The first initial–boundary value problem for Hessian equations of parabolic type on Riemannian manifolds

Nonlinear Analysis 143 (2016) 45–63

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Nonlinear Analysis www.elsevier.com/locate/na

The first initial–boundary value problem for Hessian equations of parabolic type on Riemannian manifolds Gejun Bao, Weisong Dong, Heming Jiao ∗ Department of Mathematics, Harbin Institute of Technology, Harbin, 150001, China

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Article history: Received 13 September 2015 Accepted 9 May 2016 Communicated by Enzo Mitidieri MSC: 35B45 35R01 35K20 35K96

abstract In this paper, we are concerned with the first initial–boundary value problem for a class of fully nonlinear parabolic equations on Riemannian manifolds. As usual, the establishment of the a priori C 2 estimates is our main part. Based on these estimates, the existence of classical solutions is proved under conditions which are nearly optimal. © 2016 Elsevier Ltd. All rights reserved.

Keywords: Riemannian manifolds Fully nonlinear parabolic equations First initial–boundary value problem a priori estimates

1. Introduction In this paper, we study the Hessian equations of parabolic type of the form f (λ(∇2 u + χ), −ut ) = ψ(x, t)

(1.1)

in MT = M × (0, T ] ⊂ M × R satisfying the boundary condition u = ϕ,

on PMT ,

(1.2)

where (M, g) is a compact Riemannian manifold of dimension n ≥ 2 with smooth boundary ∂M and M := M ∪ ∂M , PMT = BMT ∪ SMT is the parabolic boundary of MT with BMT = M × {0} and SMT = ∂M × [0, T ], f is a symmetric smooth function of n + 1 variables, ∇2 u denotes the Hessian of u(x, t) with respect to x ∈ M , ut = ∂u ∂t is the derivative of u(x, t) with respect to t ∈ [0, T ], χ is a smooth (0, 2) 1 , . . . , λ n ) denotes the eigenvalues of ∇2 u + χ with respect to the metric g. ¯ and λ(∇2 u + χ) = (λ tensor on M ∗ Corresponding author. E-mail addresses: [email protected] (G. Bao), [email protected] (W. Dong), [email protected] (H. Jiao).

http://dx.doi.org/10.1016/j.na.2016.05.005 0362-546X/© 2016 Elsevier Ltd. All rights reserved.

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We assume f to be defined in an open convex cone Γ ⊂ Rn+1 with vertex at the origin satisfying Γn+1 ≡ {λ ∈ Rn+1 : each component λi > 0, 1 ≤ i ≤ n + 1} ⊆ Γ ̸= Rn+1 and furthermore, Γ is invariant under interchange of any two λi , i.e. it is symmetric. In this work, f is assumed to satisfy the following structural conditions as in [3] (see [9] also): ∂f > 0 in Γ , 1 ≤ i ≤ n + 1, ∂λi f is concave in Γ

fi ≡

(1.3) (1.4)

and δψ,f ≡ inf ψ − sup f > 0, MT

where sup f ≡ sup lim sup f (λ).

∂Γ

∂Γ

λ0 ∈∂Γ

(1.5)

λ→λ0

In this work we are interested in the existence of classical solutions to (1.1)–(1.2). Recent research on the Hessian equations of elliptic type (see [9,7]): f (λ(∇2 u + χ)) = ψ(x)

(1.6)

provides some ideas to deal with our Eq. (1.1) under nearly minimal restrictions on f . 1/k The most typical examples of f satisfying (1.3)–(1.5) are f = σk and f = (σk /σl )1/(k−l) , 1 ≤ l < k ≤ n + 1, defined in the G˚ arding cone Γk = {λ ∈ Rn+1 : σj (λ) > 0, j = 1, . . . , k}, where σk are the elementary symmetric functions  σk (λ) = λi1 . . . λik ,

k = 1, . . . , n + 1.

i1 <···
When f = σn+1

, Eq. (1.1) can be written as the parabolic Monge–Amp`ere equation: − ut det(∇2 u + χ) = ψ n+1 ,

(1.7)

which was introduced by Krylov in [19] when χ = 0 in Euclidean space. Instead of the determinant in (1.7), Ren [25] studied equations of the form − ut f (λ(∇2 u)) = ψ(x, t).

(1.8)

Our interest to study (1.1) is from their natural connection to the deformation of surfaces by some curvature functions. For example, Eq. (1.7) plays a key role in the study of contraction of surfaces by Gauss–Kronecker curvature (see Firey [5] and Tso [28]). For the study of more general curvature flows, the reader is referred to [1,2,14,24] and their references. (1.7) is also relevant to a maximum principle for parabolic equations (see Tso [29]). In [23], Lieberman studied the first initial–boundary value problem of Eq. (1.1) when χ ≡ 0 and ψ may depend on u and ∇u in a bounded domain Ω ⊂ Rn+1 under various conditions. Jiao and Sui [18] considered parabolic Hessian equations of the form f (λ(∇2 u + χ)) − ut = ψ(x, t)

(1.9)

on Riemannian manifolds under an additional condition which was introduced in [10] Tλ ∩ ∂Γ σ is a nonempty compact set, ∀λ ∈ Γ and sup f < σ < f (λ),

(1.10)

∂Γ

where ∂Γ σ = {λ ∈ Γ : f (λ) = σ} is the boundary of Γ σ = {λ ∈ Γ : f (λ) > σ} and Tλ denote the tangent plane at λ of ∂Γ f (λ) , for σ > sup∂Γ f and λ ∈ Γ . Eq. (1.9) in domains of Rn was also studied by Ivochkina

