and

dc ≤ C(1 + k), dt

we conclude that w and hence k are uniformly bounded. Thus k is uniformly bounded from above.


607

The upper bound on k controls the speed of the support center. Using c(t) as the origin, there exist positive 5 and d0 independent of t such that dist(c(t), γ (·, s)) ≥ d0 for all s in [t, t + 5]. We may represent γ as polar graphs, arguing as in the previous section to obtain a uniform positive lower bound for k. By parabolic theory, we also obtain uniform estimates on all derivatives of k in θ and t. Thus we have the following lemma. Lemma 5.5. The curvature of the solution of (5.5)F and (5.2), F ∈ [F , F¯ ], is ¯ pinched between two positive constants for all time. Furthermore, derivatives of the curvature are also uniformly bounded for all time. However, the validity of Lemma 5.5 does not imply that the flow converges. In fact, if ∂γ /∂t approaches zero and the flow converges to a convex curve as t approaches infinity, then the limit curve must satisfy g(θ)k + F = 0. This implies that g must satisfy

2π

g(θ)eiθ dθ = 0

(5.7)

0

or, equivalently, the physical condition g=

d 2f +f dθ 2

for some f . Thus unless (5.7) is fullfilled, one does not expect the convergence of the flow. We modify the flow in the following manner. Let D = c = (c1 , c2 ) : c = |c|(cos θ, sin θ), 0 ≤ |c| < −F /β(θ) and consider the map from D to ᏾n given by 2π g(θ)eiθ dθ . c " −→ β(θ)(c1 cos θ + c2 sin θ) + F 0 It is readily verified that the map is a diffeomorphism between D and the plane. As a result, we can find a unique point c∗ in D that satisfies 2π g(θ)eiθ dθ . (5.8) (0, 0) = ∗ β(θ)(c1 cos θ + c2∗ sin θ) + F 0 Let h be the support function of (5.1)F and (5.2) for F ∈ [F , F¯ ]. We set

hˆ = h − c∗ , (cos θ, sin θ) t.

¯

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Then β where

∂ hˆ = − g kˆ + Fˆ , ∂t

Fˆ = F + c∗ , (cos θ, sin θ) β

ˆ Geometrically, γˆ and kˆ = k is the curvature of γˆ , the convex curve determined by h. is a shift-back of γ in the direction c∗ /|c∗ | over a distance |c∗ |t. ˆ Consider the functional I for h:

2π

I (t) = 0

ˆ t)dθ 1 g(θ )h(θ, − ∗ β(θ ) c , (cos θ, sin θ) + F 2

2π

0

ˆ

∂h − hˆ 2 dθ. ∂θ

By (5.8), the first integral in I is independent of the choice of the origin. Consequently, it is bounded by a constant multiple of the diameter of γˆ , which is uniformly bound by (5.4). The second term in I , 1 − 2

2π

0

ˆ

∂h 1 2π 2 ˆ −h = ∂θ 2 0

hˆ dθ, kˆ

is the area enclosed by γˆ . Therefore, I is uniformly bounded for all t. Now dI =− dt

ˆ ˆ 2 gk + F ≤ 0. β Fˆ kˆ

(5.9)

By Lemma 5.5, it follows that ∂ hˆ sup (θ, t) → 0 ∂t θ

as t → ∞.

(5.10)

To furnish the last step in proving convergence, we consider the “modified support center” 2π g(θ)β(θ) ˆ ˆ d(t) = h(θ, t)(cos θ, sin θ)dθ Fˆ 2 (θ) 0 ˆ may not lie inside γˆ (·, t). of γˆ . Notice that d(t) ˆ Lemma 5.6. There exists d(∞) ∈R2 such that ˆ = d(∞). ˆ lim d(t)

t→∞


609

Proof. By the necessary condition in the classical Minkowski problem and (5.8), 2π eiθ (0, 0) = ˆ t) k(θ, 0 2π geiθ ! =− ˆ 0 Fˆ 1 + ∂∂th βˆ F ˆ 2 2π iθ 2π ∂h ge gβeiθ ∂ hˆ =− + +O 2 ˆ ˆ ∂t ∂t F F 0 0 2π 2 iθ ∂ hˆ gβe ∂ hˆ +O = ∂t Fˆ 2 ∂t 0 ˆ for sufficiently small ∂ h/∂t. Therefore, by (5.9), ˆ 2π gβeiθ ∂ hˆ d d(t) dt = Fˆ 2 ∂t 0 2π ˆ 2 ∂h ≤C ∂t 0 dI ≤ C . dt Consequently,

