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Research article

On estimates for augmented Hessian type parabolic equations on Riemannian manifolds

  • Received: 28 November 2021 Revised: 11 May 2022 Accepted: 06 June 2022 Published: 07 July 2022
  • The author extends previous results to general classes of equations under weaker assumptions obtained in 2016 by Bao, Dong and Jiao concerning the study of the regularity of solutions for the first initial-boundary value problem for parabolic Hessian equations on Riemannian manifolds.

    Citation: Yang Jiao. On estimates for augmented Hessian type parabolic equations on Riemannian manifolds[J]. Electronic Research Archive, 2022, 30(9): 3266-3289. doi: 10.3934/era.2022166

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  • The author extends previous results to general classes of equations under weaker assumptions obtained in 2016 by Bao, Dong and Jiao concerning the study of the regularity of solutions for the first initial-boundary value problem for parabolic Hessian equations on Riemannian manifolds.



    Let (Mn,g) be a compact Riemannian manifold of dimension n2 with smooth boundary M and ˉM:=MM. Define MT=M×(0,T]M×R, PMT=BMTSMT is the parabolic boundary of MT with BMT=M×{0} and SMT=M×[0,T]. In [1], the authors derived C2 estimates for solutions of the first initial-boundary value problem of parabolic Hessian equations in the form

    f(λ(2u+χ(x,t)),ut)=ψ(x,t), (1.1)

    where f is a symmetric smooth function of n+1 variables.

    In this paper, we apply an exponential barrier from [2] where Jiang-Trudinger treat the corresponding elliptic problems in Rn to study (1.1) in the general augmented Hessian form

    f(λ(2u+A(x,t,u)),ut)=ψ(x,t,u) (1.2)

    in MT with boundary condition

    u=φ on PMT, (1.3)

    where 2u+A(x,t,u) is called augmented Hessian, u and 2u denote the gradient and the Hessian of u(x,t) with respect to xM respectively, ut=Dtu is the derivative of u(x,t) with respect to t[0,T], A[u]=A(x,t,u) is a (0,2) tensor on ¯M which may depend on t[0,T] and u, and

    λ(2u+A[u])=(λ1,,λn)

    denotes the eigenvalues of 2u+A[u] with respect to the metric g.

    As in [3], throughout the paper we assume A[u] is smooth on ¯MT for uC(¯MT), ψC(T¯M×[0,T]). We shall write ψ=ψ(x,t,p) for (x,p)T¯M and t[0,T]. Note that for fixed (x,t)¯MT and pTxM,

    A(x,t,p):TxM×TxMR

    is a symmetric bilinear map. We shall use the notation

    Aξη(x,t,):=A(x,t,)(ξ,η),ξ,ηTxM.

    For a function vC2(MT), we write A[v]:=A(x,t,v), Aξη[v]:=Aξη(x,t,v) and ψ[u]:=ψ(x,t,u).

    There are many different A in conformal geometry, the optimal transportation satisfies, the isometric embedding, reflector design and other research fields, we recommend readers see subsection 3.8 in [4] and references therein for the Monge-Ampère type equations arising in applications.

    We are concerned in this work with the a priori estimates of admissible solutions to (1.2) with boundary condition. The use of the exponential barrier allows us to relax the concavity assumption of A to Ma-Trudinger-Wang conditions(see [5]). By the perturbation method of subsolutions in [2] (see Remark 2.2 in [6] for details), we can obtain strict subsolutions from non-strict subsulutions which simplifies the proofs and relaxes some restrictions to f in the estimates of |ut|.

    Our treatment here will also work for parabolic equations in the form

    f(λ(2u+A(x,t,u)))ut=ψ(x,t,u) (1.4)

    with slight modification. Note that we do not require a priori bound of |ut| in the study of (1.4).

    The idea of this paper is mainly from Guan-Jiao [7] and Jiang-Trudinger [2] where those authors studied the second order estimates for the elliptic counterpart of (1.2):

    f(λ(2u+A(x,u,u)))=ψ(x,u,u). (1.5)

    The first initial-boundary value problem for equation of form (1.4) in Rn with A0 and ψ=ψ(x,t) was studied by Ivochkina-Ladyzhenskaya in [8] (when f=σ1/nn) and [9]. In recent years, Jiao-Sui [10] treated the case that Aχ(x,t) and ψ=ψ(x,t) on Riemannian manifolds and Jiao [3] extend their results to the form

    f(λ(2u+A(x,t,u)))ut=ψ(x,t,u,u)

    by the method using in the corresponding elliptic problems.

    Krylov in [11] treated (1.2) in the parabolic Monge–Ampère form

    utdet(2u+A)=ψn+1

    in Rn, where A0 and ψ=ψ(x,t). In [12], Lieberman studied the first initial–boundary value problem of (1.2) when A=0 and ψ may depend on u and u in a bounded domain under various conditions.

    For the elliptic Hessian equations, we refer the readers to Li [13], Urbas [14,367–377], Guan [15,16], Guan-Jiao [17], Jiang-Trudinger [2] and their references.

    Following [18], in which the authors studied the corresponding elliptic equations in Rn, fC(Γ)C0(¯Γ) is assumed to be defined on Γ, where Γ is an open, convex, symmetric proper subcone of Rn+1 with vertex at the origin and

    Γ+{λRn+1: each component λi>0}Γ,

    and to satisfy the following structure conditions in this paper:

    fifλi>0 in Γ,  1in+1, (1.6)
    f is concave in Γ, (1.7)

    and

    δψ,finfMTψsupΓf>0,  wheresupΓfsupλ0Γlim supλλ0f(λ). (1.8)

    Typical examples are f=σ1/kk and f=(σk/σl)1/(kl), 1l<kn, defined in the cone

    Γk={λRn:σj(λ)>0,j=1,,k}

    and f=(Mk)1/(nk) defined in

    Mk={λRn:λi1++λik>0},

    where σk(λ) are the kth elementary symmetric functions and Mk are the p-plurisubharmonic functions defined by

    σk(λ)=i1<<ikλi1λik,1kn

    and

    Mk(λ)=i1<<ik(λi1++λik),1kn

    respectively. When k=n, f=σ1nn is the famous Monge-Ampère equation arising in many research fields such as conformal geometry, optimal transportation, isometric embedding and reflector designs, see the survey [4] and references therein.