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and Ladyzhenskaya in [15] (for the Monge–Amp`ere case) and [16]. A generalization of (1.9) was considered in [17]. Guan, Shi and Sui [12] extended the work of [18] using the idea of [7]. As we know, works about the elliptic Hessian equations usually provide useful techniques to deal with the parabolic version. The reader is referred to [21,30,8,9,7,10,11] and their references for examples. The motivation to assume that f is defined in the cone Γ is due to the consideration that many equations are elliptic (or parabolic) with respect to solutions in a cone but they are not in general (see the examples above). We mean an admissible function by u ∈ C 2 (MT ) satisfying (λ(∇2 u + χ), −ut ) ∈ Γ in MT , where C k (MT ) denotes the space of functions defined on MT which are k-times continuously differentiable with respect to x ∈ M and [k/2]-times continuously differentiable with respect to t ∈ (0, T ] and [k/2] is the largest integer not greater than k/2. We see that (1.1) is parabolic for admissible solutions (see [3]). We first recall the following notations  sup |∇β Dtr u|, |u|C k (MT ) = |β|+2r≤k MT

|u|C k+α (MT ) = |u|C k (MT ) +

sup

sup

|β|+2r=k

(x,s),(y,t)∈MT (x,s)̸=(y,t)

|∇β Dtr u(x, s) − ∇β Dtr u(y, t)|  α |x − y| + |s − t|1/2

and C k+α (MT ) denotes the subspace of C k (MT ) defined by C k+α (MT ) := {u ∈ C k (MT ) : |u|C k+α (MT ) < ∞}. Our main result is stated in the following theorem. Theorem 1.1. Suppose that ψ ∈ C ∞ (MT ), ϕ ∈ C ∞ (PMT ) for 0 < T < ∞, and there exists a function Θ ∈ C 2 (BMT ) such that Θ = −ϕt on ∂M × {0} and (λ(∇2 ϕ(x, 0) + χ(x)), Θ(x)) ∈ Γ

for all x ∈ M

(1.11)

and that f (λ(∇2 ϕ(x, 0) + χ(x)), −ϕt (x, 0)) = ψ(x, 0) for all x ∈ ∂M.

(1.12)

In addition to (1.3)–(1.5), assume that 

fj (λ) ≥ ν0 1 +

n+1 

 fi (λ)

for any λ ∈ Γ with λj < 0,

(1.13)

i=1

for some positive constant ν0 , n+1  i=1

n+1    fi λi ≥ −K0 1 + fi ,

∀λ ∈ Γ

(1.14)

i=1

for some constant K0 ≥ 0 and that there exists an admissible subsolution u ∈ C 2 (MT ) satisfying  2  f (λ(∇ u + χ), −ut ) ≥ ψ(x, t) in MT , u=ϕ on SMT ,   u≤ϕ on BMT .

(1.15)

Then there exists a unique admissible solution u ∈ C ∞ (MT ) of (1.1)–(1.2). Remark 1.2. Conditions (1.13) and (1.14) are only used to derive the gradient estimates which are commonly used, see [23,13,8,22,26,30] for examples. It would be an interesting problem to establish the gradient estimates without (1.13) and (1.14).

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If f > sup∂Γ f > −∞ in Γ , it is easy to show that n+1 

fi λi ≥ sup f ≥ −K0 ,

∀λ ∈ Γ

∂Γ

i=1

by (1.3) and (1.4). Remark 1.3. As in [7], the existence of u is useful to construct some barrier functions which are crucial to our estimates. We can prove the short time existence as Theorem 15.9 in [23]. So without of loss of generality, we may assume that ϕ is defined on M × [0, t0 ] for some small constant t0 > 0 and f (λ(∇2 ϕ(x, 0) + χ(x)), −ϕt (x, 0)) = ψ(x, 0)

for all x ∈ M .

(1.16)

As usual, the main part of this paper is to derive the a priori C 2 estimates. We see that (1.1) is uniformly parabolic after establishing the C 2 estimates by (1.3) and (1.5). The C 2,α estimates can be obtained by applying Evans–Krylov theorem (see [4,20]). Finally Theorem 1.1 can be proved as Theorem 15.9 of [23]. The rest of this paper is devoted to the a prior C 2 estimates for admissible solutions of (1.1)–(1.2). In Section 2, we introduce some notations and one useful lemma. C 1 estimates are derived in Section 3. An a priori bound for |ut | is obtained in Section 4. In Sections 5 and 6 we deal with the global and boundary estimates for second order derivatives respectively. 2. Preliminaries Let F be the function defined by F (A, τ ) = f (λ(A), τ ) for A ∈ Sn , τ ∈ R with (λ(A), τ ) ∈ Γ , where Sn is the set of n × n symmetric matrices. It was shown in [3] that (1.4) implies the concavity of F . Throughout this paper ∇ denotes the Levi-Civita connection of (M, g). For simplicity we shall use the notations U = ∇2 u + χ, U = ∇2 u + χ and under an orthonormal local frame e1 , . . . , en , Uij ≡ U (ei , ej ) = ∇ij u + χij ,

U ij ≡ U (ei , ej ) = ∇ij u + χij .

Thus, (1.1) can be written in the form locally F (U, −ut ) = f (λ(Uij ), −ut ) = ψ.

(2.1)

Let F ij = F ij,kl =

∂F (U, −ut ), ∂Aij

∂2F (U, −ut ), ∂Aij ∂Akl

F ij,τ =

Fτ =

∂F (U, −ut ) ∂τ

∂2F (U, −ut ), ∂Aij ∂τ

F ττ =

∂2F (U, −ut ). ∂τ 2

By (1.3) we see that F τ > 0 and {F ij } is positive definite. We shall also denote the eigenvalues of {F ij } by f1 , . . . , fn when there is no possible confusion. We note that {Uij } and {F ij } can be diagonalized simultaneously and that     i , 2 , F ij Uij = fi λ F ij Uik Ukj = fi λ i ij

i

ijk

1 , . . . , λ n ). where λ({Uij }) = (λ We write µ(x, t) = (λ(U (x, t)), −ut (x, t)), λ(x, t) = (λ(U (x, t)), −ut (x, t))

i

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and νλ ≡ Df (λ)/|Df (λ)| is the unit normal vector to the level hypersurface ∂Γ f (λ) for λ ∈ Γ . Since 1 ) such K ≡ {µ(x, t) : (x, t) ∈ MT } is a compact subset of Γ , there exist uniform constants β ∈ (0, 2√n+1 that νµ(x,t) − 2β1 ∈ Γn+1 ,

∀(x, t) ∈ MT

(2.2)

where 1 = (1, . . . , 1) ∈ Rn+1 (see [7]). We need the following lemma which is Lemma 2.1 in [7]. (An general version can be found in [12].) Lemma 2.1. Suppose that |νµ − νλ | ≥ β. Then there exists a uniform constant ε > 0 such that n+1 

n+1    fi (λ)(µi − λi ) ≥ ε 1 + fi (λ) .

i=1

(2.3)

i=1

Define the linear operator L locally by  Lv = F ij ∇ij v − F τ vt ,

for v ∈ C 2 (MT ).

ij

By Lemma 2.1 and Lemma 6.2 of [3] it is easy to derive that when |νµ(x,t) − νλ(x,t) | ≥ β,    F ii + F τ . L(u − u) ≥ ε 1 +

(2.4)

If |νµ − νλ | < β, we have νλ − β1 ∈ Γn+1 . It follows that fi ≥ √

n+1  β fj , n + 1 j=1

∀1 ≤ i ≤ n + 1.

