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 n≥2 with smooth boundary ∂M and ˉM:=M∪∂M. Define MT=M×(0,T]⊂M×R, PMT=BMT∪SMT 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 x∈M 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 u∈C∞(¯MT), ψ∈C∞(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 p∈T∗xM,
A(x,t,p):T∗xM×T∗xM→R |
is a symmetric bilinear map. We shall use the notation
Aξη(x,t,⋅):=A(x,t,⋅)(ξ,η),ξ,η∈T∗xM. |
For a function v∈C2(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 A≡0 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 A≡χ(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 A≡0 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, f∈C∞(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+≡{λ∈Rn+1: each component λi>0}⊆Γ, |
and to satisfy the following structure conditions in this paper:
fi≡∂f∂λi>0 in Γ, 1≤i≤n+1, | (1.6) |
f is concave in Γ, | (1.7) |
and
δψ,f≡infMTψ−sup∂Γf>0, wheresup∂Γf≡supλ0∈∂Γlim supλ→λ0f(λ). | (1.8) |
Typical examples are f=σ1/kk and f=(σk/σl)1/(k−l), 1≤l<k≤n, defined in the cone
Γk={λ∈Rn:σj(λ)>0,j=1,…,k} |
and f=(Mk)1/(nk) defined in
Mk={λ∈Rn:λi1+⋯+λik>0}, |
where σk(λ) are the kth elementary symmetric functions and Mk are the p-plurisubharmonic functions defined by
σk(λ)=∑i1<…<ikλi1⋯λik,1≤k≤n |
and
Mk(λ)=∏i1<⋯<ik(λi1+⋯+λik),1≤k≤n |
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 ϕ∈C2(MT) is a admissible function and (ˆx,ˆt)∈MT is a local minimum of ¯u−ϕ.
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
n∑i,j,k,lBijpk,pl(x,t,p)ξiξjηkηl≥0(>0) |
for all (x,t,p)∈M×[0,T]×Rn, ξ,η∈T∗xM 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 u∈C4(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 Θ∈C2(BMT) such that Θ=−φt on ∂M×{0} and
(λ(∇2φ(x,0)+A[φ(x,0)]),Θ(x))∈Γ, ∀x∈ˉM, | (1.13) |
and that
f(λ(∇2φ(x,0)+A[φ(x,0)]),−φt(x,0))=ψ[φ(x,0)], ∀x∈∂M, | (1.14) |
for each (x,t)∈SMT and p∈T∗xˉM. 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 u∈C4(ˉMT) be an admissible solution of (1.2) in MT with u≥u_ 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 ¯u≥φ 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 u∈C4(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=∂F∂hij(U,−ut),Fτ=∂F∂τ(U,−ut), |
Fij,kl=∂2F∂hij∂hkl(U,−ut),Fij,τ=∂2F∂hij∂τ(U,−ut),Fτ,τ=∂2F∂2τ(U,−ut) |
and, under a local frame e1,…,en,
Uij≡U(ei,ej)=∇iju+Aij(x,t,∇u), |
∇kUij≡∇U(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)t≡∇ijut+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 x∈M only, while the meanings of Aijt and Aijpl, etc are obvious. Similarly we can calculate ∇klUij=∇k∇lUij−Γmkl∇mUij, etc.
It is convenient to express the regular condition of −A in the equivalent form as in [26],
−Aijpkplξiξjηkηl≥−2¯λ|ξ||η|g(ξ⋅η), | (2.1) |
for all ξ,η∈Rn, 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(ϵ∑Fii|η|2+1ϵFijηiηj). | (2.2) |
Define the linear operator L locally by
Lv=Fij∇ijv+(FijAijpk−ψpk)∇kv−Fτvt |
for v∈C2(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∈(0,12√n) such that
νμ−2β1∈Γ+,∀x∈ˉMT, | (2.3) |
where νλ:=Df(λ)/|Df(λ)| is the unit normal vector to the level hypersurface ∂Γf(λ) for λ∈Γ and 1=(1,…,1)∈Rn+1.