    We define a function u(x,t) to be admissible if (λ(2u+A[u]),ut)Γ in M×[0,T]. It is shown in [18] that (1.6) ensures that Eq (1.2) is parabolic for admissible solutions. (1.7) means that the function F defined by F(A,τ)=f(λ[A],τ) is concave for (A,τ) with (λ[A],τ)Γ, where A is in the set of n×n symmetric matrices Sn×n. Moreover, when {Uij} is diagonal so is {Fij}, and the following identities hold

    FijUij=fiλi,FijUikUkj=fiλ2i,λ(U)=(λ1,,λn).

    We define a function ¯u to be a admissible viscosity supersolution of (1.2) if

    f(λ(2ϕ(ˆx,ˆt)+A(ˆx,ˆt,ϕ(ˆx,ˆt)),ϕt(ˆx,ˆt))ψ(ˆx,ˆt,ϕ(ˆx,ˆt))

    whenever ϕC2(MT) is a admissible function and (ˆx,ˆt)MT is a local minimum of ¯uϕ.

    In this paper we assume that there exists an admissible function u_C2(ˉMT) satisfying

    {f(λ(2u_+A[u_]),u_t)ψ(x,t,u_) in M×[0,T],u_=φ on M×[0,T],u_φ on M×{0}. (1.9)

    A (0,2) tensor B is called regular (strictly regular), if

    ni,j,k,lBijpk,pl(x,t,p)ξiξjηkηl0(>0)

    for all (x,t,p)M×[0,T]×Rn, ξ,ηTxM and g(ξ,η)=0.

    The regular condition, well known as MTW condition, was first introduced by Ma, Trudinger and Wang in [5] for the study of optimal transportation in its strict form, and used in [2,19] and other relevant problems. It is natural to consider MTW conditions instead of normal concavity assumptions on A. Examples in [5] shows that there exists a tensor A, without convexity respect to p, derived from special cost functions satisfying this regular condition. There are many results about MTW conditions, see, for instance, [20,21,22,23,24,25] and references therein.

    We now begin to formulate the main theorems of this paper.

    Theorem 1. Let uC4(ˉMT) be an admissible solution of (1.2). Suppose (1.6)(1.8) and (1.9) hold. Assume, in addition, that

    ψ(x,t,p)isconvexinp, (1.10)
    Aξξ(x,t,p)isregular, (1.11)

    then

    maxˉMT|2u|C1(1+maxPMT|2u|), (1.12)

    where C1>0 depends on |u|C1(ˉMT), |ut|C0(ˉMT) and |u_|C2(ˉMT). Suppose that u also satisfies the boundary condition (1.3) and, in addition, assume that there exists a function ΘC2(BMT) such that Θ=φt on M×{0} and

    (λ(2φ(x,0)+A[φ(x,0)]),Θ(x))Γ,  xˉM, (1.13)

    and that

    f(λ(2φ(x,0)+A[φ(x,0)]),φt(x,0))=ψ[φ(x,0)],  xM, (1.14)

    for each (x,t)SMT and pTxˉM. Then there exists C2>0 depending on |u|C1(ˉMT), |ut|C0(ˉMT), |u_|C2(ˉMT) and |φ|C4(PMT) such that

    maxPMT|2u|C2. (1.15)

    Combining with the gradient estimates and the estimates of |ut|, we can prove the following theorem immediately.

    Theorem 2. Let uC4(ˉMT) be an admissible solution of (1.2) in MT with uu_ in MT and u=φ on PMT. Suppose (1.6)(1.11) and (1.13)(1.14) hold. Assume, in addition, for every C>0, there is a constant R=R(C) such that

    f(R11)>C, (1.16)

    where 11=(1,,1)Rn+1. Assume also there exist a bounded admissible viscosity supersolution ¯u of (1.2) satisfying ¯uφ on PMT. Then we have

    |u|C2(ˉMT)C, (1.17)

    where C>0 depends on n, M and |u_|C2(ˉMT) under the additional assumptions (3.1)(3.4) in Section 3.

    The assumptions of the existence of bounded viscosity supersolution and the additional conditions (3.1)–(3.4) are only used to derive C0 and C1 estimates. (1.16) is used in the estimates of |ut| and can be dropped if u_ is strict subsolution. Both (1.16) and (3.4) hold for many operators such as the famous Monge-Ampère operator or more general k-Hessian operator σ1/kk.

    The outline of this paper is as follows. In Section 2, we present some preliminaries and give a proof of Lemma 4. The solution bound and the gradient bound are derived in Section 3 while an a priori estimates for ut is obtained in Section 4. Finally we establish the global and boundary C2 estimates in Sections 5 and 6 respectively.

    Throughout the paper denotes the Levi-Civita connection of (Mn,g).

    Let uC4(ˉMT) be an admissible solution of Eq (1.2). For simplicity we shall denote U:=2u+A(x,t,u) and U_:=2u_+A(x,t,u_). Moreover, we denote,

    Fij=Fhij(U,ut),Fτ=Fτ(U,ut),
    Fij,kl=2Fhijhkl(U,ut),Fij,τ=2Fhijτ(U,ut),Fτ,τ=2F2τ(U,ut)

    and, under a local frame e1,,en,

    UijU(ei,ej)=iju+Aij(x,t,u),
    kUijU(ei,ej,ek)=kiju+kAij(x,t,u)kiju+Aijk(x,t,u)+Aijpl(x,t,u)klu,
    (Uij)t(U(ei,ej))t=(iju)t+Aijt(x,t,u)+Aijpl(x,t,u)(lu)tijut+Aijt(x,t,u)+Aijpl(x,t,u)lut,

    where Aij=Aeiej and Aijk denotes the partial covariant derivative of A when viewed as depending on xM only, while the meanings of Aijt and Aijpl, etc are obvious. Similarly we can calculate klUij=klUijΓmklmUij, etc.