(2.5)

3. The C 1 estimates n+1 n+1 First we note that Γ ⊂ {λ ∈ Rn+1 : i=1 λi > 0}. Indeed, if there exists λ ∈ Γ such that i=1 λi < 0, we can conclude that Γ = Rn+1 since it is a symmetric and convex cone which contradicts the n+1 fact that Γ ̸= Rn+1 . Thus, Γ ⊂ {λ ∈ Rn+1 : i=1 λi ≥ 0} and by the openness of Γ , we have  n+1 Γ ⊂ {λ ∈ Rn+1 : i=1 λi > 0}. It follows that u is a subsolution of  △h − ht + tr(χ) = 0, in MT , (3.1) h = ϕ, on PMT since u is admissible. Let h be the solution of (3.1). It follows from the maximum principle that u ≤ u ≤ h on MT . Therefore, we have sup |u| + sup |∇u| ≤ C0 , MT

(3.2)

PMT

where C0 depends on |u|C 1 (MT ) and |h|C 1 (MT ) . For the global gradient estimates, we can prove the following maximum principle. Theorem 3.1. Suppose that (1.3), (1.4), (1.13) and (1.14) hold. Let u ∈ C 3 (MT ) be an admissible solution of (1.1) in MT . Then sup |∇u| ≤ C1 (1 + sup |∇u|), MT

PMT

where C1 depends on |ψ|C 1 (MT ) , |u|C 0 (MT ) and other known data.

(3.3)

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Proof. Set W =

weφ ,

sup (x,t)∈MT

|∇u|2 2

where w = and φ is a function to be determined. It suffices to estimate W and we may assume that W is achieved at (x0 , t0 ) ∈ MT − PMT . Choose a smooth orthonormal local frame e1 , . . . , en about x0 such that ∇ei ej = 0 at x0 and U (x0 , t0 ) is diagonal. We see that the function log w + φ attains its maximum at (x0 , t0 ). Therefore, at (x0 , t0 ), we have ∇i w + ∇i φ = 0, for each i = 1, . . . , n, w wt + φt ≥ 0 w

(3.4) (3.5)

and ∇ii w  ∇i w 2 − + ∇ii φ ≤ 0. w w

(3.6)

Differentiating the Eq. (1.1), we get n 

F ii ∇k Uii − F τ ∇k ut = ∇k ψ

for k = 1, . . . , n

(3.7)

i=1

and n 

F ii (Uii )t − F τ utt = ψt .

(3.8)

i=1

Note that ∇i w =



∇k u∇ik u,

wt =



k

∇k u(∇k u)t ,

k

∇ii w =



(∇ik u)2 + ∇k u∇iik u

(3.9)



k

and that ∇ijk u − ∇jik u =



l Rkij ∇l u,

(3.10)

l l where Rkij = g im Rmjkl and Rmjkl = g(R(ek , el )ej , em ) are coefficients of the curvature tensor. We have, by (3.5), (3.7), (3.9) and (3.10),   F ii ∇ii w ≥ ∇k uF ii ∇iik u i

i,k

 =



∇k uF

ii



∇kii u −



i,k

≥



l Riik ∇l u

l ii

∇k uF ∇k Uii − C|∇u|2



F ii

i,k

≥ −C|∇u| − C|∇u|2



≥ −C|∇u| − C|∇u|2



F ii +



F τ ∇k u∇k ut

k

F ii − wF τ φt ,

provided |∇u| is sufficiently large. Combining (3.4), (3.6) and (3.11), we obtain   C −C F ii − F ii (∇i φ)2 + Lφ. 0≥− |∇u|

(3.11)

(3.12)

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Let φ = δv 2 , where v = u + supMT |u| + 1 and δ is a small positive constant to be chosen. Thus, choosing δ sufficiently small such that 2δ − 4δ 2 v 2 ≥ c0 > 0 for some uniform constant c0 , and by (1.14), we see   Lφ − F ii (∇i φ)2 = 2δvLu + (2δ − 4δ 2 v 2 ) F ii (∇i u)2     ≥ c0 (3.13) F ii (∇i u)2 − Cδ 1 + F ii + F τ . It follows from (3.12) and (3.13) that     c0 F ii (∇i u)2 ≤ C 1 + F ii + F τ ,

(3.14)

provided |∇u| is sufficiently large. We may assume |∇1 u(x0 , t0 )| = max1≤i≤n |∇i u(x0 , t0 )|. It follows that |∇u(x0 , t0 )| ≤ n|∇1 u(x0 , t0 )|. Recalling that {Uij } is diagonal, by (3.4), we have U11 = −2δvw + n 2δ

∇k uχ1k <0 ∇1 u



|χ1k |. Then we can derive from (1.13) that    F 11 ≥ ν0 1 + F ii + F τ . √ Therefore, we obtain a bound |∇u(x0 , t0 )| ≤ Cn/ c0 ν0 by (3.14) so that (3.3) is proved. provided w >

maxM¯

k



Remark 3.2. We see that in the proof of Theorem 3.1, we do not need the existence of u. By (3.2) and (3.3), the C 1 estimates are established. 4. The estimates for |ut | In this section, we derive the estimates for |ut |. Theorem 4.1. Suppose that (1.3), (1.4) and (1.15) hold. Let u ∈ C 3 (MT ) be an admissible solution of (1.1) in MT . Then there exists a positive constant C2 depending on |u|C 1 (MT ) , |u|C 2 (MT ) , |ψ|C 2 (MT ) and other known data such that sup |ut | ≤ C2 (1 + sup |ut |).