For fixed (x0,t0), we consider two cases: (i) |νμ−νλ|≥β and (ii) |νμ−νλ|<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 μ∈K and λ∈Γ with |νμ−νλ|≥β,
∑fi(μi−λi)≥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_ij−Uij)≥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 u∈C2(ˉMT) be an admissible solution of Eq (1.2) Suppose |νμ−νλ|≥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_ij−Uij]−Fτ[u_t−ut]+AijpkDk(u_−u)−Aij(x,t,Du_)+Aij(x,t,Du)}−ψpk∇k(u_−u)≥F(U_,−u_t)−F(U,−ut)−ψpk∇k(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=θ∇u+(1−θ)∇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
ϵ∑Fii−12FijAijpk,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)≥ϵ2∑Fii |
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 νλ−β1∈Γ+. Thus we derive
Fii≥β√n+1∑Fii∀1≤i≤n+1. | (2.10) |
Remark 1. If u_ is a strict subsolution or M=Rn, then we can derive (2.6) without the assumption |νμ−νλ|≥β. Actually, when M=Rn, let d(x)=dist(x,∂M), by consider u_+aebx1 and u_+a(ebd−1) 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,
p⋅∇xψ(x,t,p),p⋅∇xAξξ(x,t,p)/|ξ|2≤ˉψ1(x,t)(1+|p|γ), | (3.1) |
|p⋅Dpψ(x,t,p)|,|p⋅DpAξξ(x,t,p)|/|ξ|2≤ˉψ2(x,t)(1+|p|γ) | (3.2) |
and
|Aξη(x,t,p)|≤ˉψ3(x,t)|ξ||η|(1+|p|γ1)∀ξ,η∈T∗xˉM | (3.3) |
hold for some functions ˉψ1,ˉψ2,ˉψ3≥0, 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 u∈C3(ˉMT) be an admissible solution of (1.2). Suppose (1.6)–(1.7) and (3.1)–(3.3) hold. Assume, in addition, that
fj≥ν0(1+n+1∑i=1fi)for anyλ∈Γwithλj<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 ϕ∈C2(ˉMT) is a positive function to be determined. Suppose |∇u|ϕ−a achieves a positive maximum at an interior point (x0,t0)∈ˉMT−PMT 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 i≠j 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=∇lu∇ilu|∇u|2−a∇iϕϕ=0 | (3.6) |
and
Fτvt≥0≥Fii∇iiv. | (3.7) |
Thus, by (3.6) and (3.7), we have
0≥Fii∇iiv−Fτvt=Fii∇ii(log|∇u|)−Fτ(log|∇u|)t−aFii∇iilogϕ+aFτ(logϕ)t=1|∇u|2Fii∇ilu∇ilu+∇lu|∇u|2(Fii∇iilu−Fτ∇lut)+a−2a2ϕ2Fii(∇iϕ)2−aϕFii∇iiϕ. | (3.8) |
Differentiating both sides of Eq (1.2) with respect to x, we obtain, at (x0,t0),
Fii∇kUii−Fτ∇kut=ψk+ψpj∇kju | (3.9) |
for all k=1,…,n.
Let ϕ=−u+supˉMTu+1. Note that, at (x0,t0), ∇iju=∇iju and
∇ijku−∇jiku=Rlkij∇lu. | (3.10) |
By (3.1), (3.2), (3.6), (3.9) and (3.10), we have
∇lu|∇u|2(Fii∇iilu−Fτ∇lut)=∇lu|∇u|2Fii(∇liiu−Rkiil∇ku−Fτ∇lut)≥∇lu|∇u|2Fii(∇lUii−∇l(Aii)−Fτ∇lut)−C≥−C(1+|∇u|γ−2)(1+∑Fii). | (3.11) |
Therefore, by substituting (3.11) into (3.8), we have
0≥1|∇u|2Fii∇ilu∇ilu+a−2a2ϕ2Fii(∇iu)2+aϕFii∇iiu−C(1+|∇u|γ−2)(1+∑Fii). | (3.12) |
Notice that
1|∇u|2Fii∇iiu∇iiu+aϕFii∇iiu≥−a2|∇u|24ϕ2∑Fii. |
It follows from (3.12) that
