α-Gauss Curvature flows with flat sides

J. Differential Equations 254 (2013) 1172–1192

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α -Gauss Curvature flows with flat sides Lami Kim a,∗ , Ki-ahm Lee a,b , Eunjai Rhee a a b

Department of Mathematical Sciences, Seoul National University, Seoul 151-747, Republic of Korea School of Mathematics, Korea Institute for Advanced Study, Seoul 130-722, Republic of Korea

a r t i c l e

i n f o

Article history: Received 20 June 2012 Available online 13 November 2012

a b s t r a c t In this paper, we study the deformation of the 2-dimensional convex surfaces in R3 whose speed at a point on the surface is proportional to α -power of positive part of Gauss Curvature. First, for 12 < α  1, we show that there is smooth solution if the initial data is smooth and strictly convex and that there is a viscosity solution with C 1,1 -estimate before the collapsing time if the initial surface is only convex. Moreover, we show that there is a waiting time effect which means the flat spot of the convex surface will persist for a while. We also show the interface between the flat side and the strictly convex side of the surface remains smooth on 0 < t < T 0 under certain necessary regularity and non-degeneracy initial conditions, where T 0 is the vanishing time of the flat side. © 2012 Elsevier Inc. All rights reserved.

1. Introduction We are concerned with the regularity of the α -Gauss Curvature flow with flat sides, which is associated to the free boundary problem. This flow explains the deformation of compact convex subjects moving with collision from any random angle. The probability of impact at any point P on the surface Σ is proportional to the α -Gauss Curvature K α . Then the deformation of the surface Σ can be described by the flow

∂X (x, t ) = − K α (x, t )ν (x, t ), ∂t X (x, 0) = X 0 (x) where

*

ν denotes the unit outward normal to the hypersurface and α > 0.

Corresponding author. E-mail addresses: [email protected] (L. Kim), [email protected] (K.-a. Lee), [email protected] (E. Rhee).

0022-0396/$ – see front matter © 2012 Elsevier Inc. All rights reserved. http://dx.doi.org/10.1016/j.jde.2012.10.012

(1.1)

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Now we are going to summarize the known results for the evolution of the strictly convex surfaces following (1.1) [5,6]. Let X (·,·) : S n × [0, T ) → Rn+1 be an embedding and set Σt = X ( S n , t ). Since the volume is decreasing in time and vanishes at finite time, there is the first time, T 0 , when vol(Σt ) becomes zero. Let us assume Σ0 be strictly convex and smooth. Then Σt is also smooth and strictly convex for 0 < t < T 0 for α = 1 [16]. If we rescale Σt to Σ˜ t so that vol(Σ˜ t ) = vol(Σ0 ), then Σ˜ t converges to a sphere, for α = 1 and n = 2 [1] or for α = n1 and n  2 [5]. It is also true if α > n1 and if the initial surface is sufficiently close to the sphere [5]. And Σt converges to a point if 1 ], n

α ∈ ( n+1 2 , n1 ],

or if α ∈ (0, provided Σt has bounded isoperimetric ratio which is the ratio between the radius of the inner sphere and that of outer sphere [2]. Various applications of (1.1) have been discussed at [2]: rolling stone on the hyperplanes (α = 1 1 [14,15]), the gradient flows of the mean width in and n = 2 [11]), the affine normal flows (α = n+ 2

1 L p -norm (α = p − [2]), and image process (α = 14 [3]). 1 Now let Σ0 be convex and smooth. For α > 0, there is a viscosity solution, Σt , for 0 < t < T 0 which has uniform Lipschitz bound [2]. The convex viscosity solution, Σt , has a uniform C 1,1 -estimate for 0 < t < T 0 , for α = 1 and n = 2 [1] or for 12 < α  1 and n = 2 [12]. For α = 1 and n = 2, the C δ∞ regularity of the strictly convex part of the surface and the smoothness of the interface between the strictly convex part and flat spot have been proved in [10]. The dynamics and degeneracy of the diffusion vary depending on α . If α is smaller than n1 , it becomes more singular and the solution gets regular instantaneously. On the other hand, if α is greater than n1 , it becomes degenerate and has waiting time effect which means that the flat spot of the surface stays for a while [2,4]. Waiting time and finite speed of propagation caused by the degeneracy have been studied at the other well-known degenerate equations: the Porous Medium Equation

u t = u m

( u  0, m > 1 )

and Parabolic p-Laplace Equation





ut = ∇ · |∇ u | p −2 ∇ u . 1.1. The balance of terms In this paper, we are going to study the regularity of Σt , when the initial surface, Σ0 , has a flat spot for n = 2. We will assume for simplicity that the initial surface Σ0 has only one flat spot, namely that at t we have Σt = Σt1 ∪ Σt2 where Σt1 is the flat spot and Σt2 is strictly convex part of Σt . The intersection between two regions is the free boundary Γt = Σt1 ∩ Σt2 . The lower part of the surface Σ0 can be written as a graph z = f (x). And similarly we can write the lower part of Σt as z = f (x, t ) for x ∈ Ω ⊂ Rn where Ω is an open subset of Rn . The function f (x, t ) satisfies α -Gauss Curvature flow:

ft =

[det( D 2 f )]α (1 + |∇ f |2 )

α (n+2)−1

.

(1.2)

2

Let’s consider rotationally symmetric case first to see the balance between terms for n = 2. If f = f (r ) is rotationally symmetric, (1.2) can be written as

ft =

f rα f rrα r α (1 + f r2 )

4α −1 2

.

(1.3)

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Let r = γ (t ) be the equation of the free boundary Γ ( f ) = ∂{ f = 0}. The speed of boundary is given by

ft

γt = −

fr

=−

f rα −1 f rrα r α (1 + f r2 )

4α −1 2

.

The regularity comes from the non-degenerate finite speed of the free boundary before the flat β spot converges to a lower dimensional singularity at a focusing time. When f = (r − 1)+ at a given time t, for r ≈ 1,

|γt | = s(α −1)(β−1) sα (β−2) ≈ 1 −1 for s = r − 1, which implies β = 32α α −1 . −1 For the general f = f (x, y , t ), let f = β1 g β for β = 32α α −1 . The equation for this pressure g will be

gt =

[ g det( D 2 g ) + θ(α )( g x2 g y y + g 2y g xx − 2g x g y g xy )]α (1 + g 2β−2 |∇ g |2 )

(1.4)

4α −1 2

for θ(α ) = β − 1 = 2αα−1 . Assuming g τ = 0 at the boundary, the speed of boundary will be

γt = − for a tangential direction

gt gν

= −θ(α )α g ν2α −1 g τατ

(1.5)

τ and a normal direction ν to ∂Ω .

1.2. Conditions for f Condition 1.1. Set Λ( f ) = { f = 0} and Γ ( f ) = ∂Λ( f ). (I) (Non-degeneracy condition) Our basic assumption on the initial surface is that the function f 3 α −1

vanishes of the order dist( X , Λ( f )) 2α−1 and that the interface Γ ( f ) is strictly convex so that the 1

interface moves with finite non-degenerate speed. Namely, setting g = (β f ) β , we assume that at time t = 0 the function g satisfies the following non-degeneracy condition: at t = 0,





0 < λ <  D g ( X ) <

1

λ

0 < λ2 < D 2τ τ g ( X ) <

and

1

λ2

(1.6)

for all X ∈ Γ0 and some positive number λ > 0, where D 2τ τ denotes the second order tangential derivative at Γ . Then the initial speed of free boundary has the speed, at t = 0,

0 < λ4α −1 < |γt | <

1

λ4 α − 1

.