    It is convenient to express the regular condition of A in the equivalent form as in [26],

    Aijpkplξiξjηkηl2¯λ|ξ||η|g(ξη), (2.1)

    for all ξ,ηRn, where ¯λ is a non-negative function in C0(¯MT×Rn), depending on pA. Hence, we have, for any non-negative symmetric matrix Fij and ϵ(0,1],

    FijAijpkplηkηl¯λ(ϵFii|η|2+1ϵFijηiηj). (2.2)

    Define the linear operator L locally by

    Lv=Fijijv+(FijAijpkψpk)kvFτvt

    for vC2(MT).

    A crucial lemma was proved by Jiang-Trudinger for elliptic type equations in Lemma 2.1(ii) in [2] for M=Rn, we extend their results to the parabolic case. Note that their perturbation of non-strict subsolution, which make a non-strict subsolution to be strict, only holds near the boundary in the Riemannian manifolds case. Therefore we shall apply a classification technique from [7] to deal with global estimates.

    Let μ(x,t)=λ(2u_(x,t)+A[u_]) and note that {μ(x,t):(x,t)MT} is a compact subset of positive cone Γ+ since (1.6). There exists uniform constant β(0,12n) such that

    νμ2β1Γ+,xˉMT, (2.3)

    where νλ:=Df(λ)/|Df(λ)| is the unit normal vector to the level hypersurface Γf(λ) for λΓ and 1=(1,,1)Rn+1.

    For fixed (x0,t0), we consider two cases: (i) |νμνλ|β and (ii) |νμνλ|<β. In case (i), we shall modify Jiang-Trduinger's Lemma 2.1 [2]. First, we need the following lemma, its proof can be found in Lemma 4 [27].

    Lemma 3. Let K be a compact subset of Γ and β>0. There is a constant ϵ>0 such that, for any μK and λΓ with |νμνλ|β,

    fi(μiλi)f(μ)f(λ)+ϵ(1+fi(λ)). (2.4)

    It follows from Lemma 6.2 in [18] and Lemma (2.4) that

    Fij(U_ijUij)F(U_,u_t)F(U,ut)+ϵ(1+Fii+Fτ). (2.5)

    We now prove the crucial lemma for case (i).

    Lemma 4. Let uC2(ˉMT) be an admissible solution of Eq (1.2) Suppose |νμνλ|β. Assume F satisfies (1.6)(1.7) and (1.9)(1.11) hold. Then there exist positive constants K and ϵ, depending on MT, A, |u|C1(ˉMT) and |u_|C1(ˉMT) such that

    Lη>ϵ(1+Fii+Fτ), (2.6)

    where η=eK(u_u).

    Proof. By (2.5), we have

    L(u_u)=Fij{[U_ijUij]Fτ[u_tut]+AijpkDk(u_u)Aij(x,t,Du_)+Aij(x,t,Du)}ψpkk(u_u)F(U_,u_t)F(U,ut)ψpkk(u_u)12FijAijpk,pl(x,t,ˆp)Dk(u_u)Dl(u_u)+ϵ(1+Fii+Fτ)12FijAijpk,pl(x,t,ˆp)Dk(u_u)Dl(u_u)+ϵ(1+Fii+Fτ) (2.7)

    by Taylor's formula and the convexity of ψ, where ˆp=θu+(1θ)u_ for some θ(0,1). Thus

    LeK(u_u)=KeK(u_u)[L(u_u)+KFijDi(u_u)Dj(u_u)]KeK(u_u){12FijAijpk,pl(x,t,ˆp)Dk(u_u)Dl(u_u)+KFijDi(u_u)Dj(u_u)+ϵ(1+Fii+Fτ)}. (2.8)

    Since A is regular, by (2.2), we obtain

    ϵFii12FijAijpk,pl(x,t,ˆp)Dk(u_u)Dl(u_u)+KFijDi(u_u)Dj(u_u)(ϵ¯λϵ12|D(u_u)|2)Fii+(K¯λ2ϵ1)FijDi(u_u)Dj(u_u)ϵ2Fii

    by successively fixing ϵ1 and K.

    Therefore, by (2.8), we have

    LeK(u_u)KeK(u_u)(ϵ2(1+Fii+Fτ))ϵ0(1+Fii+Fτ) (2.9)

    for some positive constant ϵ0.

    Next, in case (ii), we have νλβ1Γ+. Thus we derive

    Fiiβn+1Fii1in+1. (2.10)

    Remark 1. If u_ is a strict subsolution or M=Rn, then we can derive (2.6) without the assumption |νμνλ|β. Actually, when M=Rn, let d(x)=dist(x,M), by consider u_+aebx1 and u_+a(ebd1) for interior and near boundary respectively in Rn, a strict subsolution can be derived from a non-strict one, see remark 2.2 in [6]. Then (2.6) will be obtained by Jiang-Trudinger's proof with a little modification.

    In this section, we derive the gradient estimates. We introduce the following growth conditions: When |p| is sufficiently large,

    pxψ(x,t,p),pxAξξ(x,t,p)/|ξ|2ˉψ1(x,t)(1+|p|γ), (3.1)
    |pDpψ(x,t,p)|,|pDpAξξ(x,t,p)|/|ξ|2ˉψ2(x,t)(1+|p|γ) (3.2)

    and

    |Aξη(x,t,p)|ˉψ3(x,t)|ξ||η|(1+|p|γ1)ξ,ηTxˉM (3.3)

    hold for some functions ˉψ1,ˉψ2,ˉψ30, and constants γ(0,4) and γ1(0,2).

    By the existence of viscosity supersolution ¯u and classical subsolution u_, we have

    maxˉMT|u|C.

    Since u is admissible, we have

    0<u+trA(x,t,u)ut.