0≥a−2a2ϕ2Fii(∇iu)2−a2|∇u|24ϕ2∑Fii−C(1+|∇u|γ−2)(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+∑l≥2∇luA1l∇1u≤−aϕ|∇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),
f1≥ν0(1+n∑i=1fi+Fτ) |
and a bound |∇u(x0,t0)|≤C3 follows from (3.13) by choosing a sufficiently small such that
a−2a2ϕ2⋅ν0n−a24ϕ2≥c1>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 u∈C3(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)∈ˉMT−PMT. 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
Fτvt≥0≥Fii∇iiv=Fij∇iiv+(FijAijpk−ψpk)∇kv. | (4.4) |
Thus, by (4.3) and (4.4), we have
0≥Fii∇iiv−Fτvt+(FijAijpk−ψpk)∇kv=Fii∇iilog(−ut)−Fτ(log(−ut))t+Fii∇iiϕ−Fτϕt+(FijAijpk−ψpk)∇k(log(−ut)+ϕ)=1ut(Fii∇iiut−Fτutt+(FijAijpk−ψpk)∇kut)+Lϕ−Fii(∇iϕ)2. | (4.5) |
By differentiating equation (1.2) with respect to t, we get
Fii(Uii)t−Fτutt=ψt+ψpk(∇ku)t. | (4.6) |
It follows from (4.5) and (4.6) that
0≥1ut((ψt−FiiAiit)−Fii(∇iϕ)2+Lϕ≥Cut(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 δ≪b≪1 are positive constants to be determined. By straightforward calculations, we have
∇iϕ=δ1+α∑k∇ku∇iku+δ∇iu+b∇iη, |
ϕt=δ1+α∑k∇ku(∇ku)t+δut+bηt, |
∇iiϕ=δ1+α∑k(∇iku)2+δ1+α∑k∇ku∇iiku+δ∇iiu+b∇iiη. |
It follows that
Lϕ≥δ1+α∇ku(Fii∇iiku−Fτ(∇ku)t+FijAijpl∇klu−ψpl∇klu)+δ1+α2FiiU2ii−Cδ1+α∑Fii+δLu+bLη≥−Cδ1+α(1+∑Fii)+δ1+α2FiiU2ii+δLu+bLη | (4.8) |
and
(∇iϕ)2≤Cδ2(1+α)U2ii+Cb2 | (4.9) |
since b≫δ. Thus, (4.7) becomes, by (4.8) and (4.9),
bLη+δ1+α4FiiU2ii+δLu≤−Cut(1+∑Fii)+Cδ1+α(1+∑Fii)+Cb2∑Fii. | (4.10) |
We first consider case (i): |νμ−νλ|≥β. Note that
δFiiUii≥−δ1+α4FiiU2ii−δ1−α∑Fii. |
It follows from that
δ1+α4FiiU2ii+δLu≥−Cδ(1+∑Fii)+δ1+α4FiiU2ii+δFiiUii−δFτut≥−Cδ1−α(1+∑Fii) | (4.11) |
since ut(x0,t0)<0. Therefore, by (4.10) and (4.11), we have
bLη≤−Cut(1+∑Fii)+Cδ1−α(1+∑Fii)+Cb2∑Fii. | (4.12) |
Choosing b and δ such that bϵ0−Cδ1−α−Cb2≥b1>0 for a positive constant b1, then a upper bound of −ut(x0,t0) derived by (2.6).
Case (ii): |νμ−νλ|<β. We see that (2.10) holds. Note that
δ1+α8FiiU2ii+δFiiUii≥−2δ1−α∑Fii |
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)}≥−C∑Fii | (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+α8FiiU2ii−δFτut≤−Cut(1+∑Fii)+Cδ(1+∑Fii)+C(δ1−α+b+b2)∑Fii≤−Cut(1+∑Fii)+Cδ1−α+C∑Fii. | (4.14) |
Recalling that ut<0, we get
FiiUii−Fτut≥ut(∑Fii+Fτ)+14ut(FiiU2ii+Fτu2t). |
Therefore, by the concavity of f, we have
−ut(∑Fii+Fτ)≥f(−ut1)−f(λ(U),−ut)+FiiUii−Fτut≥ut(∑Fii+Fτ)+14ut(FiiU2ii+Fτu2t)+f(−ut1)−ψ[u], | (4.15) |
where 1=(1,…,1)∈Rn+1.
Note that limt→∞f(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
−Fτut≥−2γ0ut(∑Fii+Fτ)≥−γ0ut(∑Fii+Fτ)+γ0b2+γ08ut(FiiU2ii+Fτu2t)≥−γ0ut∑Fii+γ0b2+γ08utFiiU2ii, | (4.18) |
where γ0:=β2√n+1>0.