(4.1)

PMT

MT

Proof. We first show that sup(−ut ) ≤ C(1 + sup |ut |)

(4.2)

PMT

MT

for which we set W = sup(−ut )eφ , MT

where φ is a function to be chosen. We may assume that W is attained at (x0 , t0 ) ∈ MT − PMT . As in the proof of Theorem 3.1, we choose an orthonormal local frame e1 , . . . , en about x0 such that ∇ei ej = 0 and {Uij (x0 , t0 )} is diagonal. We may assume −ut (x0 , t0 ) > 0. At (x0 , t0 ) where the function log(−ut ) + φ achieves its maximum, we have ∇i ut + ∇i φ = 0, for each i = 1, . . . , n, ut utt + φt ≥ 0, ut

(4.3) (4.4)

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and 0≥



F ii

i

 ∇ u  ∇ u 2 ii t i t + ∇ii φ . − ut ut

(4.5)

Combining (4.3)–(4.5), we find 0≥

  1  ii F ∇ii ut − F τ utt − F ii (∇i φ)2 + Lφ. ut

(4.6)

By (3.8) and (4.6), Lφ ≤ −

ψt  ii + F (∇i φ)2 . ut

(4.7)

1+α

Fix a positive constant α ∈ (0, 1) and let φ = δ 2 |∇u|2 + δu + b(u − u), where δ ≪ b ≪ 1 are positive constants to be determined. By straightforward calculations, we have  ∇i φ = δ 1+α ∇k u∇ik u + δ∇i u + b∇i (u − u), k

φt = δ

1+α



∇k u(∇k u)t + δut + b(u − u)t ,

k

∇ii φ = δ 1+α



∇ik u

2

+ δ 1+α

k



∇k u∇iik u + δ∇ii u + b∇ii (u − u).

k

It follows that, in view of (3.7) and (3.10),   δ 1+α   F ii Uii2 Lφ ≥ δ 1+α ∇k u F ii ∇iik u − F τ (∇k u)t + 2 i k   ii +δ F ∇ii u − δF τ ut − Cδ 1+α F ii + bL(u − u)   δ 1+α   ≥ −Cδ 1+α 1 + F ii Uii2 + δLu + bL(u − u). F ii + 2

(4.8)

Next, (∇i φ)2 ≤ Cδ 2(1+α) Uii2 + Cb2

(4.9)

since b ≫ δ. Thus, we can derive from (4.7)–(4.9) that bL(u − u) +

    C δ 1+α  ii 2 F Uii + δLu ≤ − + Cδ 1+α 1 + F ii + Cb2 F ii , 4 ut

(4.10)

provided δ is sufficiently small. Now we use the idea of [7] to consider two cases: (i) |νµ0 − νλ0 | ≥ β and (ii) |νµ0 − νλ0 | < β, where µ0 = µ(x0 , t0 ) and λ0 = λ(x0 , t0 ). In case (i), by Lemma 2.1, we see that (2.4) holds. Since −ut (x0 , t0 ) > 0, we have    δLu = δ F ii Uii − F ii χii − F τ ut   ≥δ F ii Uii − Cδ F ii  δ 1+α  ii 2 F Uii − Cδ F ii . 4

(4.11)

    C + Cδ 1−α 1 + F ii + Cb2 F ii . ut

(4.12)

≥ −δ 1−α



F ii −

Combining (4.11) and (4.10), we have bL(u − u) ≤ −

By (2.4), we can obtain a bound −ut (x0 , t0 ) ≤

Cδ 1−α bε

provided δ ≪ b ≪ 1.

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In case (ii), we see that (2.5) holds. By (4.10), we find   δ 1+α  ii 2 F Uii + δ F ii Uii − F τ ut 4     C F ii . F ii + C(δ + b2 ) ≤ − + Cδ 1+α 1 + ut

bL(u − u) +

(4.13)

Note that   δ 1+α  ii 2 F Uii + δ F ii Uii ≥ −2δ 1−α F ii 8

(4.14)

L(u − u) ≥ 0

(4.15)

and

by the concavity of F . Therefore, by (4.13)–(4.15), we have  C δ 1+α  ii 2 F Uii − δF τ ut ≤ − + Cδ 1+α + C F ii . 8 ut

(4.16)

By the concavity of f , recalling that ut < 0, we get    − ut F ii + F τ ≥ f (−ut 1) − f (λ(U ), −ut ) + F ii Uii − F τ ut ≥ f (−ut 1) − f (λ(U ), −ut )    1  ii 2 + ut F ii + F τ + F Uii + F τ u2t , 4ut

(4.17)

where 1 = (1, . . . , 1) ∈ Rn+1 . Note that there exists a constant R > 0 depending only on |u|C 2 (MT ) such that |(λ(U ), −ut )| ≤ R on MT . It follows from (1.3) that f (2R1) − f (λ(U ), −ut ) ≥ f (2R1) − f (R1) := 2b0 .

(4.18)

We may assume −ut (x0 , t0 ) > 2R for otherwise we are done. Therefore, combining (4.17) and (4.18), we obtain    1  ii 2 F Uii + F τ u2t . − ut F ii + F τ ≥ b0 + (4.19) 8ut It follows from (2.5) and (4.19) that   γ0  ii 2 F ii + F τ + γ0 b0 + F Uii + F τ u2t 8ut  7 γ0  ii 2 = −γ0 ut F ii − γ0 ut F τ + γ0 b0 + F Uii 8 8ut  γ0  ii 2 ≥ −γ0 ut F ii + γ0 b0 + F Uii , 8ut

− F τ ut ≥ −γ0 ut

where γ0 :=

√β 2 n+1



(4.20)

> 0.

Substituting (4.20) in (4.16) we obtain  δ 1+α 8

+

  δγ0   ii 2 C F Uii − δγ0 ut F ii + δγ0 b0 ≤ − + Cδ 1+α + C F ii . 8ut ut

Choose δ sufficiently small such that δγ0 b0 − Cδ 1+α ≥ c1 > 0

(4.21)

G. Bao et al. / Nonlinear Analysis 143 (2016) 45–63

54

for some constant c1 . We can derive from (4.21) that −ut (x0 , t0 ) ≤ max

γ

0 , δα

C C , δγ0 c1

and therefore (4.2) holds. Similarly, we can show sup ut ≤ C(1 + sup |ut |),

(4.22)

PMT

MT

by setting W = sup ut eφ MT

and φ =

δ

1+α

2

|∇u|2 − δu + b(u − u).