(1.7)

(II) (Before focusing of flat spot) Let T be any number on 0 < T < T 0 , so that the flat side Σt1 is nonzero. Since the area is non-zero, Σt1 contains a disc D ρ0 for some ρ0 > 0. We may assume that





D ρ0 = X ∈ R2 : | X |  ρ0 ⊂ Σt1

for 0  t  T 0 .

(1.8)

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(III) (Graph on a neighborhood of the flat spot Σt1 ) We will also assume, without loss of generality, throughout the paper that

max f (·, t )  2,

0  t  T0

x∈Ω(t )

(1.9)

where Ω(t ) = { X = (x, y ) ∈ R2 : | D f |( X , t ) < ∞}. Set

  Ω P (t ) = x ∈ R2 : f (x, y , t )  f ( P ) .

(1.10)

1.3. The concept of regularity Let’s assume P 0 = (x0 , y 0 , t 0 ) is an interface point and t 0 is sufficiently small. Then condition (1.6) is satisfied at t 0 for small constant c. We can assume

gx( P 0)  c > 0

for some c > 0

(1.11)

by rotating the coordinates. Also by transforming the free-boundary to a fixed boundary near P 0 , we can obtain the map x = h( z, y , t ) where ( z, y , t ) is around Q 0 = (0, y 0 , t 0 ) and then the free-boundary g = 0 is transformed into the fixed boundary z = 0. From the calculation on g (h( z, y , t ), y , t ) = z, we have the fully nonlinear degenerate equation

ht = −

{ z(h zz h y y − h2zy ) − θ(α )h z h y y }α { z2(β−1) + h2z + z2(β−1) h2y }

4α −1 2

z>0

,

(1.12)

implying that under (1.6) and initial regularity conditions, the linearized operator

√

h˜ t = za11 h˜ zz + 2 za12 h˜ zy + a22 h˜ y y + b1 h˜ z + b2 h˜ y where (ai j ) is strictly positive and b1  ν > 0 for some

(1.13)

ν > 0. 2

Definition 1.2. For the Riemannian metric ds with ds2 = dzz + dy 2 , let distance between Q 1 = ( z1 , y 1 ) √ √ and Q 2 = ( z2 , y 2 ) in the metric s be s( Q 1 , Q 2 ) = | z1 − z2 | + | y 1 − y 2 | and the parabolic √ distance √ √ between Q 1 = ( z1 , y 1 , t 1 ) and Q 2 = ( z2 , y 2 , t 2 ) be s( Q 1 , Q 2 ) = | z1 − z2 | + | y 1 − y 2 | + |t 1 − t 2 |. γ Then we define C s , γ ∈ (0, 1), as the space of Hölder continuous functions with respect to the met2+γ

ric s and C s

as the space of all functions h with

h, h z , h y , ht , zh zz ,

√

γ

zh zy , h y y ∈ C s .

Remark 1.3. When we consider the equation

ht = zh zz + h y y + ν h z

(1.14)

on the half-space with ν > 0, which doesn’t have the other condition of h on z = 0, the Riemannian metric ds decides the diffusion of the equation. 2+γ

Remark 1.4. If the transformed function h ∈ C s

2+γ

, we say that g ∈ C s

around the interface Γ .

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1.4. Main theorems Now we are going to state main theorems when the initial surface is just convex having a flat spot. 1 2

< α  1. If Σ0 is convex, then any viscosity solution Σt of (1.1) is C 1,1 for 0 < t < T 0 . Moreover the strictly convex part, Σt2 , is smooth for 0 < t < T 0 . Theorem 1.5. Let us assume

The following short time existence of C s∞ -solution with a flat spot has be essentially proved in [8] since the linearized equation for h, (5.3), is in the same class of operators considered in [8] because of the conditions, (1.6), as in [8]. Therefore their Schauder theory can be applied to (5.3) and then the application of implicit function theorem gives the short time existence as in [8]. 1

Theorem 1.6 (Short Time Regularity). (See [8].) For 12 < α  1, assume that g = (β f ) β is of class C 2+γ up to the interface z = 0 at time t = 0, for some 0 < γ < 1, and satisfies Conditions 1.1 for f . Then there exists a time T > 0 such that the α -Gauss Curvature Flow (1.1) admits a solution Σ(t ) on 0  t  T . In addition the 1

function g = (β f ) β is smooth up to the interface z = 0 on 0 < t  T . In particular the junction Γ (t ) between the strictly convex and the flat side will be a smooth curve for all t in 0 < t  T . One of the main results in this paper is the following long time regularity of the solution. 1

Theorem 1.7 (Long Time Regularity). Under the assumptions of Theorem 1.6, the function g = (β f ) β remains smooth up to the interface z = 0 on 0 < t < T for all T < T 0 . And the interface Γt between the strictly convex and the flat side will be smooth curve for all t in 0 < t < T 0 . To show Theorem 1.7, we follow the main steps in [10]. But the exponent α creates large number of nontrivial terms especially in the estimate of the second derivatives. New quantities have been considered to absorb the effect of terms depending on (1 − α ) in Lemma 4.3. Optimal regularity and Aronson–Bénilan type estimate have been proved in Lemmas 4.4 and 4.5. 2. Convex surface 2.1. Evolution of the metric and curvature The metric and second fundamental form can be defined by

 gi j =

∂X ∂X , ∂ xi ∂ xi



 and h i j = −

with respect to local coordinates {x1 , . . . , xn } of Σt and Weingarten map is given by

∂2 X ,ν ∂ xi ∂ x j



ν is the outward unit normal to Σt . Also the

h ij = g ik hkj , and then

σk =



1i 1 <···
H = trace(h) =

σ1 =



1i n λi ,

K = det(h) =

σn =

where λ1 , . . . , λn are the eigenvalues of the Weingarten λ1 λ2 · · · λn , and map. The evolution of the metric, second fundamental form, and curvature are the following. Lemma 2.1. Let X (x, t ) be a smooth solution of (1.1). Then we have the following. The proof can be referred to Chapter 2, [17]. Let 2 denote K α (h−1 )kl ∇k ∇l .

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∂ gi j α ∂ t = −2K h i j , ∂N ij ∂ Kα ∂ X (ii) ∂ t = − g = −∇ j K α ∂∂xXj , ∂ xi ∂ x j ∂ hi j (iii) ∂ t = α 2h i j + α 2 K α (h−1 )kl (h−1 )mn ∇i hkl ∇ j hmn

(i)

− α K α (h−1 )km (h−1 )nl ∇i hmn ∇ j hkl + α K α Hh i j − (1 + 2α ) K α h jl hli , (iv) ∂∂Kt = α 2 K + α (α − 1) K α −1 (h−1 )i j ∇i K ∇ j K + K α +1 H , α (v) ∂ ∂Kt = α 2 K α + α K 2α H , (vi) ∂∂Ht = α 2 H + α 2 K α −2 g i j ∇i K ∇ j K − α K α g i j (h−1 )km (h−1 )nl ∇i hmn ∇ j hkl + α K α H 2 + (1 − 2α ) K α | A |2 .