Without loss of generality, we suppose −ut≥γ0δ−α for fixed δ. Substituting (4.18) in (4.14) we derive
(−δγ0ut−C)∑Fii+δγ0b2−Cδ1−α≤−Cut(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
supˉMTut≤C4(1+supPMT|ut|) | (4.20) |
by letting
ϕ=δ1+α2|∇u|2−δu+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]∀x∈ˉM. | (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 u∈C4(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)∈¯MTmaxξ∈TxM,|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)∈ˉMT−PMT. 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
Fτvt≥0≥Fii∇iiv. | (5.3) |
Thus, by (5.3), we have
0≥Fii∇iiv−Fτvt=Fii∇ii(logU11)−Fτ(logU11)t+Fii∇iiϕ−Fτϕt=−1U211Fii∇iU211+1U11(Fii∇iiU11−Fτ(U11)t)+Fii∇iiϕ−Fτϕt. | (5.4) |
Differentiating Eq (1.2) twice, we obtain, by (1.10), (3.9). (3.10) and (5.2),
Fii∇11Uii+Fij,kl∇1Uij∇1Ukl−2Fij,τ∇1Uij∇1ut+Fτ,τ(∇1ut)2−Fτ∇11ut≥−CU11+ψpkpl∇1ku∇1lu+ψpk∇11lu≥−CU11−U11ψpk∇kϕ. | (5.5) |
Note that the regular condition of A means Aiip1p1≤0 for i≠1. Therefore by (3.9) and (5.2), we have
Fii(∇iiA11−∇11Aii)≥Fii(A11pk∇iiku−Aiipk∇11ku)−CU11∑Fii+Fii(A11pipiU2ii−Aiip1p1U211)≥U11FiiAiipk∇kϕ+FτA11pk∇kut−CU11∑Fii−CU11−C∑i≥2FiiU2ii. | (5.6) |
Note that
∇ijklv−∇klijv=Rmljk∇imv+∇iRmljk∇mv+Rmlik∇jmv+Rmjik∇lmv+Rmjil∇kmv+∇kRmjil∇mv. |
Thus we have
∇iiU11≥∇11Uii+∇iiA11−∇11Aii−CU11. | (5.7) |
It follows from (5.5), (5.6) and (5.7) that
Fii∇iiU11−Fτ(U11)t≥Fii∇11Uii−Fτ∇11ut−CU11∑Fii−Fii(∇iiA11−∇11Aii)−Fτ(A11)t≥−Fij,kl∇1Uij∇1Ukl−2Fij,τ∇1Uij∇1ut+Fτ,τ(∇1ut)2+U11(FiiAiipk−ψpk)∇kϕ−C∑i≥2FiiU2ii−CU11(1+∑Fii). | (5.8) |
Thus, by (5.4) and (5.8), we have, at (x0,t0),
Lϕ≤CU11∑i≥2FiiU2ii+C(1+∑Fii)+E, | (5.9) |
where
E=1U211Fii(∇iU11)2+1U11(Fij,kl∇1Uij∇1Ukl−2Fij,τ∇1Uij∇1ut+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<1≤b. We find, at (x0,t0),
∇iϕ=δ∇ju∇iju+b∇iη=δ∇iuUii−δ∇juAij+b∇iη, | (5.10) |
ϕt=δ∇ju(∇ju)t+bηt, | (5.11) |
∇iiϕ≥δ2U2ii−Cδ+δ∇ju∇iiju+b∇iiη. | (5.12) |
From (3.10) and (3.9), we derive
Fii∇ju∇iiju≥Fii∇ju(∇jUii−∇jAii)−C|∇u|2∑Fii≥(ψpk−FiiAiipk)∇ju∇jku+Fτ∇ju∇j(ut)−C(1+∑Fii). | (5.13) |
Therefore,
Lϕ≥bLη+δ2FiiU2ii−Cδ(1+∑Fii). | (5.14) |
Next, by (5.10) we get
(∇iϕ)2≤Cδ2(1+U2ii)+2b2(∇i(u_−u))2≤Cδ2U2ii+Cb2. | (5.15) |
Now we estimate E as in [16] and [17] (see [1] for details). Let
J={i:Uii≤−sU11},K={i:Uii>−sU11}, |
where 0<s≤1/3 is a fixed number. Using an inequality of Andrews [30] and Gerhardt [31], we have, by (5.15),
−Fij,kl∇1Uij∇1Ukl≥∑i≠jFii−FjjUjj−Uii(∇1Uij)2≥2∑i≥2Fii−F11U11−Uii(∇1Ui1)2≥2(1−s)(1+s)U11∑i∈K(Fii−F11)((∇iU11)2−CU211/s). | (5.16) |
Thus, we obtain
E≤1U211∑i∈JFii(∇iU11)2+C∑i∈KFii+CF11U211∑i∈K(∇iU11)2≤∑i∈JFii(∇iϕ)2+C∑Fii+CF11∑(∇iϕ)2≤Cb2∑i∈JFii+Cδ2∑FiiU2ii+C∑Fii+C(δ2U211+b2)F11. | (5.17) |
Therefore, by (5.9), (5.14), (5.15) and (5.17), we have
bLη≤(Cδ2+CU11−δ2)FiiU2ii+Cb2∑i∈JFii+C(δ2U211+b2)F11+C(1+∑Fii). | (5.18) |
Case (i): |νμ−νλ|≥β. It follows from (2.6) and (5.18) that
(bε−C)(1+∑Fii)≤(Cδ2+CU11−δ2)FiiU2ii+Cb2∑i∈JFii+C(δ2U211+b2)F11. |
Choosing b sufficiently large such that bε−C≥bε2, we have
bε2(1+∑Fii)≤(Cδ2+CU11−δ2)FiiU2ii+Cb2∑i∈JFii+C(δ2U211+b2)F11. |
and we can get a bound U11(x0,t0)≤C by choosing δ sufficiently small since |Uii|≥sU11 for i∈J. Thus we derive a bound of U11(x0,t0) and therefore (5.1) holds.