    The boundary gradient estimates are derived by subsolution u_ for the lower bound and by (3.3) with the method of Lemma 10.1 in [12] for the upper bound.

    Theorem 5. Let uC3(ˉMT) be an admissible solution of (1.2). Suppose (1.6)(1.7) and (3.1)(3.3) hold. Assume, in addition, that

    fjν0(1+n+1i=1fi)for anyλΓwithλj<0, (3.4)

    where ν0 is a uniform positive constant. Then

    maxˉMT|u|C3(1+maxPMT|u|), (3.5)

    where C3 is a positive constant depending on |u|C0(ˉMT) and other known data.

    Proof. Let ϕC2(ˉMT) is a positive function to be determined. Suppose |u|ϕa achieves a positive maximum at an interior point (x0,t0)ˉMTPMT where a<1 is a constant. Choose a smooth orthonormal local frame e1,,en about (x0,t0) such that eiej=0 at (x0,t0) if ij and {Uij} is diagonal. Define v=log|u|alogϕ, then the function v also attains its maximum at (x0,t0) where, for i=1,,n,

    iv=luilu|u|2aiϕϕ=0 (3.6)

    and

    Fτvt0Fiiiiv. (3.7)

    Thus, by (3.6) and (3.7), we have

    0FiiiivFτvt=Fiiii(log|u|)Fτ(log|u|)taFiiiilogϕ+aFτ(logϕ)t=1|u|2Fiiiluilu+lu|u|2(FiiiiluFτlut)+a2a2ϕ2Fii(iϕ)2aϕFiiiiϕ. (3.8)

    Differentiating both sides of Eq (1.2) with respect to x, we obtain, at (x0,t0),

    FiikUiiFτkut=ψk+ψpjkju (3.9)

    for all k=1,,n.

    Let ϕ=u+supˉMTu+1. Note that, at (x0,t0), iju=iju and

    ijkujiku=Rlkijlu. (3.10)

    By (3.1), (3.2), (3.6), (3.9) and (3.10), we have

    lu|u|2(FiiiiluFτlut)=lu|u|2Fii(liiuRkiilkuFτlut)lu|u|2Fii(lUiil(Aii)Fτlut)CC(1+|u|γ2)(1+Fii). (3.11)

    Therefore, by substituting (3.11) into (3.8), we have

    01|u|2Fiiiluilu+a2a2ϕ2Fii(iu)2+aϕFiiiiuC(1+|u|γ2)(1+Fii). (3.12)

    Notice that

    1|u|2Fiiiiuiiu+aϕFiiiiua2|u|24ϕ2Fii.

    It follows from (3.12) that

    0a2a2ϕ2Fii(iu)2a2|u|24ϕ2FiiC(1+|u|γ2)(1+Fii). (3.13)

    Without loss of generality we may consider 1u(x0,t0)1n|u(x0,t0)|>0. Recall that Uij(x0,t0) is diagonal. By (3.3) and (3.6), we have

    U11=aϕ|u|2+A11+l2luA1l1uaϕ|u|2+C(1+|u|γ1)<0 (3.14)

    provided |u| is sufficiently large. The appearance of A1l in the first line is due to the diagonality of {Uij}. Therefore, by (3.4),

    f1ν0(1+ni=1fi+Fτ)

    and a bound |u(x0,t0)|C3 follows from (3.13) by choosing a sufficiently small such that

    a2a2ϕ2ν0na24ϕ2c1>0

    holds for some uniform constant c1.

    Remark 2. This assumptions follow from [7] and [3]. (3.3) with γ1(0,2) is more of a technical condition here. Actually, it will be better to obtain gradient estimates with quadratic growth conditions, i.e γ1=2, see examples in [4]. The reason why we need (3.3) is the regular assumption of A which make us can not use barrier η=eK(u_u) in gradient estimates. From the proof of Lemma 4 you can see the proof of the barrier is based on the gradient estimates. This requirement also occurs in Theorem 1.3 (ii) in [28].

    (3.4) is a natural assumption satisfied by many operators such as the k-Hessian operator σ1kk. It is commonly used in deriving gradient estimate, for example in [29].

    In this section, we derive the estimates for |ut|.

    Theorem 6. Suppose that (1.6)(1.7), (1.9) and (1.16) hold, A=A(x,t,u) and ψ=ψ(x,t,u). Let uC3(ˉMT) be an admissible solution of (1.2)-(1.3) in MT. Then there exists a positive constant C2 depending on |u|C1(ˉMT), |u_|C2(ˉMT), |ψ|C2(ˉMT) and other known data such that

    supˉMT|ut|C4(1+supPMT|ut|). (4.1)

    Proof. We first show that

    supˉMT(ut)C4(1+supPMT|ut|) (4.2)

    for which we set

    W=supˉMT(ut)eϕ,

    where ϕ is a positive function to be chosen.

    We may assume that W is attained at (x0,t0)ˉMTPMT. As in the proof of Theorem 5, we choose an orthonormal local frame e1,,en about x0 such that eiej=0 and {Uij(x0,t0)} is diagonal. We may assume ut(x0,t0)>0. Define v=log(ut)+ϕ. At (x0,t0), where the function v achieves its maximum, we have, for i=1,n,

    iv=iutut+iϕ=0 (4.3)

    and

    Fτvt0Fiiiiv=Fijiiv+(FijAijpkψpk)kv. (4.4)

    Thus, by (4.3) and (4.4), we have

    0FiiiivFτvt+(FijAijpkψpk)kv=Fiiiilog(ut)Fτ(log(ut))t+FiiiiϕFτϕt+(FijAijpkψpk)k(log(ut)+ϕ)=1ut(FiiiiutFτutt+(FijAijpkψpk)kut)+LϕFii(iϕ)2. (4.5)

    By differentiating equation (1.2) with respect to t, we get

    Fii(Uii)tFτutt=ψt+ψpk(ku)t. (4.6)

    It follows from (4.5) and (4.6) that

    01ut((ψtFiiAiit)Fii(iϕ)2+LϕCut(1+Fii)Fii(iϕ)2+Lϕ. (4.7)

    Fix a positive constant α(0,1) and let ϕ=δ1+α2|u|2+δu+bη, where η=eK(u_u) as in Lemma 4 and δb1 are positive constants to be determined. By straightforward calculations, we have

    iϕ=δ1+αkkuiku+δiu+biη,
    ϕt=δ1+αkku(ku)t+δut+bηt,
    iiϕ=δ1+αk(iku)2+δ1+αkkuiiku+δiiu+biiη.