2.2. Curvature estimates Now we are going to show the regularity of Σt . The following lemma was proved in [2,12]. Lemma 2.2. (See [2,12].) Let Σ0 be strictly convex and α > 0. Then: (i) There is a constant C > 0 such that

sup

x∈Σ, 0


K α (x, t )  C (α ) = max sup K α (x, 0), x∈Σ

2α + 1

2α 

2αρ0

.

(ii) infx∈Σ, 0 0 where C (t 0 ) is some constant for 0 < t 0 < T 0 . (iii) There is a unique viscosity solution Σt . Lemma 2.3. Set ψ(x, t ) = x, ν and let B R 0 (0) be a ball of radius R 0 about the origin and P = H , where Σ0 is contained in B R 0 (0) and R 2 = max( R 0 2 , R 0 ). Then there exists a constant C = ψ+4R 2 −|x|2 C (supx∈Σ, 0t < T 0 K α , R ) > 0 for

1 2

< α  1 such that sup

x∈Σ, 0t < T 0

H (x, t )  C .

Proof. Since |x| is decreasing, ψ + 4R 2 − |x|2 is positive and then we have

 kl ∂ 2 |x| = 2|x|2 + 2K α x, ν − 2K α h−1 gkl . ∂t By using ∇i P = 0 at the maximum point, we can obtain

α2 P =

α2 H ψ

+ 4R 2

− |x|2

+

α H 2|x|2 (ψ + 4R 2 − |x|2 )2

−

α H 2ψ (ψ + 4R 2 − |x|2 )2

and then since ∇i ∇ j P  0 at the maximum point, we get

(1 − α ) H 2|x|2 ∂ H ((2α + 1) K α + 2K α ψ) H 2 (α K α ψ + 2K α −1 ) + − P 2 2 2 2 2 2 ∂t (ψ + 4R − |x| ) (ψ + 4R − |x| ) (ψ + 4R 2 − |x|2 )2  i j  −1 kl  −1 mn  km  −1 nl  αKα + αg h h ∇i hkl ∇ j hmn − g i j h−1 h ∇i hmn ∇ j hkl ψ + 4R 2 − |x|2  α 2  1 α K H + (1 − 2α ) K α | A |2 (2.1) + 2 2 ψ + 4R − |x| at the maximum point. Now, we can estimate the fourth term of (2.1) by the following inequality

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 km  −1 nl ∇i hkl ∇ j hmn − g i j h−1 h ∇i hmn ∇ j hkl 11 22 2  11 22 2    = (α − 1) h−1 ∇1 h11 + h−1 ∇1 h22 + h−1 ∇2 h11 + h−1 ∇2 h22  11  −1 22  11  −1 22   (∇2 h11 )2 + (∇1 h22 )2 h {∇1 h11 ∇1 h22 + ∇2 h11 ∇2 h22 } − 2 h−1 h + 2 h −1  11  −1 22    −(∇1 h22 )2 − P ∇1 h22 ∇1 |x|2 − ∇1 ψ − (∇2 h11 )2 h  2 h −1   − P ∇2 h11 ∇2 |x|2 − ∇2 ψ

2  11  −1 22  h˜  2 |x| − x, ν 2 P 2 h 1−  2 h −1

α g i j h −1

kl 

h −1

mn

2

where h˜ = min{h11 , h22 }. So

 ˜    2α K α −1 (1 − h2 )2  2 ∂ |x| − ψ 2 + K α 4(1 − α ) R 2 − (1 − α )|x|2 P 2 2 ∂t ψ + 4R − |x|  α α −1 + (1 − 2α ) K ψ − 2K P2 +



1

ψ

+ 4R 2

− |x|2

(1 − α )2|x|2 +



 α 2(2α − 1) K α +1 K P+ . ψ + 4R 2 − |x|2

2ψ + (2α + 1)

1 2

< α  1, we can make the coefficient of P 2 be negative, which can be achieved if we consider η small enough. The reason is if we begin with ηΣ0 for any given Σ0 , we can make K  ηC 02 where C 0 For

is some constant depending on initial surface, which comes from Lemma 2.2, and |x|2  η2 , R 2  η2 , and ψ  η for sufficiently small η . Then the first term and second term of coefficient of P 2 are O (η2−2α ) and the third term is negative with K α ψ = O (η1−2α ) for η small enough. This implies ∂P 1 2 1 2 α ∂ t  − 2 P + C where C = C (supx∈Σ, 0t < T 0 K , R ) and then if − 2 P + C < 0, it is a contradiction. So P is bounded and hence H is bounded before Σ shrinks a point. 2 2.3. Strictly convexity away from the flat spot To apply Harnack principle, let us introduce new coordinate defined on the sphere S n . If Σt is strictly convex, ν (x, t ) is a one-to-one map from Σt to S n , which means for each z ∈ S n , there is X (x, t ) = ν −1 ( z, t ). K ( z, t ) denotes Gauss Curvature K at ν −1 ( z, t ). If Σt is convex, we still use the same coordinate ( z, t ) for strictly convex part Σt2 by using approximates with strictly convex surfaces. Also the support function S ( z, t ) of the strictly convex surface is given as



S ( z, t ) = z, X





ν −1 (z), t , for (z, t ) ∈ S n × [0, T 0 ].

Lemma 2.4. Assume that the flat spot, Σ01 , is a part of the plane orthogonal to en+1 . For any η > 0, there is a constant c η > 0 such that

K α ( z, t )  c η for z ∈ Sη := { z ∈ S n and  z + en+1   η > 0}. Proof. We can immediately obtain the result from the Harnack estimate in [7]:

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For any points z1 , z2 ∈ Sη and times 0  t 1 < t 2

K α ( z2 , t 2 ) K α ( z1 , t 1 )

e

−Θ/4



t2

−(1+(2α )−1 )−1

t1

t

2 where Θ = Θ( z1 , z2 , t 1 , t 2 ) = infγ t 2 |dt γ (t )|m (t ) dt and the infimum is taken over all paths γ in Σ 1 whose graph (γ (t ), t ) joins ( z1 , t 1 ) to ( z2 , t 2 ). The short time existence of smooth surfaces implies that, for z ∈ Sη , X ( z, t ) is the strictly convex part, Σt2 , for 0  t  δ0 for some δ0 > 0. Therefore we can take 0 < δ0  t 1 < t 2  T , which implies K α ( z2 , t 2 )  c 1 K α ( z1 , t 1 )  c η for some c 1 , c η > 0 and then the conclusion. 2

We finally know (1.2) is uniformly parabolic, which comes from Lemmas 2.2–2.3. And then we can show that Σt is C ∞ on the point being away from flat spot. Corollary 2.5. Under the same condition of Lemma 2.4, Σ˜ t2 := { X ( z, t ) ∈ Σt2 : z ∈ S2η } is smooth. Proof. Let λi be the eigenvalues of (h ij ). From the convexity, λi  0. And from the upper bound of Mean Curvature and the lower bound of Gauss Curvature, λ1 + · · · + λn < C 1 and K = λ1 · · · λn > c 2 . Now we have

C 1  λi 

c2

Π j =i λ j



c2 C 1n−1

> 0.