Case (ii): |νμ−νλ|<β. For every fixed C>0, choosing δ sufficiently small such that δ4−Cδ2≥δ0>0. Without loss of generality, suppose U11≥Cδ0 for otherwise we are done. Then (5.18) becomes
bLη+δ4FiiU2ii≤Cb2∑i∈JFii+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
δ4FiiU2ii≥2c2|ˆλ|2(∑Fii+Fτ)≥c2|ˆλ|2(∑Fii+Fτ)+c2b3|ˆλ|, |
where c2=δβ8√n+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 φ∈C4(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 u−u_=0 on SMT we have
∇αβ(u−u_)=−∇n(u−u_)Π(eα,eβ),∀1≤α,β<non SMT, | (6.2) |
where Π denotes the second fundamental form of ∂M. Therefore,
|∇αβu|≤C,∀1≤α,β<nonSMT. | (6.3) |
Let ρ(x) and d(x) denote the distance from x∈M 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
Sij−U_ij=ab2h∇id∇jd+abh∇ijd+abhAijpk(x,t,ˆp1)∇kd, |
where ˆp1=∇u_+θ1abh∇d 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 Fij0∇id∇jd≥c3>0 since |∇d(x)|≡1. Thus, we derive
˜M≥Fij0(ab2h∇id∇jd+abh∇ijd+abhAijpk(x,t,˜p)∇kd)−abhψpk(x,t,ˆp2)∇kd≥ab2hc3−abC>0, |
where b>C/c3≥C/hc3 and ˆp2=∇u_+θ2abh∇d 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 |νμ−νλ|≥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+A2ρ2−A3∑l<n|∇l(u−φ)|2, | (6.4) |
where
v=1−η=1−eK(u_−u) |
and A1, A2, A3 are positive constants to be chosen. By differentiating Eq (1.2) and
∇ij(∇ku)=∇ijku+Γlik∇jlu+Γljk∇ilu+∇∇ijeku, |
we obtain, by straightforward calculation,
L(∇k(u−φ))≤C(1+∑fi|λi|+∑fi+Fτ),∀1≤k≤n, | (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 Ψ≥K1ρ2 in ¯MδT and
LΨ≤−K1(1+fi|λi|+∑fi+Fτ)in¯MδT. | (6.6) |
Proof. First by Lemma 4, we have
Lv≤−ε(1+∑fi+Fτ)inMδT. | (6.7) |
Similar to Proposition 2.19 of [16], we can show that
∑l<nFijUilUjl≥12∑i≠rfiλ2i, | (6.8) |
for some index r. It follows that
∑l<nL|∇l(u−φ)|2≥∑l<nFijUilUjl−C(1+∑fi|λi|+∑Fii+Fτ)≥12∑i≠rfiλ2i−C(1+∑fi|λi|+∑Fii+Fτ). | (6.9) |
We first consider the case that λr≥0. Notice that
Lv=−LeK(u_−u)=−KeK(u_−u)[L(u_−u)+KFijDi(u_−u)Dj(u_−u)]≥a0∑fiλi−C(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)Lv−BLv+CA2(1+∑fi+Fτ)−A32∑i≠rfiλ2i+CA3(1+fi|λi|+∑fi+Fτ)≤−(A1+B)ε(1+∑fi+Fτ)−a0Bfiλi+CA3fi|λi|−A32∑i≠rfiλ2i+C(B+A2+A3)(1+∑fi+Fτ)≤−(A1+B)ε(1+∑fi+Fτ)+2a0B∑i≠rfi|λi|−A32∑i≠rfiλ2i−(a0B−CA3)fi|λi|+C(B+A2+A3)(1+∑fi+Fτ). | (6.10) |
Notice that
A32∑i≠rfiλ2i≥2a0B∑i≠rfi|λi|−2(a0B)2A3∑fi. | (6.11) |
Thus, we derive from (6.10) and (6.11) that
LΨ≤−(A1+B)ε(1+∑fi+Fτ)−(a0B−CA3)fi|λi|+C(B+A2+A3)(1+∑fi+Fτ)+2(a0B)2A3∑fi. | (6.12) |
If λr<0, similarly to (6.12), we have
LΨ≤−(A1+B)ε(1+∑fi+Fτ)−(a1B−CA3)fi|λi|+C(B+A2+A3)(1+∑fi+Fτ)+2(a1B)2A3∑fi, | (6.13) |
where a1=supPMTKeK(u_−u).