    It follows that

    Lϕδ1+αku(FiiiikuFτ(ku)t+FijAijplkluψplklu)+δ1+α2FiiU2iiCδ1+αFii+δLu+bLηCδ1+α(1+Fii)+δ1+α2FiiU2ii+δLu+bLη (4.8)

    and

    (iϕ)2Cδ2(1+α)U2ii+Cb2 (4.9)

    since bδ. Thus, (4.7) becomes, by (4.8) and (4.9),

    bLη+δ1+α4FiiU2ii+δLuCut(1+Fii)+Cδ1+α(1+Fii)+Cb2Fii. (4.10)

    We first consider case (i): |νμνλ|β. Note that

    δFiiUiiδ1+α4FiiU2iiδ1αFii.

    It follows from that

    δ1+α4FiiU2ii+δLuCδ(1+Fii)+δ1+α4FiiU2ii+δFiiUiiδFτutCδ1α(1+Fii) (4.11)

    since ut(x0,t0)<0. Therefore, by (4.10) and (4.11), we have

    bLηCut(1+Fii)+Cδ1α(1+Fii)+Cb2Fii. (4.12)

    Choosing b and δ such that bϵ0Cδ1αCb2b1>0 for a positive constant b1, then a upper bound of ut(x0,t0) derived by (2.6).

    Case (ii): |νμνλ|<β. We see that (2.10) holds. Note that

    δ1+α8FiiU2ii+δFiiUii2δ1αFii

    and

    LeK(u_u)=KeK(u_u)[L(u_u)+KFijDi(u_u)Dj(u_u)]KeK(u_u){12FijAijpk,pl(x,t,ˆp)Dk(u_u)Dl(u_u)+KFijDi(u_u)Dj(u_u)}CFii (4.13)

    by the concavity of F and ψ, where C depends on |u|C1(ˉMT) and other known data. We have, by (4.10),

    δ1+α8FiiU2iiδFτutCut(1+Fii)+Cδ(1+Fii)+C(δ1α+b+b2)FiiCut(1+Fii)+Cδ1α+CFii. (4.14)

    Recalling that ut<0, we get

    FiiUiiFτutut(Fii+Fτ)+14ut(FiiU2ii+Fτu2t).

    Therefore, by the concavity of f, we have

    ut(Fii+Fτ)f(ut1)f(λ(U),ut)+FiiUiiFτutut(Fii+Fτ)+14ut(FiiU2ii+Fτu2t)+f(ut1)ψ[u], (4.15)

    where 1=(1,,1)Rn+1.

    Note that limtf(t1)=supΓf>supˉMTψ[u]. It follows from (1.6) that

    f(ut1)ψ[u]f(ut1)supˉMTψ[u]:=2b2 (4.16)

    provided ut(x0,t0) is big enough, where b2 is a positive constant. Therefore, by (4.15) and (4.16), we have

    ut(Fii+Fτ)b2+18ut(FiiU2ii+Fτu2t). (4.17)

    It follows from (2.10) and (4.17) that

    Fτut2γ0ut(Fii+Fτ)γ0ut(Fii+Fτ)+γ0b2+γ08ut(FiiU2ii+Fτu2t)γ0utFii+γ0b2+γ08utFiiU2ii, (4.18)

    where γ0:=β2n+1>0.

    Without loss of generality, we suppose utγ0δα for fixed δ. Substituting (4.18) in (4.14) we derive

    (δγ0utC)Fii+δγ0b2Cδ1αCut(1+Fii). (4.19)

    By (1.16), we see that b2 can be sufficiently large, then a bound is derived from (4.19) and therefore (4.2) holds.

    Similarly, we can show

    supˉMTutC4(1+supPMT|ut|) (4.20)

    by letting

    ϕ=δ1+α2|u|2δu+b(u_u).

    Combining (4.2) and (4.20), the proof is finished.

    Remark 3. If u_ is a strict subsolution, then Theorem 6 follows without (1.16). In face, in this case we have (2.6) holds without classification. Let W=supˉMT|ut|eaϕ and ϕ=η in Lemma 2.6, the theorem will be proved easily.

    By (1.13) and (1.14) we can the short time existence as Theorem 15.9 in [12]. 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(2φ(x,0)+A[φ]),φt(x,0))=ψ[φ]xˉM. (4.21)

    Since that ut=φt on SMT and (4.21), we can obtain the estimate

    supˉMT|ut|C5. (4.22)

    In this section, we derive the global estimates for the second order derivatives. In particular, we prove the following maximum principle.

    Theorem 7. Let uC4(ˉMT) be an admissible solution of (1.2) in MT. Suppose that (1.6)(1.7) and (1.9)(1.11) hold. Then

    supˉMT|2u|C1(1+supPMT|2u|), (5.1)

    where C1>0 depends on |u|C1(ˉMT), |u_|C1(ˉMT), |ut|C0(ˉMT), |ψ|C2(ˉMT) and other known data.

    Proof. Set

    W=max(x,t)¯MTmaxξTxM,|ξ|=1(ξξu+Aξξ(x,t,u))eϕ,

    as in [7], where ϕ is a function to be determined. It suffices to estimate W. We may assume W is achieved at (x0,t0)ˉMTPMT. Choose a smooth orthonormal local frame e1,,en about x0 such that iej=0, and {Uij} is diagonal at (x0,t0). We assume U11(x0,t0)Unn(x0,t0) and, without loss of generality, we assume U11>1.