Combining (4.2) and (4.22), we can see that (4.1) holds.



Since that ut = ϕt on SMT and (1.16), we can derive the estimate sup |ut | ≤ C3 ,

(4.23)

MT

where the constant C3 depends on C2 in (4.2) and |ϕ|C 2 (MT ) . 5. Global estimates for second order derivatives In this section, we derive the global estimates for the second order derivatives. In particular, we prove the following maximum principle. Theorem 5.1. Let u ∈ C 4 (MT ) be an admissible solution of (1.1) in MT . Suppose that (1.3), (1.4) and (1.15) hold. Then sup |∇2 u| ≤ C4 (1 + sup |∇2 u|),

(5.1)

PMT

MT

where C4 > 0 depends on |u|C 1 (MT ) , |ut |C 0 (MT ) , |ψ|C 2 (MT ) and other known data. Proof. Set W =

max

max

(x,t)∈MT ξ∈Tx M,|ξ|=1

(∇ξξ u + χ(ξ, ξ))eφ ,

where φ is a function to be determined. We may assume W is achieved at (x0 , t0 ) ∈ MT − PMT and ξ0 ∈ Tx0 M . Choose a smooth orthonormal local frame e1 , . . . , en about x0 as before such that ξ0 = e1 , ∇ei ej = 0, and {Uij (x0 , t0 )} is diagonal. We see that W = U11 (x0 , t0 )eφ(x0 ,t0 ) . We may also assume that U11 ≥ · · · ≥ Unn at (x0 , t0 ). Since the function log(U11 ) + φ attains its maximum at (x0 , t0 ), we have, at (x0 , t0 ), ∇i U11 + ∇i φ = 0 U11

for each i = 1, . . . , n,

(∇11 u)t + φt ≥ 0, U11

(5.2) (5.3)

and 0≥



F ii

∇ U  ∇ U 2  ii 11 i 11 − + ∇ii φ . U11 U11

(5.4)

G. Bao et al. / Nonlinear Analysis 143 (2016) 45–63

Therefore, by (5.3) and (5.4), we find  ∇ U 2   1  ii i 11 Lφ ≤ − . F ∇ii U11 − F τ (∇11 u)t + F ii U11 U11 By the formula    m m m ∇ijkl v − ∇klij v = Rljk ∇im v + ∇i Rljk ∇m v + Rlik ∇jm v m

m



+

m Rjik ∇lm v

+

m

55

(5.5)

m



m Rjil ∇km v

m

+



m ∇k Rjil ∇m v,

(5.6)

m

we have ∇ii U11 ≥ ∇11 Uii − CU11 .

(5.7)

F ij ∇11 Uij − F τ ∇11 ut + F ij,kl ∇1 Uij ∇1 Ukl + F τ τ (∇1 ut )2 − 2F ij,τ ∇1 Uij ∇1 ut = ∇11 ψ ≥ −C.

(5.8)

Differentiating Eq. (1.1) twice, we have

It follows from (5.5), (5.7) and (5.8) that Lφ ≤

 C +C F ii + E, U11

(5.9)

where E=

   ∇ U 2  1  ij,kl i 11 . F ∇1 Uij ∇1 Ukl − 2 F ij,τ ∇1 Uij ∇1 ut + F τ τ (∇1 ut )2 + F ii U11 U 11 ij ijkl

E can be estimated as in [9] using an idea of Urbas [30] to which the following inequality proved by Andrews [1] and Gerhardt [6] is crucial. Lemma 5.2. For any symmetric matrix η = {ηij } we have   ∂2f  fi − fj 2 F ij,kl ηij ηkl = ηii ηjj + ηij . ∂λ ∂λ λ − λ i j i j ij ijkl

i̸=j

The second term on the right hand side is nonpositive if f is concave, and is interpreted as a limit if λi = λj . ˆ by To proceed we define the (n + 1) × (n + 1) matrix U   U (x , t ) 0 ij 0 0 ˆ= U . 0 −ut (x0 , t0 ) Let J = {i : 3Uii ≤ −U11 } and K = {i : 3Uii > −U11 }. Therefore, by Lemma 5.2, we have   − F ij,kl ∇1 Uij ∇1 Ukl + 2F ij,τ ∇1 Uij ∇1 ut − F τ τ ∇1 ut ∇1 ut ij

ijkl



≥

1≤i,j≤n,i̸=j

F ii − F jj (∇1 Uij )2 Ujj − Uii

 F ii − F 11 (∇1 Ui1 )2 U11 − Uii 2≤i≤n 3  ii ≥ (F − F 11 )(∇1 Ui1 )2 2U11 ≥2

i∈K

1  ii ≥ (F − F 11 )((∇i U11 )2 − C), U11 i∈K

(5.10)

G. Bao et al. / Nonlinear Analysis 143 (2016) 45–63

56

where the last inequality is derived from the fact that 2 2 (∇i U11 )2 ≤ (∇1 U1i )2 + C 3 3 which follows from (3.10). Thus, by (5.10) and (5.2), we obtain C  ii F 11  1  ii F (∇i U11 )2 + 2 F + 2 (∇i U11 )2 E ≤ 2 U11 U11 U11 i∈J i∈K i∈K    ii 2 ii 11 2 ≤ F (∇i φ) + C F +F (∇i φ) . i∈J

i∈K

(5.11)

(5.12)

i∈K

Let δ|∇u|2 + b(u − u), 2 where δ ≪ 1 ≪ b are positive constants to be determined. We have φ=

∇i φ ≤ δ∇i uUii + Cb. Thus, we can derive from (5.12) that    2 E ≤ Cb2 F ii + Cδ 2 F ii Uii2 + C F ii + C(δ 2 U11 + b2 )F 11 . i∈J

(5.13)

i∈K

On the other hand, by (3.7) and (3.10),    Lφ = δ F ii (∇ik u)2 + δ ∇k uF ii ∇iik u − δ ∇k uF τ (∇k u)t + bL(u − u) ik

≥δ



ik

k



F ii Uii2 + bL(u − u) − Cδ 1 +





F ii .