It implies there are 0 < λ  Λ < ∞ such that

 i j λ|ξ |2  K α h−1 ξi ξ j  Λ|ξ |2 and the support function S ( z, t ) satisfies a uniformly parabolic equation in Σ˜ t2 . Therefore S ( z, t ) is C 2,γ and then C ∞ in Σ˜ t2 through the standard bootstrap argument using the Schauder theory. 2 2.4. Proof of Theorem 1.5 Recall that | A |2 is the square sum of principle curvatures of a given surface. First, we approximate the initial surface Σ0 with strictly convex smooth functions, Σ0,ε whose | A 0,ε |2 is uniformly bounded by 2| A 0 |2 of Σ0 . Then there are smooth solutions Σt ,ε of (1.1) [12], and | A 0,ε |2  2H ε2 < 8| A 0 |2 < C uniformly. As ε → 0, Σt ,ε converges to a viscosity solution Σt as in [1]. | A t |2 of Σt will be uniformly bounded, which implies that Σt is C 1,1 . And for any X ∈ Σt2 , there is a small η > 0 such that ν X + en+1   η > 0 and then X ∈ Σ˜ t2 . Since Σ˜ t2 is smooth at X , so is Σt2 . 2.5. A waiting time effect We now are going to show the flat spot of the convex surface will persist for some time. Lemma 2.6. Let Σ0 be convex. For 12 < α  1, there is a waiting time of flat spot: if P 0 ∈ intn (Σ0 ∩ Π) where Π is an n-dimensional plane and intn ( A ) is the interior of A with respect to the topology in Π , there is t 0 > 0 such that P 0 ∈ intn (Σt ∩ Π) for 0 < t < t 0 . | X − P |μ Proof. Let h+ = C + ( T −t0)γ for

1

and C + = ( μ2α (μ−1)α ) 2α−1 . Then h+ is a supersolution of (1.2). Now we are going to compare the solution f with h+ . From C 1,1 -estimate of f , f t is bounded and then there is a ball B ρ0 ( P 0 ) ⊂ intn (Σ0 ∩ Π) and t 0 > 0 such that f ( X , t )  h+ ( X , t ) on ∂ B ρ0 ( P 0 ) for 0  t  t 0 and f ( X , 0)  h+ ( X , 0). From the comparison principle, we have f ( X , t )  h+ ( X , t ) for ( X , t ) ∈ B ρ0 ( P 0 ) × [0, t 0 ), which implies f ( P 0 , t ) = 0 and P 0 ∈ Σt for 0 < t < t 0 . 2

μ=

4α , 2α −1

γ=

1 , 2α −1

γ

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3. Optimal gradient estimate near free interface 3.1. Finite and non-degenerate speed of level sets From using the differential Harnack inequalities, we can show that the free-boundary Γ (t ) has finite and non-degenerate speed as in [10]. As Theorem 1.6, we assume that z = f (x, t ) is a solution 1

of (1.2) and C 1,1 on Ω(t ) for all 0 < t  T and g = (β f ) β is smooth up to the interface Γ (t ) on 0 < t  τ for some τ < T . Let us consider the function

f  (x, t ) =

 (1 − A  )(4α −1)/2 (1 +  )4α  f (1 +  )x, (1 − A  )t 2 α − 1 (1 + B  )

(3.1)

and then the consequences of [10] can be applied to our equation by the similar ways. We may assume condition (1.8) and let r = γ (θ, t ) be the interface Γ (t ) and r = γε (θ, t ) be the ε -level set of the function f with 0  θ < 2π by expressing in polar coordinates. Then

˜ , B˜ , C˜ > 0 such that Lemma 3.1. There exist constants A , B , C > 0 and A t −t 0

e − B + AT γ (θ, t 0 )  γ (θ, t )  e

−

t −t 0 Ct 0

γ (θ, t 0 )

(3.2)

γε (θ, t 0 )

(3.3)

and

e

t −t 0 ˜ B + AT

−˜

γε (θ, t 0 )  γε (θ, t )  e

−

t −t 0

˜ Ct 0

for all 0 < t 0  t  T , 0  θ < 2π . In particular, the free-boundary r = γ (θ, t ) and the ε -level set r = γε (θ, t ) of f for each ε > 0 move with finite and non-degenerate speed on 0  t  T . 3.2. Gradient estimates 1

Throughout this subsection, we will assume that g = (β f ) β is solution of (1.4) and smooth up to the interface on 0  t  T , condition (1.6) and also is satisfied with

max g (x, t )  2,

x∈Ω(t )

for 0  t  T ,

(3.4)

which comes from (1.9). We now will show that the gradient | D g | has the bound from above and below. Lemma 3.2 (Optimal Gradient estimates). With the same assumptions of Theorem 1.6 and (3.4), there is a positive constant C 0 such that

|D g|  C0,

on 0  g (·, t )  1, 0  t  T .

Moreover if (1.8) is satisfied and if g is smooth up to the interface on 0  t  T , then there is a positive constant c 0 such that

| D g |  c0 ,

on g (·, t ) > 0, 0  t  T .

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Proof. (i) First, we are going to show the upper bound of ∇ g. Suppose that f is approximated by f ε of (1.2) which is a decreasing sequence of solutions satisfying the positivity, strictly convexity and 1

smoothness on {x ∈ R2 : | D f ε (x)| < ∞} for 0  t  T . Set g ε = (β f ε ) β . We can choose the f ε ’s such that | D g ε |  C 0 at t = 0, on {x: 0  g ε  1} and | D g ε |  C 0 at g ε = 1, 0  t  T , for some uniform constant C 0 . Then the last estimate comes from (1.9) and (3.4). 1

Let’s denote g ε by g for convenience of notation, where g = (β f ) β is a strictly positive and | D g |2

g2 +g2

smooth solution of (1.4) with convex f . Let us apply the maximum principle to X = 2 = x 2 y and assume X has an interior maximum at the point P 0 = (x0 , y 0 , t 0 ). By rotating the coordinates, we can assume g x > 0 and g y = 0 at P 0 . Then we have X t  0 by using the facts that X x = X y = 0, X xx  0 and X y y  0 are satisfied at P 0 . On the other hand |∇ g | is bounded at t = 0 from the condition on the initial data and on { g = 1}, |∇ g | =

|∇ f | g β−1

= |∇ f | is bounded since f is convex. Hence

X  C˜ , on 0  g  1, 0  t  T , provided that X  C˜ at t = 0 and g = 1, 0  t  T so that

|D g|  C0,

on 0  g (·, t )  1, 0  t  T .

(ii) Now we are going to show the lower bound of the gradient. Consider

X = xg x + yg y . Using the maximum principle as in (i), we have that

X t  −C X where C is a constant depending on

(3.5)

ρ0 and d dt

X





γ (t ), t  −C X

(3.6)

at an interior or boundary minimum point P 0 of X . Then

min

{ g (·,t )>0}

X (t ) 

min

{ g (·,0)>0}

X (0)e −Ct

for all 0  t  T by Gronwall’s inequality, and it implies the desired estimate.

2

Theorem 3.3. Under the same assumptions of Lemma 3.2, there exist positive constants C 1 , C 2 and pending only on ρ0 and the initial data, for which

−C 2  (γε )t (θ, t )  −C 1 < 0,

for 0  t  T and 0 < ε < ε0 .

ε0 , de-

(3.7)

4. Second derivative estimate 1

Throughout this section, we will assume that g = (β f ) β is solution of (1.4) and smooth up to the interface on 0  t  T , conditions (1.8) and (1.9). 4.1. Decay rate of α -Gauss Curvature Under the same conditions with Sections 3.1 and 3.2, we will show a priori bounds of the Gauss Curvature K = det( D 2 f )/(1 + | D f |2 ) and the second derivatives of f and g.