    Define v=logU11+ϕ. At (x0,t0), where the function v attains its maximum, we have, for each i=1,,n,

    iv=iU11U11+iϕ=0 (5.2)

    and

    Fτvt0Fiiiiv. (5.3)

    Thus, by (5.3), we have

    0FiiiivFτvt=Fiiii(logU11)Fτ(logU11)t+FiiiiϕFτϕt=1U211FiiiU211+1U11(FiiiiU11Fτ(U11)t)+FiiiiϕFτϕt. (5.4)

    Differentiating Eq (1.2) twice, we obtain, by (1.10), (3.9). (3.10) and (5.2),

    Fii11Uii+Fij,kl1Uij1Ukl2Fij,τ1Uij1ut+Fτ,τ(1ut)2Fτ11utCU11+ψpkpl1ku1lu+ψpk11luCU11U11ψpkkϕ. (5.5)

    Note that the regular condition of A means Aiip1p10 for i1. Therefore by (3.9) and (5.2), we have

    Fii(iiA1111Aii)Fii(A11pkiikuAiipk11ku)CU11Fii+Fii(A11pipiU2iiAiip1p1U211)U11FiiAiipkkϕ+FτA11pkkutCU11FiiCU11Ci2FiiU2ii. (5.6)

    Note that

    ijklvklijv=Rmljkimv+iRmljkmv+Rmlikjmv+Rmjiklmv+Rmjilkmv+kRmjilmv.

    Thus we have

    iiU1111Uii+iiA1111AiiCU11. (5.7)

    It follows from (5.5), (5.6) and (5.7) that

    FiiiiU11Fτ(U11)tFii11UiiFτ11utCU11FiiFii(iiA1111Aii)Fτ(A11)tFij,kl1Uij1Ukl2Fij,τ1Uij1ut+Fτ,τ(1ut)2+U11(FiiAiipkψpk)kϕCi2FiiU2iiCU11(1+Fii). (5.8)

    Thus, by (5.4) and (5.8), we have, at (x0,t0),

    LϕCU11i2FiiU2ii+C(1+Fii)+E, (5.9)

    where

    E=1U211Fii(iU11)2+1U11(Fij,kl1Uij1Ukl2Fij,τ1Uij1ut+Fτ,τ(1ut)2).

    Let η=eK(u_u). Define

    ϕ=δ|u|22+bη,

    where b and δ are undetermined constants such that 0<δ<1b. We find, at (x0,t0),

    iϕ=δjuiju+biη=δiuUiiδjuAij+biη, (5.10)
    ϕt=δju(ju)t+bηt, (5.11)
    iiϕδ2U2iiCδ+δjuiiju+biiη. (5.12)

    From (3.10) and (3.9), we derive

    FiijuiijuFiiju(jUiijAii)C|u|2Fii(ψpkFiiAiipk)jujku+Fτjuj(ut)C(1+Fii). (5.13)

    Therefore,

    LϕbLη+δ2FiiU2iiCδ(1+Fii). (5.14)

    Next, by (5.10) we get

    (iϕ)2Cδ2(1+U2ii)+2b2(i(u_u))2Cδ2U2ii+Cb2. (5.15)

    Now we estimate E as in [16] and [17] (see [1] for details). Let

    J={i:UiisU11},K={i:Uii>sU11},

    where 0<s1/3 is a fixed number. Using an inequality of Andrews [30] and Gerhardt [31], we have, by (5.15),

    Fij,kl1Uij1UklijFiiFjjUjjUii(1Uij)22i2FiiF11U11Uii(1Ui1)22(1s)(1+s)U11iK(FiiF11)((iU11)2CU211/s). (5.16)

    Thus, we obtain

    E1U211iJFii(iU11)2+CiKFii+CF11U211iK(iU11)2iJFii(iϕ)2+CFii+CF11(iϕ)2Cb2iJFii+Cδ2FiiU2ii+CFii+C(δ2U211+b2)F11. (5.17)

    Therefore, by (5.9), (5.14), (5.15) and (5.17), we have

    bLη(Cδ2+CU11δ2)FiiU2ii+Cb2iJFii+C(δ2U211+b2)F11+C(1+Fii). (5.18)

    Case (i): |νμνλ|β. It follows from (2.6) and (5.18) that

    (bεC)(1+Fii)(Cδ2+CU11δ2)FiiU2ii+Cb2iJFii+C(δ2U211+b2)F11.

    Choosing b sufficiently large such that bεCbε2, we have

    bε2(1+Fii)(Cδ2+CU11δ2)FiiU2ii+Cb2iJFii+C(δ2U211+b2)F11.

    and we can get a bound U11(x0,t0)C by choosing δ sufficiently small since |Uii|sU11 for iJ. Thus we derive a bound of U11(x0,t0) and therefore (5.1) holds.

    Case (ii): |νμνλ|<β. For every fixed C>0, choosing δ sufficiently small such that δ4Cδ2δ0>0. Without loss of generality, suppose U11Cδ0 for otherwise we are done. Then (5.18) becomes

    bLη+δ4FiiU2iiCb2iJFii+C(δ2U211+b2)F11+C(1+Fii). (5.19)

    Next, let ˆλ:=λ(U(x0,t0)). In the view of (4.15)–(4.17), we have

    |ˆλ|(Fii+Fτ)b3, (5.20)

    where b3:=12(f(|ˆλ|1)supˉMTψ[u])>0 provided |ˆλ| is large enough. By (2.10) and (5.20), we have

    δ4FiiU2ii2c2|ˆλ|2(Fii+Fτ)c2|ˆλ|2(Fii+Fτ)+c2b3|ˆλ|,

    where c2=δβ8n+1. Therefore, it follows from (4.13) and (5.19) that

    c2|ˆλ|2(Fii+Fτ)+c2b3|ˆλ|Cδ2U211F11+C(1+Fii). (5.21)

    Then a bound for U11 is derived since δ(0,1) and U11|ˆλ|.