Checking (6.12) and (6.13), we can choose A1≫A2≫A3≫1 and A1−B≫a1B≥a0B≫A2≫A3 in (6.12) and (6.13) such that (6.6) holds and Ψ≥K1ρ2 in MδT.
By (6.5) and (6.6), we can use Lemma 8 to choose suitable δ, N and A1≫A2≫A3≫1 such that in MδT, L(Ψ±∇α(u−ϕ))≤0, and Ψ±∇α(u−ϕ)≥0 on PMδT. Then it follows from the maximum principle that Ψ±∇α(u−ϕ)≥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 △u−ut+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]=(λ′1,…,λ′n−1) denote the eigenvalues of the (n−1)×(n−1) matrix {Uαβ}1≤α,β≤(n−1) and ψ[u]=ψ(⋅,⋅,∇u). Define
˜F(Uαβ,−ut)≡limR→+∞f(λ′({Uαβ}),R,−ut) |
and consider
m≡min(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
c≡min(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=∂˜F∂rαβ(Uαβ(x0,t0),−ut(x0,t0)), |
˜Fτ0=∂˜F∂τ(Uαβ(x0,t0),−ut(x0,t0)). |
Note that σαβ=Π(eα,eβ) on ∂M and by (6.2), we have, at (x0,t0),
∇n(u−u_)˜Fαβ0σαβ≥˜F(U_αβ,−u_t)−˜F(Uαβ,−ut)+˜Fτ0(u_t−ut)+˜Fαβ0(Aαβ[u]−Aαβ[u_])≥c2+H[u]−H[u_]≥c2+Hpn∇n(u−u_), | (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 Hpnpn≤0 and
0<∇n(u−u_)<c4 |
for some positive constant c4. It follows from (6.16) that, at (x0,t0),
κ−Hpn≥c2c4>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)>c5on∂MδT∩¯SMT | (6.18) |
for some small positive constant c5, where ∇′φ=(∇1φ,…,∇n−1φ).
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(∇αβu−∇αβu(x0,t0))+˜Fτ0φt−˜Fτ0φt(x0,t0)=˜Fαβ0Uαβ−ψ[u]−˜Fτ0ut−˜Fαβ0Uαβ(x0,t0)+ψ[u](x0,t0)+˜Fτ0ut(x0,t0)≥˜F(Uαβ,−ut)−ψ[u]−m≥0 | (6.20) |
on ¯SMT. It follows from (6.2), (6.19) and (6.20) that
−ϑ(∇n(u−φ)−∇n(u−φ)(x0,t0))≥˜Fαβ[∇n(u−φ)(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(u−φ)−∇n(u−φ)(˜x0)≤ϑ−1Θ(x,t)≤l(˜x−˜x0)+˜C(ρ2+(t−t0)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(u−φ)−∇n(u−φ)(˜x0)−l(˜x−˜x0)−˜C(t−t0)2. |
By extending φ smoothly to the interior near the boundary to be constant in the normal direction, By (6.5), we have
LΦ≤C(1+∑fi+∑fi|λi|+Fτ). |
We see from (6.20) and (6.2) that Φ≥0 on ¯SMT and Φ(x0,t0)=0. Therefore, by the compatibility condition (1.14), we have, when δ is sufficiently small, Ψ≥0 on PMδ.
Therefore, by Lemma 8, we can choose suitable Ψ such that
{L(Ψ−Φ)≤0 inMδT,Ψ−Φ≥0 onPMδT. | (6.23) |
By the maximum principle we find Ψ≥Φ 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,…,n−1, Lemma 1.2 in [18] means λ′i=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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