ˆ − d(s)| ˆ |d(t) ≤ C I (t) − I (s) → 0

as t, s → ∞. Lemma 5.7. γˆ (·, t) is uniformly bounded in the plane. Proof. For each t, we can write

hˆ = h + l, (cos θ, sin θ) ,

where h is positive and l ∈R2 . It suffices to show that the l’s are uniformly bounded. By rotating axes, we may assume that l1 = |l| and l2 = 0. Then 2π

gβ ˆ d1 (t) = h + l · (cos θ, sin θ) cos θdθ 2 ˆ F 0 2π 2π gβ gβ h cos θdθ + |l| cos2 θdθ. = Fˆ 2 Fˆ 2 0 0 Thus

by (5.4) and Lemma 5.6.

|l| ≤ C |dˆ1 (t)| + L(t) ˆ ≤ C 1 + |d(∞)| + L0

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Now, we may use the Blaschke selection theorem and Lemma 5.5 to conclude that, for any sequence {γˆ (·, tj )}, tj → ∞, there exists a subsequence converging smoothly to a stationary solution, that is, a curve that satisfies gk + Fˆ = 0.

(5.11)

To show that the convergence is uniform, it is sufficient to show that all limit curves are identical. Recall that all stationary solutions are the same up to translations. Suppose γˆ (·, tj ) and γˆ (·, sj ) converge to γ1 and γ2 , respectively. The support functions of γ1 and γ2 , h1 and h2 , satisfy h1 − h2 = l · (cos θ, sin θ) for some l. From 2π gβ (l1 cos θ + l2 sin θ)(cos θ, sin θ)dθ Fˆ 2 0 2π 2π gβ gβ (cos θ, sin θ)h1 dθ − (cos θ, sin θ)h2 dθ = 2 ˆ F Fˆ 2 0 0 ˆ j ) − lim d(s ˆ j) = lim d(t tj →∞

tj →∞

= (0, 0), we conclude that l = (0, 0). Finally, we show that F = F¯ . Let γ and γ¯ be the modified flow of (5.1)F for ¯ ¯ by Lemma 5.1 (b), F = F and F¯ , respectively. On one hand, ¯

γ (∞) = lim γ (·, t) t→∞ ¯ ¯ = lim γ¯ (·, t)

(5.12)

t→∞

= γ¯ (∞). ¯ However, on the other hand, consider the equation satisfied by h − h: ∂ h − h¯ ¯

∂t

 g = k k¯  β¯

2

∂ h − h¯ ¯

∂θ 2



¯

1 + h − h¯  + (F¯ − F ). ¯ β ¯

¯ If F¯ > F , then minθ (h − h)(t) becomes positive and is nondecreasing for t > 0. Hence ¯ ¯ ¯ h(θ, ∞) > h(θ, ∞), contradicting (5.12). So we must have F¯ = F . In summarizing, ¯ ¯ we have proved the following result. Proposition 5.8. For every convex initial curve γ0 , there exists F ∗ < 0 such that the following hold.


611

(a) The flow (5.1)F shrinks to a point in finite time when F > F ∗ . (b) The flow (5.1)F ∗ , modified to γˆ , converges to a solutions of (5.11) smoothly. (c) The flow (5.1)F expands to infinity as t → ∞ when F < F ∗ . There is an alternative formulation for what we have proved in this section. Consider the equation (5.1) subject to the initial condition γ (·, t) = λγ0 ,