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Lemma 4.1. For the same hypotheses of Theorem 1.6 and (1.8), there exists a positive constant c such that

c

Kα α

g 2α−1

 c −1 ,

on 0  t  T

(4.1)

for K = det D 2 f /(1 + | D f |2 ). Proof. We will only consider the bound of (4.1) around the interface. It suffices to show the bound 2 α −1

α

of gt from gt = K α /((1 + | D f |2 ) 2 g 2α−1 ) because | D f | is bounded around { g = 0}. For r = γε (θ, t ) which is the ε -level set of g in polar coordinates,

gt = − gr · γ˙ε (θ, t ) since g (γε (θ, t ), θ, t ) = ε . Then since the level set of g is convex, we know that c < gr < c −1 and −C 2  γ˙ε (θ, t )  −C 1 < 0 for 0  t  T from Lemma 3.2 and Theorem 3.3 implying that C 1 c < gt < C 2 c −1 , so the proof is complete. 2 Corollary 4.2. Under the assumptions of Lemma 4.1, the solution g of (1.4) satisfies the bound

c  g t  c −1 .

(4.2)

4.2. Upper bound of the curvature of level sets Lemma 4.3. With the assumptions of Theorem 1.6 and condition (1.8), there exists a constant C > 0 such that

0 < gτ τ  C with τ denoting the tangential direction to the level sets of g. Proof. Strictly convexity of the level sets of g directly implies g τ τ > 0. We will obtain the bound from above by using the maximum principle on





X = g 2y g xx − 2g x g y g xy + g x2 g y y + g ( g xx + g y y ) + θ|∇ g |2 .

(4.3)

Let ν and τ denote the outward normal and tangential direction to the level sets of g respectively. Then we can write X as







X = g + g ν2 g τ τ + gg νν + θ g ν2



(4.4)

since g τ = 0. We have also known that

0 < c  g ν  c −1

on g > 0, 0  t  T

for some c > 0, depending on ρ0 and the initial data. Also g ( g xx + g y y ) + θ|∇ g |2 is bounded since f ∈ C 1,1 . Hence an upper bound on X will imply the desired upper bound on g τ τ . We will apply the maximum principle on the evolution of X . The term ( g ( g xx + g y y ) + θ|∇ g |2 ) on X will control the sign of error terms. Corollary 4.2 implies

X C

at g = 0,

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1183

1

since we know that X = θ1 gtα + θ|∇ g |2 at the free-boundary g = 0. Then we can assume that X has its space–time maximum at an interior point P 0 = (x0 , y 0 , t 0 ). Let’s assume that

g τ = g y = 0 and

g ν = g x > 0 at P 0

(4.5)

without loss of generality, since X is rotationally invariant. Also let’s consider the following transformation

g˜ (x, y ) = g (μ, η) where

g μη (x0 , y 0 ,t 0 ) , which is similar in the transformation g μμ (x0 , y 0 ,t 0 ) g g g˜ x = g μ − ηg μη = g μ > 0, g˜ y = g η = 0, and ηη

μ = x and η = y − ax with a =

tion 4.1 of [13]. Then we can obtain

 ( g˜ i j ) =

g μμ −

to Proposi-



2 g μη g ηη

0

0

g ηη

at P 0 . Here g˜ y y = g ηη > 0 and g˜ xx < 0 at P 0 . Hence equation is maintained with this change of coordinate. Also we can drop off the third derivative term of g˜ because it is changed under the perfect square of the third derivative of g. Hence we can assume

g˜ xy = 0

(4.6)

at P 0 without loss of generality. Then we will proceed with the function g instead of g˜ for convenient of notation. From (4.3), we get









X = g + g x2 g y y + gg xx + θ g x2 ,

at P 0 .

At the maximum point P 0 , we also have X x = 0 and X y = 0 implying that

g xy y = −

gg xxx + 2g x det D 2 g + (2θ + 1) g x g xx + g x g y y g+

g x2

and

g yyy = −

gg xxy g + g x2

.

(4.7)

We are going to compute the evolution equation of X from the evolution equation of g to find a contradiction saying that

0  X t < 0 at P 0 , when X > C > 0 for some constant C . This implies that X  C , on 0  t  T . First we will consider the following simpler case that f satisfies the evolution



f t = det D 2 f

α

1

for the convenience of the reader. Then g = (β f ) β satisfies the equation





gt = g det D 2 g + θ g 2y g xx − 2g x g y g xy + g x2 g y y

α

We differentiate twice Eq. (4.8) to obtain the evolution of X . Set





K g = g det D 2 g + θ g 2y g xx − 2g x g y g xy + g x2 g y y , I = 1 + g 2β−2 |∇ g |2 ,

and

J = g + |∇ g |2 .

.

(4.8)

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L. Kim et al. / J. Differential Equations 254 (2013) 1172–1192

Let L denote the operator

L X := X t − α K gα −1











gg y y + θ g 2y X xx − 2( gg xy + θ g x g y ) X xy + gg xx + θ g x2 X y y .

Then after many tedious calculations, we have that at the maximum point P 0 ,

LX = A +

where

1

(1 + 2γ )2 ( g + g x2 )2 K g2

B

(4.9)

γ = θ − 1 and 2 A = −4g 2 g xxy −

+ In addition, B = 0 if



4g 3

g xxx +

g x2 + g

g 2y y   g x2

+g

− 2g x2 − gg xx

2

6g x g xx + 3g x g xx g y y

2

g

   + 3g xx g x2 + g g x2 − gg xx .

(4.10)

γ = 0, otherwise 







2 B = − B 1 g 2 g x2 + g g xxy − g 3 B 1 ( g xxx + B 11 )2 + (1 + 2γ )2 g + g x2

2

K g2

 E1 E2

.

(4.11)

Here 2

2+3γ 1+2γ



B 1 = 4(1 + 2γ ) K g

1+γ 1 + 2γ

1+γ 1+2γ

+ γ (1 + γ ) K g



γ

1+2γ



− Kg 



g x2 + g g y y − gg xx + (1 + γ ) g x2

2

and set Z = g x2 g y y so that



E 2 = 4(1 + 2γ )3 gg x6 g x2 + g

2

6+13γ 1+2γ

Kg

Z4

γ (1 + γ )(1 + 32 γ ) 1+2γ − K g (1 + 2γ )2



 3 51++112γγ 6 + g γ (1 + γ )(1 + 2γ ) g x2 g x2 + g K g Z + l.o.t.   4 6  g E 11 Z + γ E 12 Z + l.o.t. , where l.o.t. means lower order terms. We may assume that P 0 lies close to the free-boundary and γ

1 +2 γ

that K g

<

1 2

by considering a scaled solution g λ (x, t ) = λ 4 γ +5 γ +2

1 −γ+ 2

4γ +3

g (λx, λ 2γ +1 t ) as g at the beginning

1 +γ

 || K g1|| ∞ ( 12 ) γ . Then on g  1, we have A is negative in (4.10) since g xx L is negative and E 11 , E 12  δ0 ( g x , K g ) > 0 uniformly, which implies E 2 is positive. And we also have, of the proof with λ

in (4.11),



E 1 = −γ (1 + γ ) g x2 + g with

2



K g5 Z 8 C 8 + γ Z 7 C 7 + l.o.t.