(5.14)

Combining (5.9), (5.13) and (5.14), we obtain     δ  ii 2 C F Uii + bL(u − u) ≤ + Cb2 F ii + Cb2 F 11 + C 1 + F ii 2 U11

(5.15)

i∈J

provided δ is sufficiently small. Note that |Ujj | ≥ 13 U11 , for j ∈ J. Therefore, by (5.15), we have    δ  ii 2 F ii F Uii + bL(u − u) ≤ C 1 + 4 2 ≥ max{12Cb2 /δ, 1}. when U11

(5.16)

Now let µ0 = µ(x0 , t0 ) and λ0 = λ(x0 , t0 ). If |λ0 − µ0 | ≥ β, we can obtain a bound of U11 (x0 , t0 ) by (2.4) as in [9] when b is sufficiently large.  = λ(U (x0 , t0 )). We may assume |λ|  ≥ |ut (x0 , t0 )|. By the If |λ0 − µ0 | < β, we see that (2.5) holds. Let λ concavity of f , we have      − f (λ(U ), −ut ) + |λ| F ii + F τ ≥ f (|λ|1) F ii Uii − F τ ut    − f (λ(U ), −u ) − |λ|  ≥ f (|λ|1) F ii + F τ . (5.17) t  ≥ 2R for otherwise we are done. By (5.17) and As in the estimate of |ut | (Section 4), we may assume |λ| (4.18), we have    |λ| F ii + F τ ≥ b0 , (5.18) where b0 is the constant defined in (4.18).

G. Bao et al. / Nonlinear Analysis 143 (2016) 45–63

By (2.5), (4.15) and (5.16), we see that      2 F ii , 2c2 |λ| F ii + F τ ≤ C 1 +

57

(5.19)

where δβ c2 := √ . 8 n+1  from (5.18). Then we can derive a bound of |λ| Therefore in both cases we have U11 (x0 , t0 ) ≤ C.



6. Boundary estimates for second order derivatives In this section, we consider the estimates for second order derivatives on SMT . We may assume ϕ ∈ C (MT ). We shall establish the following estimate 4

max |∇2 u| ≤ C5 ,

(6.1)

PMT

for some positive constant C5 depending on |u|C 1 (MT ) , |ut |C 0 (MT ) , |u|C 2 (MT ) , |ϕ|C 4 (MT ) and other known data. Fix a point (x0 , t0 ) ∈ SMT . We choose smooth orthonormal local frames e1 , . . . , en around x0 such that when restricted to ∂M , en is normal to ∂M . Since u − u = 0 on SMT , we have ∇αβ (u − u) = −∇n (u − u)Π (eα , eβ ),

∀ 1 ≤ α, β < n on SMT ,

(6.2)

where Π denotes the second fundamental form of ∂M . Therefore, |∇αβ u| ≤ C,

∀ 1 ≤ α, β < n on SMT .

(6.3)

Let ρ(x) and d(x) denote the distance from x ∈ M to x0 and ∂M respectively and set MTδ = {X = (x, t) ∈ M × (0, T ] : ρ(x) < δ}. Clearly ρ(x) < δ implies that d(x) < δ. We shall use the following barrier function as in [9]. Ψ = A1 v + A2 ρ 2 − A3



|∇γ (u − ϕ)|2 ,

(6.4)

γ
where v = (u − u) + ad −

N d2 . 2

The following lemma is crucial to construct barrier functions and the idea is mainly from [7,10] (see [12] also). Lemma 6.1. Suppose that (1.3), (1.4) and (1.15) hold. Then for any constant K > 0, there exist uniform positive constants a, δ sufficiently small, and A1 , A2 , A3 , N sufficiently large such that Ψ ≥ K(d + ρ2 ) in MTδ and 

LΨ ≤ −K 1 +

n  i=1

i | + fi |λ

n  i=1

fi + F τ



in MTδ .

(6.5)

G. Bao et al. / Nonlinear Analysis 143 (2016) 45–63

58

Proof. For any fixed (x, t) ∈ MTδ , we may assume that Uij and F ij are both diagonal at (x, t). Firstly, we have (see [9] for details), n n     i | + L(∇k (u − ϕ)) ≤ C 1 + fi |λ fi + F τ , i=1

∀ 1 ≤ k ≤ n.

(6.6)

i=1

Therefore, 



L(|∇l (u − ϕ)|2 ) ≥

l
n n     i | + F ij Uil Ujl − C 1 + fi |λ fi + F τ . i

l
(6.7)

i

Using the same method of Proposition 2.19 in [9], we can show  1  2 fi λi , F ij Uil Ujl ≥ 2

(6.8)

i̸=r

l
 −ut ), where for some index r. Write µ = µ(x, t) and λ = λ(x, t) and note that µ = ( µ, −ut ) and λ = (λ, µ  = λ(U ). We shall consider two cases as before: (a) |νµ − νλ | < β and (b) |νµ − νλ | ≥ β. Case (a). By (2.5), we have n

fi ≥ √

 β  fk + F τ , n + 1 k=1

∀ 1 ≤ i ≤ n.

(6.9)

Now we make a little modification of the proof of Lemma 3.1 in [7] to show the following inequality      2 ≥ c3 2 − 1 fi λ fi + F τ fi λ (6.10) i i c3 i i̸=r

r < 0, we have for some c3 > 0. If λ 2 ≤ n λ r



2 + C, λ i

(6.11)

i̸=r

where C depends on the bound of ut since 

i − ut > 0. λ

Therefore, by (6.9) and (6.11), we have 2 ≤ nfr fr λ r

 i̸=r

√  n n + 1  2 2  fi λi + C fi λi + Cfr ≤ β i

(6.12)

i̸=r

and (6.10) holds. r ≥ 0. By the concavity of f , Now suppose λ  r ≤ fr µ fr λ r − F τ (ut − ut ) +



i ). fi ( µi − λ

(6.13)

i̸=r

Thus, by (6.9) and Schwarz inequality, we have       2  βf λ 2 2 2 τ 2 2 ≤ C f 2 µ  √ r r f ( µ + λ ) + (F ) fi + F τ ≤ fr2 λ  + f k i i r r r i n+1 k̸=r i̸=r      2 , fi λ ≤C fi + F τ fi + F τ + i i̸=r