    In this section, we establish the estimates of second order derivatives on parabolic boundary PMT. We may assume φC4(ˉMT). We shall establish the estimate

    maxPMT|2u|C2 (6.1)

    for some positive constant C2 depending on |u|C1ˉMT, |ut|C0ˉMT, |u_|C2ˉMT, |ψ|C4ˉMT, and other known data.

    Fix a point (x0,t0)SMT. We shall choose smooth orthonormal local frames e1,,en around x0 such that when restricted to M, en is the interior normal to M along the boundary when restricted to M. Since uu_=0 on SMT we have

    αβ(uu_)=n(uu_)Π(eα,eβ),1α,β<non SMT, (6.2)

    where Π denotes the second fundamental form of M. Therefore,

    |αβu|C,1α,β<nonSMT. (6.3)

    Let ρ(x) and d(x) denote the distance from xM to x0 and M respectively and set

    MδT={X=(x,t)M×(0,T]:ρ(x)<δ}.

    Now we shall use a perturbation method to obtain a strict subsolution from a non-strict one. Let s(x,t)=u_(x,t)+a(h(x)1) and S={ijs+A[s]}, where h(x)=ebd(x), a and b are constants to be determined. We wish to show ˜M=(F(S,st)ψ[s])(F(U_,u_t)ψ[u_])>0 for some a and b. Note that d is smooth near boundary and

    SijU_ij=ab2hidjd+abhijd+abhAijpk(x,t,ˆp1)kd,

    where ˆp1=u_+θ1abhd for some θ1(0,1). Therefore, if a is small enough for fixed b, s is admissible since u_ is admissible and Γ is open. Let Fij0=Fij(U_,u_t), there is a positive constant c3 such that Fij0idjdc3>0 since |d(x)|1. Thus, we derive

    ˜MFij0(ab2hidjd+abhijd+abhAijpk(x,t,˜p)kd)abhψpk(x,t,ˆp2)kdab2hc3abC>0,

    where b>C/c3C/hc3 and ˆp2=u_+θ2abhd for some θ2(0,1).

    Therefore a strict admissible subsolution with same boundary condition is derived near boundary and (2.6) holds without the assumption |νμνλ|β, see Remark 1. For convenience, we still use u_ to denote the strict subsolution below.

    For the mixed tangential-normal and pure normal second derivatives at (x0,t0), we shall use the following barrier function as in [16],

    Ψ=A1v+A2ρ2A3l<n|l(uφ)|2, (6.4)

    where

    v=1η=1eK(u_u)

    and A1, A2, A3 are positive constants to be chosen. By differentiating Eq (1.2) and

    ij(ku)=ijku+Γlikjlu+Γljkilu+ijeku,

    we obtain, by straightforward calculation,

    L(k(uφ))C(1+fi|λi|+fi+Fτ),1kn, (6.5)

    where λ=λ(2u+A[u]).

    The following lemma is crucial to construct barrier functions.

    Lemma 8. Suppose that (1.6)(1.8) and (1.9)(1.11) hold. Then for any positive constant K1 there exist uniform positive constants t,δ sufficiently small, and A1, A2, A3 sufficiently large such that ΨK1ρ2 in ¯MδT and

    LΨK1(1+fi|λi|+fi+Fτ)in¯MδT. (6.6)

    Proof. First by Lemma 4, we have

    Lvε(1+fi+Fτ)inMδT. (6.7)

    Similar to Proposition 2.19 of [16], we can show that

    l<nFijUilUjl12irfiλ2i, (6.8)

    for some index r. It follows that

    l<nL|l(uφ)|2l<nFijUilUjlC(1+fi|λi|+Fii+Fτ)12irfiλ2iC(1+fi|λi|+Fii+Fτ). (6.9)

    We first consider the case that λr0. Notice that

    Lv=LeK(u_u)=KeK(u_u)[L(u_u)+KFijDi(u_u)Dj(u_u)]a0fiλiC(1+Fii+Fτ),

    where a0=infPMTKeK(u_u).

    By (6.7), (6.8) and (6.9), we obtain, for any 0<B<A1,

    LΨ(A1+B)LvBLv+CA2(1+fi+Fτ)A32irfiλ2i+CA3(1+fi|λi|+fi+Fτ)(A1+B)ε(1+fi+Fτ)a0Bfiλi+CA3fi|λi|A32irfiλ2i+C(B+A2+A3)(1+fi+Fτ)(A1+B)ε(1+fi+Fτ)+2a0Birfi|λi|A32irfiλ2i(a0BCA3)fi|λi|+C(B+A2+A3)(1+fi+Fτ). (6.10)

    Notice that

    A32irfiλ2i2a0Birfi|λi|2(a0B)2A3fi. (6.11)

    Thus, we derive from (6.10) and (6.11) that

    LΨ(A1+B)ε(1+fi+Fτ)(a0BCA3)fi|λi|+C(B+A2+A3)(1+fi+Fτ)+2(a0B)2A3fi. (6.12)

    If λr<0, similarly to (6.12), we have

    LΨ(A1+B)ε(1+fi+Fτ)(a1BCA3)fi|λi|+C(B+A2+A3)(1+fi+Fτ)+2(a1B)2A3fi, (6.13)

    where a1=supPMTKeK(u_u).

    Checking (6.12) and (6.13), we can choose A1A2A31 and A1Ba1Ba0BA2A3 in (6.12) and (6.13) such that (6.6) holds and ΨK1ρ2 in MδT.

    By (6.5) and (6.6), we can use Lemma 8 to choose suitable δ, N and A1A2A31 such that in MδT, L(Ψ±α(uϕ))0, and Ψ±α(uϕ)0 on PMδT. Then it follows from the maximum principle that Ψ±α(uϕ)0 in MδT and therefore

    |nαu(x0,t0)|nΨ(x0,t0)C,α<n. (6.14)

    It remains to show that

    nnu(x0,t0)C (6.15)

    since uut+trA>0. We shall use an idea of Trudinger [32] to prove that there exist uniform positive constants c0, R0 such that for all R>R0, (λ[U],R,ut)Γ and

    f(λ[U],R,ut)ψ[u]+c0on¯SMT,

    which implies (6.15) by Lemma 1.2 in [18], where λ[U]=(λ1,,λn1) denote the eigenvalues of the (n1)×(n1) matrix {Uαβ}1α,β(n1) and ψ[u]=ψ(,,u). Define

    ˜F(Uαβ,ut)limR+f(λ({Uαβ}),R,ut)

    and consider

    mmin(x,t)¯SMT(˜F(Uαβ(x,t),ut(x,t))ψ[u](x,t)).