(5.13)

where λ > 0 is a parameter. Then we have the following. Proposition 5.9. There exists λ∗ > 0 such that the following hold. (a) The flow (5.1) and (5.13)λ shrinks to a point in finite time when λ < λ∗ . (b) The flow (5.1) and (5.13)λ∗ , modified to γˆ , converges to a solution of (5.11) smoothly. (c) The flow (5.1) and (5.13)λ expands to infinity as t → ∞ when λ > λ∗ . This proposition provides a global, nonlinear version of the linearized stability result for the stationary solution discussed in [5]. Roughly speaking, the set {λ∗ (γ0 )γ0 : γ0 convex} forms a submanifold of codimension 1 in the Banach manifold of convex curves. It is transversal to the dilatation vector field and separates the Banach manifold into two parts. The flow (2) brings any point in the interior of this submanifold to the origin and sends any point in the exterior of this submanifold to infinity. For points lying on the submanifold, the flow is confined to it and converges to a unique point corresponding to the stationary solution of (5.1). In fact, the stationary solution is the global minimum of I over the submanifold. 6. Convergence to a Wulff shape. According to Proposition 5.8, the flow γ (·, t) of (5.1)F , F < F ∗, and (5.2) possesses the following property: For any bounded subset K of the plane, there exists tK such that K is contained inside γ (·, t) for all t ≥ tK . In this section, we study the asymptotic behavior of this flow. Intuitively in this case, the curvature of the flow is eventually negligible. In view of (1.2), the length of γ (·, t) grows linearly. Therefore, we consider the normalization given by γ˜ (·, t) = γ (·, t)/t. It was conjectured in [4] that γ˜ (·, t) converges to the boundary of the Wulff region of −F /β: W (−F /β) = (x, y) = r(cos α, sin α) : 0 ≤ r < −F /β(α) as t → ∞. Subsequently, Soner [22] confirmed this conjecture by showing that there exist two functions a(t) and b(t) satisfying b(t) − a(t) = o t −1

as t → ∞

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CHOU AND ZHU

such that a(t)W (1/β) ⊆ interior of γ˜ (·, t) ⊆ b(t)W (1/β). This result was strengthened in [5] to the uniform convergence of the corresponding support functions. Notice that it is in fact valid for a class of nonconvex curves. We can establish more precise convergence when γ0 is convex and the polar graph of β is uniformly convex. The latter condition means that the Wulff region W (1/β) has a uniformly convex boundary. One can directly verify that this is equivalent to the inequality 1 d2 1 + 2 > 0. β dθ β

(6.1)

Proposition 6.1. Let γ be a solution of (5.1)F , F < F ∗, and (5.2). Suppose that (6.1) holds. Then ˜ t) + (·) = O t −1 log t , h(·, t → ∞, uniformly, where h˜ is the support function of γ˜ . Moreover, ˜ t) → 1 − (θθ + ) k(·, uniformly, and d (n) ˜ (n) (θθ + ) k(·, t) dθ

= O(t −α ), L∞

uniformly for any n ≥ 1 and α ∈ (0, 1) as t → ∞. Here we have used = g/β and = F /β. We show at the end of this section ˜ that (6.1) is necessary for C 2 -convergence of h. By introducing the new time scale, τ = log t, the equations for h˜ and k˜ are, respectively, given by −

∂ h˜ = e−τ k˜ + + h˜ ∂τ

(6.2)

and ∂2 ∂ k˜ ˜ = k˜ 2 2 (e−τ k˜ + ) + k˜ 2 (e−τ k˜ + ) + k. ∂τ ∂θ

(6.3)


613

Lemma 6.2. There exists a k0 that depends on and only such that k˜ is uniformly bounded in [0, 2π ] × [0, ∞) if max k(θ, 0) ≤ k0 . θ

Proof. From (1.12), we have 2 dkmax ≤ max (θθ + )kmax (t) + θθ + kmax . θ dt Hence if

max (θθ + )kmax (t) + θθ + < 0, θ

kmax is decreasing in t and dkmax 2 ≤ −c0 kmax (t). dt This implies kmax (t) ≤

kmax (0) , 1 + c0 kmax (0)t

t ∈ [0, ∞).