L. Kim et al. / J. Differential Equations 254 (2013) 1172–1192









1185

γ

1+2γ

C 8 = (1 + γ )2 g x2 + g 2γ g + 3(5 + 4γ ) g x2 + γ (1 + 2γ ) K g

g2

γ    1+γ2γ 2 1+2γ − (1 + 2γ ) 15 + 2γ (7 + 2γ ) K g gg x − 3(1 + γ )(1 + 2γ )(5 + 4γ ) K g g x4 .

Now we can show C 8  δ1 ( g x , K g ) > 0 uniformly and then E 1 < 0 for sufficiently large Z . Therefore B is negative. Hence we can obtain desired result. We now return to the case of the α -Gauss Curvature Flow. Let us set

I = 1 + g 2β−2 | D g |2 ,

J = g + | D g |2





Q = g det D 2 g + θ g 2y g xx − 2g x g y g xy + g x2 g y y

and

α

.

Also let C = C ( g C 1 ,  f C 1,1 ) denote various constants and L˜ X denote the operator

L˜ X := X t − α K gα −1 I −

4α −1 2











gg y y + θ g 2y X xx − 2( gg xy + θ g x g y ) X xy + gg xx + θ g x2 X y y .

We find, after several calculations, that at the maximum point P 0 , where (4.5) and (4.6) hold, X satisfies the equality

L˜ X = I −

4α −1 2

LX −

4α − 1  2

 4α+1 4α +1 g + g x2 I − 2 Q I y y − (4α − 1) g x (θ + g y y ) I − 2 Q I x

  4α +1 4α +1 4α − 1 − 4α+1 16α 2 − 1 − 4α+3 2 2 + g −(4α − 1) I − 2 Q x I x − Q I xx + Q I x2 + I − 2 Q g xx I I 2

4

and from (4.9) and tedious computation we obtain that

L˜ X  I −

4α −1 2

A+

I−

(1 + 2γ



4α −1 2

)2 ( g

+

  2 − B 1 g 2 g x2 + g g xxy − g 3 B 1 ( g xxx + B 11 )2

g x2 )2 K g2

 2  E 1  + (1 + 2γ )2 g + g x2 K g2



E2

−

2    8γ (γ + 1) − 4α+1 α −1 2γ +3 g x  2 (γ + 1) g x2 + gg xx − g + g x2 K g g xxx I Kg g 2 (1 + 2γ )2 g + gx

+ g 2γ +1 · l.o.t. + I − I

− 4α2−1

A+

4α +1 2

I−

(1 + 2γ

K gα g xx



4α −1 2

)2 ( g

+

g x2 )2 K g2

  2  2 − B 1 g 2 g x2 + g g xxy − g 3 B 1 g xxx + B 11 + O ( g )

  2  E 1  4α +1 + (1 + 2γ )2 g + g x2 K g2 + O ( g ) + O ( g ) + I − 2 K gα g xx . E2

Here O ( g ) denotes various terms satisfying | O ( g )|  C g with constant C . We can know the first term and the second term are negative as in the case of L X and provided that X > C is sufficiently large. And then L˜ X < C with C depending on || f ||C 1,1 and || g ||C 1 on g  1, which implies that ( X − Ct )t < 0. Applying the evolution of X˜ = X − Ct with a simple trick implies the desired contradiction. Hence X˜  C where C is positive constant. 2

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4.3. Aronson–Bénilan type estimate Lemma 4.4. Under the assumptions of Theorem 1.6 and condition (1.8), there exists a constant C > 0 for which





det D 2 g  −C for a uniform constant C > 0. Proof. To establish the bound of det( D 2 g ) from below, we will use the maximum principle on the quantity

Z=

det D 2 g g x2 g y y + g 2y g xx − 2g x g y g xy

+ b|∇ g |2

with some positive constant b on { g (·, t ) > 0, 0  t  T }. Let us assume that Z becomes minimum at the interior point P 0 . We can assume g y = 0, g x > 0 and g xy = 0 at P 0 by using a similar transformation and the change of coordinate in Lemma 4.3 at P 0 . Then we have

ai j Z i j  0 for

 (ai j ) =

2α

α K gα −1 gg y y (1 + g 2α−1 g x2 )

1−4α 2



0 2α

α K gα −1 ( gg xx + θ g x2 )(1 + g 2α−1 g x2 )

0

1−4α 2

(4.12)

and Z x = Z y = 0 at the minimum point P 0 implying that

Z t  B 1 ( g xy y + B 2 )2 + A 0 +



1

(α + 1)(2α

− 1)3

O (g) Z + O Z 2



(4.13)

where

B1 =

  1 −2 α α −2 2

2α α 2α −1 g 2 2 1 + g Kg α g x + (2α − 1) gg xx x (1 − 2α )2 g x2 g y y 

 · (4α − 2) K g + (α − 1) α g x2 + (2α − 1) gg xx g y y

and

A 0 = A 0,0 g 2 b4 + E 0 + A 0,5 (α − 1) g 3 b5 + (α − 1) E 1 with

A 0,0 =

g x10 g y y (24 + 12g 2 g x2 − 21g 4 g x4 )

(4.14)

2(1 + g 2 g x2 )7/2

and 1

A 0,5 = −

2α

1

30g 1+ 2α−1 (1 − 2α )2 g x14 ( g 1−2α + g x2 )(1 + g 1+ 2α−1 g x2 )− 2 −2α K gα −1 g 2y y

(4α − 2) K g + (α − 1)α g x2 g y y

1

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1187

where E 0 = O (b3 , g 2 ) and E 1 = O (b4 , g 3 ). Here we can also show A 0,0  δ1 ( g x , g y y ) > 0 uniformly and we have

  (4α − 2) K g + (α − 1)α g x2 g y y = (4α − 2) gg xx g y y + θ g x2 g y y + (α − 1)α g x2 g y y   = (4α − 2) gg xx g y y + θ(4α − 2) + α 2 − α g x2 g y y >0

(4.15)

on g  1 since 12 < α  1. Then (α − 1) A 0,5 is nonnegative so that A 0 is positive for sufficiently large b  1. Also we can know B 1 is positive on g  1 from (4.15). This implies

Z t > 0 > −a Z with a positive constant a. By Gronwall’s inequality, we have

Z  Z 0 e −˜at where Z 0 is initial data of Z at t = 0 and a˜ is constant, which concludes the proof.

2

4.4. Global optimal regularity Let’s consider the quantity

  Z = max g D γ γ g + θ| D γ g |2 .

(4.16)

γ

Now we will show that Z is bounded from above through the next lemma. Lemma 4.5. With the same assumptions of Theorem 1.6 and condition (1.8), there exists a positive constant C = C (θ, ρ , λ,  g C 2 (∂Ω) ) with

max Z  C Ω( g )

where Ω( g ) = {x | g (x) > 0}. Proof. First, we know that Z is nonnegative from Z = β Lemma 3.2 implies

2−β

β

f

2−β

β

f γ γ and a convexity of f . Also

  Z  C θ, ρ , λ,  g C 2 (∂Ω) at g = 0, since Z = θ| D γ g |2 at the free-boundary g = 0. Then we can assume that Z has the its maximum at an interior point P 0 ∈ Ω( g ) and in a direction γ . To show the bound of Z , we consider γ as γ = λ1 ν + λ2 τ with λ21 + λ22 = 1, where ν , τ denote the outward normal and tangential directions to the level sets of g respectively. Then Z ( P 0 ) = g D γ γ g + θ| D γ g |2 and (1.4) can be rewritten as



Z ( P 0 ) = g λ21 g νν + 2λ1 λ2 g ντ + λ22 g τ τ + θλ21 g ν2 



 

2 gg νν + θ g ν2 g τ τ = gg ντ + gt 1 + g 2β−2 g ν2

and

 4α2−1  α1

.