    Note that ˜F is concave and m is monotonically increasing with respect to R, and that

    cmin(x,t)¯SMT(˜F(U_αβ(x,t),u_t(x,t))ψ[u_](x,t))>0

    when R is sufficiently large.

    We shall show m>0 and we may assume m<c/2 (otherwise we are done) and suppose m is achieved at a point (x0,t0)¯SMT. Choose local orthonormal frames around x0 as before and assume nnu(x0,t0)nnu_(x0,t0). Let σαβ=αeβ,en and

    ˜Fαβ0=˜Frαβ(Uαβ(x0,t0),ut(x0,t0)),
    ˜Fτ0=˜Fτ(Uαβ(x0,t0),ut(x0,t0)).

    Note that σαβ=Π(eα,eβ) on M and by (6.2), we have, at (x0,t0),

    n(uu_)˜Fαβ0σαβ˜F(U_αβ,u_t)˜F(Uαβ,ut)+˜Fτ0(u_tut)+˜Fαβ0(Aαβ[u]Aαβ[u_])c2+H[u]H[u_]c2+Hpnn(uu_), (6.16)

    where H[u]=˜Fαβ0Aαβ[u]ψ[u]. The last inequality is from the regularity of A and the convexity of ψ with respect to p.

    Note that A is regular, which means Aαβ is concave respect to pn and u_ is strict subsolution near the boundary, we have Hpnpn0 and

    0<n(uu_)<c4

    for some positive constant c4. It follows from (6.16) that, at (x0,t0),

    κHpnc2c4>0, (6.17)

    where κ=˜Fαβ0σαβ.

    Let ϑ(x,t)=κ(x,t)Hpn(x,t,φ(x,t),nu(x0,t0)). Since αu=αu_=αφ on ¯SMT, we derive

    ϑ(x,t)>c5onMδT¯SMT (6.18)

    for some small positive constant c5, where φ=(1φ,,n1φ).

    Next, since H is concave with respect to pn, we have

    H(x,t,φ,nu(x0,t0))H(x,t,φ,nu)Hpn(x,t,φ,nu(x0,t0))(nu(x0,t0)nu) (6.19)

    on ¯SMT.

    On the other hand, since ut=u_t=φt on ¯SMT, by the concavity of ˜F, we have

    H(x,t,φ,nu(x,t))H(x0,t0,φ(x0,t0),nu(x0,t0))+˜Fαβ0(αβuαβu(x0,t0))+˜Fτ0φt˜Fτ0φt(x0,t0)=˜Fαβ0Uαβψ[u]˜Fτ0ut˜Fαβ0Uαβ(x0,t0)+ψ[u](x0,t0)+˜Fτ0ut(x0,t0)˜F(Uαβ,ut)ψ[u]m0 (6.20)

    on ¯SMT. It follows from (6.2), (6.19) and (6.20) that

    ϑ(n(uφ)n(uφ)(x0,t0))˜Fαβ[n(uφ)(x0,t0)(σαβ(x0,t0)σαβ)+αβφ(x0,t0)αβφ]+H(x,t,φ,nu(x0,t0))H(x0,t0,φ(x0,t0),nu(x0,t0))+Hpn(x,t,φ,nu(x0,t0))(nφ(x0,t0)nφ)+˜Fτ0φt(x0,t0)˜Fτ0φt:=Θ(x,t). (6.21)

    From the form of the function Θ(x,t) in (6.21), since Θ(x0,t0)=0, we have, on MδT¯SMT,

    n(uφ)n(uφ)(˜x0)ϑ1Θ(x,t)l(˜x˜x0)+˜C(ρ2+(tt0)2), (6.22)

    where ˜x=(x,t), l is a linear function of ˜x˜x0 with l(0)=0, and the constant C depends on |u|C1 and other known data.

    Define

    Φ=n(uφ)n(uφ)(˜x0)l(˜x˜x0)˜C(tt0)2.

    By extending φ smoothly to the interior near the boundary to be constant in the normal direction, By (6.5), we have

    LΦC(1+fi+fi|λi|+Fτ).

    We see from (6.20) and (6.2) that Φ0 on ¯SMT and Φ(x0,t0)=0. Therefore, by the compatibility condition (1.14), we have, when δ is sufficiently small, Ψ0 on PMδ.

    Therefore, by Lemma 8, we can choose suitable Ψ such that

    {L(ΨΦ)0 inMδT,ΨΦ0 onPMδT. (6.23)

    By the maximum principle we find ΨΦ in MδT. It follows that nΦ(x0,t0)nΨ(x0,t0)C.

    Therefore, we have an a priori upper bound for all eigenvalues of {Uij(x0,t0)} and hence its eigenvalues are contained in a compact subset of Γ by (1.8), and we see m>0 by (1.6).

    Consequently, there exist positive c6 and R0 such that

    (λ(˜U(x,t)),R,ut(x,t))Γ

    and

    f(λ(˜U(x,t)),R,ut(x,t))ψ(x,t)+c6

    for all R>R0 and (x,t)¯SMT

    For i=1,,n1, Lemma 1.2 in [18] means λi=λi+o(1) if |Unn| tends to infinity. Therefore, we have

    f(λ(U),ut)>ψ

    for unbounded |Unn|, which leads a contradiction and therefore (6.15) holds.

    I would like to express my deep thanks to Heming Jiao for many useful discussions and for the guidance over the past years. I also wish to thank the referees for pointing out the errors, which helped me to improve the quality of this paper.



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