Lemma 6.3. Let h˜ C be the support function of the flow (5.1)F , F < 0, of which the initial curve is a large circle. Then h˜ C (·, t) + (·) → 0 uniformly as t → ∞. Proof. We integrate (6.2) to get eτ h˜ C (θ, τ ) + (θ) = h˜ C (θ, 0) + (θ) −

τ

˜ s)ds. (θ)k(θ,

0

When the initial circle is so large that d 2 (θ) + (θ) < 0, max (θ θ + )kmax (0) + θ dθ 2 it follows from Lemma 6.2 that k˜ is uniformly bounded. Consequently, we have h˜ C (·, τ ) + (·) = O τ e−τ uniformly as τ → ∞. ˜ τ ) + (·) tends to zero uniformly as τ → ∞. Lemma 6.4. h(·, Proof. Since γ (·, t) expands, we may assume, without loss of generality, that γ (·, t) is pinched between two large circles. Then the desired result follows from Lemma 6.3.

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Lemma 6.5. k˜ is uniformly bounded in [0, 2π] × [0, ∞). Proof. First, we claim that for ε > 0, there exists τ ∗ such that max k(θ, τ ∗ ) ≤ ε. θ

For, given ε > 0, we divide [0, 2π] into approximately [1/ε]-many subintervals Ij of length equal to ε. For τ > 0, we integrate τ +1 ˜ s)ds = − + h˜ (θ, τ + 1)eτ +1 + + h˜ (θ, τ )eτ (θ )k(θ, τ

to get

τ +1 τ

Ij

˜ s)dθds (θ )k(θ,

˜ ˜ τ + 1)dθ + eτ ( + h)(θ, τ )dθ, = −eτ +1 ( + h)(θ, Ij

where

is the average over Ij . Summing over Ij ’s, we get

Ij

Ij

τ +1





τ

Ij

j

= −e

˜ s)dθ + (θ )k(θ, Ij

j

+e

2π

˜ s)dθ  ds (θ)k(θ,

0

τ +1

τ



j

Ij

˜ ( + h)(θ, τ + 1)dθ +

2π

˜ ( + h)(θ, τ + 1)dθ

0

˜ ( + h)(θ, τ )dθ +

2π

˜ ( + h)(θ, τ )dθ .

0

By the mean-value theorem, there exists τ ∗ ∈ (τ, τ + 1) such that 2π (θ )k(θ, τ ∗ )dθ + (θ)k(θ, τ ∗ )dθ j

Ij

0

≤ 2e 1 +

1 ε

2π

+ h˜ (θ, τ )dθ +

2π

+ h˜ (θ, τ + 1)dθ

0

0

+ max + h˜ (θ, τ ) + + h˜ (θ, τ + 1) . θ

Since by Lemma 6.4, + h˜ tends to zero uniformly, for sufficiently large τ , we have 2π ∗ (θ )k(θ, τ )dθ + (θ)k(θ, τ ∗ )dθ < ε. j

Ij

0


615

Suppose that maxθ k(θ, τ ∗ ) is attained at θ = θ0 , which belongs to some Ij . We have ∗ maxk(θ, τ ) − k(θ, τ ∗ )dθ θ Ij ∂k ∗ ε (·, τ ) ≤ ∞ ∂θ L ≤ Cε by Lemma 2.2. Therefore max k(θ, τ ∗ ) ≤ (1 + C)ε. θ

Now, we can apply Lemma 6.2 to γ , using τ = τ ∗ as the initial time, to conclude that k˜ is uniformly bounded. To furnish higher-order convergence, we look at the equation satisfied by ˜ w = − (θθ + ) k. By a direct computation, ∂w ∂ 2w ∂w = e−τ k˜ 2 2 + e−τ B + e−τ Cw + w(1 − w), ∂τ ∂θ ∂θ

(6.4)

where

(θ θ + )θ B = 2k˜ 2 θ + , −(θ θ + ) 2(θθ + )2θ (θθ + )θθ (θθ + )w 2 + ) ( θ θ θ θ 2 C = k˜ + + + . −(θ θ + ) −(θθ + ) (θθ + )2 (θθ + )2

Therefore, by Lemma 6.5, we have dwmax ≤ Ae−τ + wmax (1 − wmax ) dτ and dwmin ≥ −Ae−τ + wmin (1 − wmin ) dτ for some constant A. It follows that w tends to 1 uniformly as τ approaches infinity. By differentiating (6.4), we see that u = ∂w/∂θ satisfies the equation ∂u ∂ 2u ∂u ∂C = e−τ A 2 + B + C u + e−τ w + (1 − 2w)u, ∂τ ∂θ ∂θ ∂θ

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where A , B , and C are uniformly bounded in [0, 2π] × [0, ∞). Since w tends to 1 uniformly, it is easy to see that for any α ∈ (0, 1), ∂w (·, τ ) = O e−ατ ∂θ

uniformly as τ → ∞.