(4.17)

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L. Kim et al. / J. Differential Equations 254 (2013) 1172–1192

Here if g g νν is not sufficiently large at P 0 , we have

  Z ( P 0 )  C θ, ρ , λ,  g C 2 (∂Ω) from Lemma 3.2, Corollary 4.2 and Lemma 4.3 implying the desired result immediately. On the other hand, if θ g ν2  g g νν at P 0 , then we get





2 gg ντ  2gg νν g τ τ  C θ, ρ , λ,  g C 2 (∂Ω) gg νν

√

implying g ντ  C (θ, ρ , λ,  g C 2 (∂Ω) ) g νν . Then we can know that Z ( P 0 ) is maximum when λ2 = 0 from Lemma 4.3 and (4.17) so that Z ( P 0 ) = g g νν + θ g ν2 . Also we get g g ντ + θ g ν g τ = 0 at P 0 implying that g ντ = 0 at P 0 . Here by the similar transformation in Lemma 4.3, we can assume

g τ = g y = 0,

g ν = g x > 0 and

g xy = 0 at P 0 .

Then we have

a i j Zi j  0 with (4.12) at the maximum point P 0 . And since g xxx =

− g x g xx −2θ g x g xx g

and g xxy = 0 at P 0 , we obtain

Zt = gt g xx + gg xxt + 2θ g x g xt

− 1+24α  −1   α   1 2α (1 − 2α )2 (α − 1) g 1−2α + gg x2  − 1 + g 2α−1 g x2 g y y θ g x2 + gg xx  2α  · g 2α−1 α (4α − 1) 2α (2α θ − 3θ + α + 1) + 2θ − 1 g x6  1  + (1 − 2α )2 (α − 1) g xx g 1−2α (α − 1) + (4α − 1) g 2 g xx     4α−1   + (2α − 1) gg x2 g xx (α − 1) 16α 2 − 5α + 1 + 3α 8α 2 − 6α + 1 g 2α−1 g xx      4α−1  + α g x4 4α 2 − 5α + 1 θ(4α − 2) + 1 + 6α 10α 2 − 9α + 2 g 2α−1 g xx at the point P 0 . Also from g xx = 2α

Zt 

Z −θ g x2 g

, we have

3+4α

   2α (1 + g 2α−1 g x2 )− 2 (Z g y y )α −Z 2α 2 − 3α + 1 (4α − 1)Z g 2α−1 + α − 1 g (α − 1)(2α − 1)  4α     4α 2α + g 2α−1 g x2 g 1−2α (α − 1)2 α − 3α 8α 2 − 6α + 1 Z 2 − (α − 1) 8α 2 − 3α + 1 Z g 1−2α   2α + α (α − 1) g x2 2(α − 1) g 1−2α − 6α Z + (α − 1) g x2

− 3+24α α 2+α 1   2α+1   2α 1   (1 − 4α ) 1 + g 2α−1 g x2 g yyZ g 2α−1 + O Z 2+α g 2α−1 + O Z 1+α , g 2α−1 . Then on g  1, Zt < 0 at P 0 since 1 − 4α < 0. Hence we can obtain the desired result.

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4.5. Decay rates of second derivatives Corollary 4.6. With the hypotheses of Lemma 4.3, there exists a positive constant c depending on ρ0 and the initial data such that (1) c  g τ τ  c −1 , f νν fτ τ −1 and (2) c  g β− 2 , g β−1  c

| f ντ | g (2β−3)/2

 c −1 for uniformly small g

with τ denoting the tangential direction to the level sets of g. Proof. (i) The upper bound of g τ τ comes from Lemma 4.3. Now we are going to show the lower bound. From Lemma 4.1, we have 1

det D 2 f  cg 2α−1 = cg 2β−3 , which implies 2 f νν f τ τ  cg 2β−3 + f ντ  cg 2β−3

(4.18)

and then

fτ τ 

cg 2β−3 f νν

 c˜ g β−1

since f νν  C g β−2 from Lemma 4.5. Since f τ τ = g β−1 g τ τ + (β − 1) g β−2 g τ2 , we conclude that

gτ τ =

fτ τ g β−1

 c˜ ,

for some positive constant c˜ depending only on the initial data and (ii) f τ τ = g β−1 g τ τ and the bound on g τ τ tell us

c

fτ τ g β−1

ρ0 .

 c −1 .

(iii) From Lemma 4.5, we have f νν  C g β−2 for uniformly small g, so we are going to show

f νν + f τ τ  cg β−2 . Let us denote by λ1 , λ2 the two eigenvalues of the matrix det D 2 f such that λ1  λ2 . Then, from Lemma 4.1, we have

c

λ1 λ2 g 2β−3

 c −1

and λ2  f τ τ  c −1 g β−1 , implying that λ1  cg β−2 for some positive constant c. Hence f νν + f τ τ  λ1 + λ2  cg β−2 > 0 as desired. 2 − f νν f τ τ . By using the bound of f νν and f τ τ , we obtain (iv) The convexity of f says 0  f ντ 2 f ντ  f νν f τ τ  c −1 g 2β−3 .

2

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L. Kim et al. / J. Differential Equations 254 (2013) 1172–1192

5. Higher regularity 5.1. Local change of coordinates For any point P 0 = P 0 (x0 , y 0 , t 0 ) at the interface Γ with 0 < t 0  T , let’s assume that n0 is the unit vector in the direction of the vector P0 = O P 0 and n0 satisfies

n0 :=

P0

|P0 |

= e1

(5.1)

by rotating the coordinates. Then we will have the following lemma as Lemma 4.6, [10] Lemma 5.1. There exist positive constants c and in (1.8), for which

c  g x ( P )  c −1

η, depending only on the initial data and the constant ρ0 and c  f xx ( P )  c −1

at all points P = (x, y , t ) with f ( P ) > 0, | P − P 0 |  η and t  t 0 under (5.1). 5.2. Class of linearized equation In this subsection, we are going to show that our transformed function h from g near the free boundary satisfies the same class of operators considered in [10] so that all the results in [10] can be applied to our equation by using the similar methods. 1 Throughout this subsection, we will assume that at time t = 0 the function g = (β, f ) β satisfies the hypotheses of Theorem 1.7 and that g is smooth up to the interface on 0  t  T for T > 0 satisfying condition (1.8). 1, γ

We will state the results of uniform C s -estimate in [10]. The readers can see detailed proofs with reference to [10]. Let P 0 = (x0 , y 0 , t 0 ) be a point on the interface curve Γ (t 0 ) at time t = t 0 , for 0 < t 0  T . We may assume, without loss of generality, that τ  t 0  T , for some τ > 0. From the short time regularity of Theorem 1.6, we know that solutions are smooth up to the interface on 0  t  2τ , for some τ depending only on the initial data. Also we may assume that condition (5.1) holds at the point P 0 by rotating the coordinates. By Lemma 5.1, g x ( P ) > 0 for all points P = (x, y , t ) with t  t 0 , sufficiently close to P 0 and then from (1.12) (see in [8], Section II),

ht = −

{ z(h zz h y y − h2zy ) − θ(α )h z h y y }α { z2(β−1) + h2z + z2(β−1) h2y }

4α −1 2

,

z > 0.