(6.14)

G. Bao et al. / Nonlinear Analysis 143 (2016) 45–63

where C may depend on the bound of |ut |. It follows that    2 ≤ C 2 + C fr λ fi λ fi + F τ r i

59

(6.15)

i̸=r

and (6.10) holds. Next, recall that d < δ in MTδ . Since |∇d| ≡ 1, in view of (6.9), we have n



F ij ∇i d∇j d ≥ (min fi )|∇d|2 = min fi ≥ √ i

i,j

i

 β  fk + F τ . n + 1 k=1

It follows that when a and δ are sufficiently small such that C(a + N d) < 2√βN , we have, n+1     fi − N F ij ∇i d∇j d Lv ≤ L(u − u) + C(a + N d) ij

βN ≤− √ 2 n+1





fk + F τ .

(6.16)

 ≥ R for R sufficiently large. By (2.5) and (5.18), we see We first suppose |λ| 

 2 ≥ √ β b0 |λ| fi λ i n+1

(6.17)

when R is sufficiently large. Note that for any σ > 0, 

i | ≤ σ fi |λ



 2 + 1 fi λ fi . i σ

(6.18)

Therefore, It follows from (6.10), (6.16) and (6.18) that for any σ > 0,       βA1 N  A3   2 i | + fi λi + CA3 1 + fi |λ fi + F τ fk + F τ + CA2 LΨ ≤ − √ fi − 2 2 n+1 i̸=r  A c       βA1 N  3 3 2 + CA2 i | + ≤− √ fk + F τ − fi λ fi + CA3 1 + fi |λ fi + F τ i 2 2 n+1     A3 c 3    2 βA1 N fi λi fk + F τ + A3 σ − ≤− √ 2 2 n+1    A3   + C A2 + fi + CA3 1 + F τ . (6.19) σ Let σ = c3 /4, we find  A c    βA1 N  3 3 2 + C(A2 + A3 ) LΨ ≤ − √ fk + F τ − fi λ fi + F τ + CA3 i 4 2 n+1      A3 βc3 b0  βA1 N A3   ≤− √ fk + F τ − √ |λ| − fi |λi | + C(A2 + A3 ) fi + F τ + CA3 2 2 n+1 8 n+1         βA1 N A3 i | + C(A2 + A3 ) ≤− √ fi + F τ − 1+ fi |λ fi + F τ (6.20) 2 2 n+1 √ by choosing R ≥ 8 n + 1C/βc3 b0 + 1.  ≤ R, by (1.3) and (1.5), we have If |λ| cR I ≤ {F ij } ≤ CR ,

cR ≤ F τ ≤ CR

G. Bao et al. / Nonlinear Analysis 143 (2016) 45–63

60

for some uniform positive constants cR , CR which may depend on R. Therefore, by (6.16), we have     LΨ ≤ C(−A1 + A2 + A3 ) 1 + fi + fi |λi | + F τ (6.21) where C depends on cR and CR . Case (b). By Lemma 2.1, we may fix a and δ sufficiently small such that v ≥ 0 in MTδ and   ε in MTδ . (6.22) fi + F τ Lv ≤ − 1 + 2 r ≥ 0. By (6.7), (6.8) and (6.22) we see for any 0 < B < A1 (see [10]), We first consider the case that λ      A3   2 i | + LΨ ≤ (A1 + B)Lv − BLv + CA2 fi − fi λi + CA3 1 + fi |λ fi + F τ 2 i̸=r         (A1 + B)ε A3   2 i + CB 1 + 1+ fi + F τ − B fi λ fi λi ≤− fi + F τ + CA2 fi − 2 2 i̸=r     i | + + CA3 1 + fi |λ fi + F τ    (A1 + B)ε  2  r − A3 fi λ 1+ fi + F τ − (B − CA3 )fr λ ≤− i 2 2 i̸=r     i |. + C(B + A2 + A3 ) 1 + fi + F τ + (B + CA3 ) fi |λ (6.23) i̸=r

Notice that 2   A3   2 i | − 2(B + CA3 ) fi λi ≥ 2(B + CA3 ) fi . fi |λ 2 A3 i̸=r

(6.24)

i̸=r

Thus, we derive from (6.23) and (6.24) that    (A1 + B)ε  i | LΨ ≤ − 1+ fi + F τ − (B − CA3 ) fi |λ 2   2(B + CA )2   3 + C(B + A2 + A3 ) 1 + fi + F τ + fi . A3

(6.25)

r < 0, similar to (6.25), we have If λ    (A1 − B)ε  i | 1+ fi + F τ − (B − CA3 ) fi |λ 2   2(B + CA )2   3 + C(B + A2 + A3 ) 1 + fi + F τ + fi . A3

LΨ ≤ −

(6.26)

Checking (6.20), (6.21), (6.25) and (6.26), we can choose A1 ≫ A2 ≫ A3 ≫ 1 and A1 − B ≫ B ≫ A3 in (6.25) and (6.26) such that (6.5) holds and Ψ ≥ K(d + ρ2 ) in MTδ . Therefore, Lemma 6.1 is proved.  By (6.6), we can use Lemma 6.1 to choose suitable A1 ≫ A2 ≫ A3 ≫ 1 such that in MTδ , L(Ψ ± ∇α (u − ϕ)) ≤ 0, and Ψ ± ∇α (u − ϕ) ≥ 0 on PMTδ . Then it follows from the maximum principle that Ψ ± ∇α (u − ϕ) ≥ 0 in MTδ and therefore |∇nα u(x0 , t0 )| ≤ ∇n Ψ (x0 , t0 ) ≤ C,

∀ α < n.