Higher-order estimates can be obtained in the same way. The proof of Proposition 6.1 is completed. ˜ Finally, we want to show that (6.1) is necessary for the smooth convergence of h. More precisely, we have the following proposition. ˜ t) converges to some h˜ in C 2 -norm, then (6.1) holds. Proposition 6.6. If h(·, Proof. From (6.3),

d k˜min ˜ −τ ˜ 2 ˜ ≥ kmin 1 + kmin min{θθ + } + e kmin min{θθ + } ; θ θ dτ ˜ t) tends to h˜ in C 2 -norm, the we know that k˜min has a positive lower bound. When h(·, convex curve determined by h˜ must have positive curvature. However, by Lemma 6.4, h˜ is equal to −F /β. Hence (6.1) holds. Appendix Let γ (·, t) be a family of immersed closed plane curves evolving according to ∂γ = F (θ, k)N, ∂t where N is the inner unit normal at γ (·, t), and θ is the angle between the unit tangent vector and the positive x-axis. We derive the evolution equations for some basic geometric quantities of γ . Let γ (u, t) : [a, b] →R2 be a fixed parametrization of γ for each t. We first compute the evolution of v ≡ ∂γ /∂u, ∂γ /∂u1/2 : ∂γ ∂ ∂γ ∂γ ∂ ∂γ ∂γ ∂ ∂v 2 =2 , =2 , =2 , (F N) = −2F v 2 k. ∂t ∂u ∂t ∂u ∂u ∂u ∂t ∂u ∂u Hence ∂v = −F kv. ∂t Notice that we have used the second of the Frenet formulas: ∂T = vkN, ∂u

∂N = −vkT . ∂u

(A.1)


Next, we find the time derivatives of T and N : ∂ 1 ∂γ ∂T = ∂t ∂t v ∂u 1 ∂v ∂γ 1 ∂ =− 2 + (F N) v ∂t ∂u v ∂u ∂F = N, ∂s where ∂/∂s = (1/v)∂/∂u is the derivative with respect to arc-length. Now, ∂N ∂N ∂T ∂F ∂N = ,N N + , T T = − N, T =− T. ∂t ∂t ∂t ∂t ∂s

617

(A.2)

(A.3)

Differentiating both sides of T = (cos θ, sin θ), where θ, 0 ≤ θ < 2π is the angle between the tangent T and the positive x-axis, and using (A.2), we have (− sin θ, cos θ)

∂F ∂θ = N. ∂t ∂s

Hence ∂F ∂θ = . ∂t ∂s

(A.4)

Now we can derive the equation for the curvature k ≡ ∂θ/∂s: ∂ 1 ∂θ 1 ∂v ∂θ 1 ∂ ∂θ ∂k = =− 2 + . ∂t ∂t v ∂u v ∂t ∂u v ∂u ∂t Using (A.1) and (A.4), we have ∂k ∂ 2 F = 2 + k 2 F. ∂t ∂s

(A.5)

It is also straightforward to derive the time derivatives of the length and (weighted) area of γ . First, for b L = ds = vdu, a

γ

we have dL = dt

b a

∂v du = − ∂t

Second, recall that 1 A=− 2

γ

F kds.

γ , Nds.

(A.6)

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We have

dA 1 ∂γ ∂N 1 ∂v =− ,N + γ, ds − γ , N du dt 2 ∂t ∂t 2 ∂t 1 1 ∂F 1 =− F ds + γ , T ds + F kγ , Nds. 2 2 ∂s 2

Using 1 2

∂F 1 γ , T ds = ∂s 2

∂F γ , T du ∂u 1 ∂γ ∂T =− F ,T +F γ, du 2 ∂u ∂u 1 1 =− F ds − F kγ , Nds, 2 2

we conclude dA =− dt

γ

F ds.

(A.7)

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Chou: Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong; [email protected] Zhu: Department of Mathematics, Zhongshan University, Guangzhou, China; stszxp@zsu. edu.cn