(5.2)

Set K h = z(h zz h y y − h2zy ) − θ(α )h z h y y and J = z2(β−1) + h2z + z2(β−1) h2y . By linearizing this equation around h, we can obtain the equation

h˜ t =

α K hα −1  J

+

4α −1 2

  (4α − 1) z2(β−1) K hα h y  − zh y y h˜ zz + 2zh zy h˜ zy + θ(α )h z − zh zz h˜ y y + h˜ y 4α +1 J

(4α − 1) K hα h z + α K hα −1 θ(α )(h2z + z2(β−1) (1 + h2y ))h y y J

4α +1 2

h˜ z .

Let us denote by Bη the box

  Bη = 0  z  η 2 , | y − y 0 |  η , t 0 − η 2  t  t 0

2

(5.3)

L. Kim et al. / J. Differential Equations 254 (2013) 1172–1192

1191

around the point Q 0 = (0, y 0 , t 0 ). We can obtain a priori bounds on the matrix

A = (ai j ) = α K hα −1



−h y y √

zh zy

√

zh zy (θ(α )h z − zh zz )



and the coefficient

b=

(4α − 1) K hα h z + α K hα −1 θ(α )(h2z + z2(β−1) (1 + h2y ))h y y J

4α +1 2

.

Then we obtain the following. Lemma 5.2. There exist positive constants η , λ and ν , depending only on the initial data and the constant ρ0 in (1.8) such that

λ|ξ |2  ai j ξi ξ j  λ−1 |ξ |2 ,

∀ξ = 0

and

|b|  λ−1 and b  ν > 0 on the box Bη . Notice that b  ν > 0 comes from the decay rates of second derivatives, Corollary 4.6, and ˜ := (˜ai j ) and b˜ i , Aronson–Bénilan type estimate, Lemma 4.4. Similarly, we can get the bound of A i = 1, 2, to be the coefficients

a˜ i j =

ai j

( z2(β−1)

+ h2z

+ z2(β−1) h2y )

4α −1 2

and

b˜ 1 = b −

α K hα −1 J h y y J

4α +1 2

and b˜ 2 =

(4α − 1) z2(β−1) K hα h y J

4α +1 2

of Eq. (5.3). Lemma 5.3. There exist constants stant ρ0 in (1.8), for which

η > 0, λ > 0 and ν > 0, depending only on the initial data and the conλ|ξ |2  a˜ i j ξi ξ j  λ−1 |ξ |2 ,

∀ξ = 0

and

|b˜ i |  λ−1 and b˜ 1  ν > 0 on the box Bη . 5.3. Regularity theory Recall Definition 1.2. Then Lemma 5.3 tells us the linearized equation (5.3) is in the same class of operators considered in Lemma 5.2 of [10] and [9]. We are now in a position to show the uniform Hölder bounds of the first order derivatives ht , h y 2, γ

and h z of h on Bη . In [10], the authors have obtained the C s

regularity of h on the box.

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L. Kim et al. / J. Differential Equations 254 (2013) 1172–1192

Lemma 5.4. There exist numbers γ and μ in 0 < γ , μ < 1, and positive constants η and C , depending only on the initial data and ρ0 , such that

h y C 2+γ (B η )  C , s

2

ht C 2+γ (B η )  C , s

and

h z C sμ (B η )  C . 2

2

Following the proof Theorem 6.10 of [10], we will have the following theorem. Theorem 5.5. With the same assumptions of Theorem 1.6 and condition (1.8) which satisfies at T < T c , there exist constants 0 < α0 < 1, C < ∞ and η > 0, depending only on the initial data and ρ0 , for which x = h(x, y , t ) fulfills

hC 2+γ (B )  C s

η

on Bη = {0  z  η2 , | y − y 0 |  η, t 0 − η2  t  t 0 } for P 0 = (x0 , y 0 , t 0 ) with 0 < τ < t 0 < T , which is any free-boundary point holding condition (5.1). Proof of Theorem 1.7. From the short time existence of Theorem 1.6, there exists a maximal time T > 0 such that g is smooth up to the interface on 0 < t < T . Assuming that T < T 0 , we will show 2+γ

, up to the interface z = 0, for some γ > 0, that at time t = T , the function g (·, T ) is of class C s and satisfies the non-degeneracy conditions (1.6). Hence by Theorem 1.6, there exists a time T  > 0 2+γ

such that g is of class C s , for all τ < T + T  , and hence C ∞ up to the interface, by Theorem 9.1 in [8]. This will contradict the fact that T is maximal, proving the theorem. From Lemma 3.2 and Corollary 4.6, the functions g (·, t ) satisfy conditions (1.6), for all 0  t < T , with constant c indepen2+γ

regularity of g, on 0  t  T , up to dent of t. Hence, it will be enough to establish the uniform C s the interface, whose proof follows the same line of argument in [10]. 2 Acknowledgment Ki-Ahm Lee was supported by Basic Science Research Program through the National Research Foundation of Korea (NRF) grant funded by the Korea Government (MEST) (2010-0001985). References [1] B. Andrews, Gauss curvature flow: The fate of the rolling stones, Invent. Math. 138 (1) (1999) 151–161. [2] B. Andrews, Motion of hypersurfaces by Gauss curvature, Pacific J. Math. 195 (1) (2000) 1–34. [3] L. Alvarez, F. Guichard, P.L. Lions, J.M. Morel, Axioms and fundamental equations of image processing, Arch. Ration. Mech. Anal. 123 (1993) 199–257. [4] D. Chopp, L.C. Evans, H. Ishii, Waiting time effects for Gauss curvature flow, Indiana Univ. Math. J. 48 (1) (1999) 311–334. [5] B. Chow, Deforming convex hypersurfaces by the nth root of the Gaussian curvature, J. Differential Geom. 22 (1) (1985) 117–138. [6] B. Chow, Deforming convex hypersurfaces by the square root of the scalar curvature, Invent. Math. 87 (1987) 63–82. [7] B. Chow, On Harnack’s inequality and entropy for the Gaussian curvature flow, Comm. Pure Appl. Math. 44 (4) (1991) 469–483. [8] P. Daskalopoulos, R. Hamilton, The free boundary on the Gauss curvature flow with flat sides, J. Reine Angew. Math. 510 (1999) 187–227. [9] P. Daskalopoulos, K. Lee, Hölder regularity of solutions to degenerate elliptic and parabolic equations, J. Funct. Anal. 201 (2003) 341–379. [10] P. Daskalopoulos, K. Lee, Worn stones with flat sides all time regularity of the interface, Invent. Math. 156 (2004) 445–493. [11] W. Firey, Shapes of worn stones, Mathematica 21 (1974) 1–11. [12] L. Kim, K. Lee, α -Gauss curvature flows, in preparation. [13] O. Savin, The obstacle problem for Monge–Ampere equation, Calc. Var. Partial Differential Equations 22 (2005) 303–320. [14] G. Sapiro, A. Tannenbaum, Affine invariant scale-spaces, Int. J. Comput. Vis. 11 (1993) 25–44. [15] G. Sapiro, A. Tannenbaum, On affine plane curve evolution, J. Funct. Anal. 119 (1994) 79–120. [16] K. Tso, Deforming a hypersurface by its Gauss–Kronecker curvature, Comm. Pure Appl. Math. 38 (1985) 867–882. [17] X.-P. Zhu, Lectures on Mean Curvature Flows, Stud. Adv. Math., vol. 32, Amer. Math. Soc., 2002.