It remains to establish the estimate sup ∇nn u ≤ C SMT

(6.27)

G. Bao et al. / Nonlinear Analysis 143 (2016) 45–63

61

˜ (x, t) be the restriction of U (x, t) to Tx ∂M , the tangent since −ut + △u + trχ ≥ 0. For (x, t) ∈ SMT , let U ′ ˜ space of ∂M at x, and λ (U (x, t)) be the eigenvalues with respect to the induced metric. Similarly one can ˜ (x, t) and λ′ (U ˜ (x, t)). The proof of (6.27) is similar to that of the elliptic case using an idea of define U Trudinger [27] (see [9,12]), so we only provide a sketch here. Define ˜ , −ut ) := lim f (λ′ (U ˜ ), R, −ut ) F˜ (U R→∞

on SMT . We shall show that the following quantity   ˜ , −ut )(x, t) − ψ(x, t) F˜ (U m := min (x,t)∈SMT

is positive. Without loss of generality we assume that m is finite. It is easy to see that   ˜ , −u )(x, t) − ψ(x, t) > 0, ~ := min F˜ (U t (x,t)∈SMT

and we remark that ~ may be +∞. We may assume m < ~/2 (otherwise we are done). Suppose m is achieved at a point (x0 , t0 ) ∈ SMT . Choose a local orthonormal frame e1 . . . , en around x0 as before, and therefore ˜ = {Uαβ }, where 1 ≤ α, β ≤ n − 1. locally U From (6.2) we see Uαβ = U αβ − ∇n (u − u)σαβ

(6.28)

on SMT , where σαβ = ⟨∇α eβ , en ⟩, since σαβ = Π (eα , eβ ) on SMT . Let F˜0αβ and F˜0τ be the derivatives of ˜αβ (x0 , t0 ) and −ut (x0 , t0 ) respectively. By the concavity of F and that u = ut = ϕt on F˜ with respect to U t SMT , we have, at (x0 , t0 ), ∇n (u − u)F˜0αβ σαβ = F˜0αβ (U αβ − Uαβ ) ≥ F˜ (U αβ , −ut ) − F˜ (Uαβ , −ut ) ~ = F˜ (U αβ , −ut ) − ψ − m ≥ ~ − m ≥ . 2 Setting η =



αβ

(6.29)

F˜0αβ σαβ which is well defined in MTδ , by (6.29) we obtain η(x0 ) ≥

~ ≥ ϵ1 ~ > 0 2∇n (u − u)(x0 , t0 )

(6.30)

for some uniform constant ϵ1 > 0. Next, since F˜ is concave, we have     αβ  F˜0 Uαβ − Uαβ (x0 , t0 ) − F˜0τ ut − ut (x0 , t0 ) + ψ(x0 , t0 ) − ψ(x, t) αβ

≥ F˜ (Uαβ , −ut ) − ψ(x, t) − m ≥ 0 on SMT . Define Φ = −η∇n (u − ϕ) + Q, where Q≡



    F˜0αβ U αβ − Uαβ (x0 , t0 ) + F˜0τ ut (x0 , t0 ) − ϕt + ψ(x0 , t0 ) − ψ(x, t).

αβ

Therefore, by (6.31) and (6.28) we see that Φ(x0 , t0 ) = 0 and Φ ≥ 0 on SMT . Note that ∇n (u − ϕ) = 0 on BMT and we can derive as in [12] that Φ(x, 0) ≥ −Cd(x)

(6.31)

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62

for (x, 0) ∈ BMTδ = MTδ ∩ M × {0}. Furthermore, by straightforward calculations, we have n n     i | + F τ . LΦ ≤ C 1 + fi + fi |λ i=1

(6.32)

i=1

Now applying Lemma 6.1 again, we get  L(Ψ + Φ) ≤ 0 in MTδ , Ψ + Φ ≥ 0 on PMTδ

(6.33)

for some A1 ≫ A2 ≫ A3 ≫ 1. Thus, by the maximum principle we have ∇n Φ(x0 , t0 ) ≥ −∇n Ψ (x0 , t0 ) ≥ −C and we obtain − C ≤ ∇n Φ(x0 , t0 ) ≤ −η(x0 )∇nn u(x0 , t0 ) + C.

(6.34)

Combining with (6.30), we see ∇nn u(x0 , t0 ) ≤

C . ϵ1 ~

By now we have got a priori upper bounds for all eigenvalues of {Uij (x0 , t0 )} and hence its eigenvalues are contained in a compact subset of Γ by (1.5). It follows that m > 0 by (1.3). Consequently, there exist positive constants c4 and R0 such that  (x, t)), R, −ut (x, t)) ∈ Γ (λ′ (U and  (x, t)), R, −ut (x, t)) ≥ ψ(x, t) + c4 f (λ′ (U for all R > R0 and (x, t) ∈ SMT and therefore (6.27) holds by Lemma 1.2 in [3]. Acknowledgment The third author is supported by the Program for Innovation Research of Science in Harbin Institute of Technology, No. PIRS OF HIT Q201501. References [1] B. Andrews, Contraction of convex hypersurfaces in Euclidean space, Calc. Var. Partial Differential Equations 2 (1994) 151–171. [2] B. Andrews, J. McCoy, Y. Zheng, Contracting convex hypersurfaces by curvature, Calc. Var. Partial Differential Equations 47 (2013) 611–665. [3] L.A. Caffarelli, L. Nirenberg, J. Spruck, Dirichlet problem for nonlinear second order elliptic equations III, functions of the eigenvalues of the Hessian, Acta Math. 155 (1985) 261–301. [4] L.C. Evans, Classical solutions of fully nonlinear, convex, second order elliptic equations, Comm. Pure Appl. Math. 25 (1982) 333–363. [5] W.J. Firey, Shapes of worn stones, Mathematika 21 (1974) 1–11. [6] C. Gerhardt, Closed Weingarten hypersurfaces in Riemannian manifolds, J. Differential Geom. 43 (1996) 612–641. [7] B. Guan, The Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds, arXiv:1403.2133. [8] B. Guan, The Dirichlet problem for Hessian equations on Riemannian manifolds, Calc. Var. Partial Differential Equations 8 (1999) 45–69. [9] B. Guan, Second order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds, Duke Math. J. 163 (2014) 1491–1524. [10] B. Guan, H.-M. Jiao, Second order estimates for Hessian type fully nonlinear elliptic equations on Riemannian manifolds, Calc. Var. Partial Differential Equations 54 (2015) 2693–2712. [11] B. Guan, H.-M. Jiao, The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds, Discrete Contin. Dyn.-A 36 (2016) 701–714.

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