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Research article Special Issues

Interior curvature bounds for a type of mixed Hessian quotient equations

  • We derive interior curvature bounds for admissible solutions of a class of mixed Hessian curvature equations subject to affine Dirichlet data. As an application, we study a Plateau type problem for locally convex Weingarten hypersurfaces.

    Citation: Weimin Sheng, Shucan Xia. Interior curvature bounds for a type of mixed Hessian quotient equations[J]. Mathematics in Engineering, 2023, 5(2): 1-27. doi: 10.3934/mine.2023040

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  • We derive interior curvature bounds for admissible solutions of a class of mixed Hessian curvature equations subject to affine Dirichlet data. As an application, we study a Plateau type problem for locally convex Weingarten hypersurfaces.



    Dedicated to Professor Neil S. Trudinger on occasion of his 80th birthday.

    One classical problem in convex geometry is the Minkowski problem, which is to find convex hypersurfaces in Rn+1 whose Gaussian curvature is prescribed as a function defined on Sn in terms of the inverse Gauss map. It has been settled by the works of Minkowski [23], Alexandrov [1], Fenchel and Jessen [28], Nirenberg [25], Pogorelov [26], Cheng and Yau [3], etc.. In smooth catagory, the Minkowski problem is equivalent to solve following Monge-Ampère equation

    det(2u+ugSn)=fonSn,

    where u is the support function of the convex hypersurface, 2u+ugSn the spherical Hessian matrix of the function u. If we take an orthonormal frame on Sn, the spherical Hessian of u is Wu(x):=uij(x)+u(x)δij, whose eigenvalues are actually the principal radii of the hypersurface.

    The general problem of finding a convex hypersurface, whose k-th symmetric function of the principal radii is the prescribed function on its outer normals for 1k<n, is often called the Christoffel-Minkowski problem. It corresponds to finding convex solutions of the nonlinear Hessian equation

    σk(Wu)=fonSn.

    This problem was settled by Guan et al [14,15]. In [16], Guan and Zhang considered a mixed Hessian equation as follows

    σk(Wu(x))+α(x)σk1(Wu(x))=k2l=0αl(x)σl(Wu(x)),xSn, (1.1)

    where α(x),αl(x)(0lk1) are some functions on Sn. By imposing some group-invariant conditions on those coefficient's functions as in [11], the authors proved the existence of solutions.

    Let M be a hypersurface of Euclidean space Rn+1 and M=graphu in a neighbourhood of some point at which we calculate. Let A be the second fundamental form of M, λ(A)=(λ1,,λn)Rn the eigenvalues of A with respect to the induced metric of MRn+1, i.e., the principle curvatures of M, and σk(λ) the k-th elementary symmetric function, σ0(λ)=1. It is natural to study the prescribing curvature problems on this aspect. In 1980s, Caffarelli, Nirenberg and Spruck studied the prescribing Weingarten curvature problem. The problem is equivalent to solve the following equation

    σk(λ)(X)=f(X),XM.

    When k=n, the problem is just the Minkowski problem; when k=1, it is the prescribing mean curvature problem, c.f. [30,33]. The prescribing Weingarten curvature problem has been studied by many authors, we refer to [2,9,11,12,13,29,37] and references therein for related works. Recently, Zhou [36] generalised above mixed prescribed Weingarten curvature equation. He obtained interior gradient estimates for

    σk(A)+α(x)σk1(A)=k2l=0αl(x)σl(A),xBr(0)Rn (1.2)

    where σk(A):=σk(λ(A)), and the coefficients satisfy αk2>0 and αl0 for 0lk3.

    Mixed Hessian type of equations arise naturally from many important geometric problems. One example is the so-called Fu-Yau equation arising from the study of the Hull-Strominger system in theoretical physics, which is an equation that can be written as the linear combination of the first and the second elementary symmetric functions

    σ1(iˉ(eu+αeu))+ασ2(iˉu)=ϕ (1.3)

    on n-dimensional compact Kähler manifolds. There are a lot of works related to this equation recently, see [6,7,27] for example. Another important example is the special Lagrangian equations introduced by Harvey and Lawson [18], which can be written as the alternative combinations of elementary symmetric functions

    sinθ([n2]k=0(1)kσ2k(D2u))+cosθ([n12]k=0(1)kσ2k+1(D2u))=0.

    This equation is equivalent to

    F(D2u):=arctanλ1++arctanλn=θ

    where λi's are the eigenvalues of D2u. It is called supercritical if θ((n2)π2,nπ2) and hypercritical if θ((n1)π2,nπ2). The Lagrangian phase operator F is concave for the hypercritical case and has convex level sets for the supercritical case, while in general F fails to be concave. For subcritical case, i.e., 0θ<(n2)π2, solutions of the special Lagrangian equation can fail to have interior estimates [24,35]. Jacob-Yau [20] initiated to study the deformed Hermitian Yang-Mills (dHYM) equation on a compact Kähler manifold (M,ω):

    Re(χu+1ω)n=cotθ0Im(χu+1ω)n,

    where χ is a closed real (1,1)-form, χu=χ+1ˉu, and θ0 is the angles of the complex number M(χ+1ω)n, u is the unknown real smooth function on M. Jacob-Yau showed that dHYM equation has an equivalent form of special Lagrangian equation. Collins-Jacob-Yau [5] solved the dHYM equation by continuity method and Fu-Zhang [8] gave an alternative approach by dHYM flow, both of which considered in the supercritical case. For more results concerning about dHYM equation and special Lagrangian equation, one can consult Han-Jin [17], Chu-Lee [4] and the references therein. Note that for n=3 and hypercritical θ(π,3π2), the special Lagrangian equation (1.3) is

    σ3(D2u)+tanθσ2(D2u)=σ1(D2u)+tanθσ0(D2u)

    which is included in (1.1).

    In this paper we derive interior curvature bounds for admissible solutions of a class of curvature equations subject to affine Dirichlet data. Let Ω be a bounded domain in Rn, and let uC4(Ω)C0,1(ˉΩ) be an admissible solution of

    {σk(λ)+g(x,u)σk1(λ)=k2l=0αl(x,u)σl(λ)inΩ,u=ϕonΩ, (1.4)

    where g(x,u) and αl(x,u)>0, l=0,1,,k2, are given smooth functions on ˉΩ×R and ϕ is affine, λ=(λ1,,λn) is the vector of the principal curvatures of graph u. u is the admissible solution in the sense that λΓk for points on the graph of u, with

    Γk={λRn|σ1(λ)>0,,σk(λ)>0}.

    For simplicity we denote F=Gkk2l=0αlGl and Gl=σl(λ)/σk1(λ) for l=0,1,,k2,k. The ellipticity and concavity properties of the operator F have been proved in [16]. Our main result is as follows.

    Theorem 1.1. Assume that for every l (0lk2), αl,gC1,1(ˉΩ×R), αl>0, and g>0 or g<0. ϕ is affine in (1.4). For any fixed β>0, if uC4(Ω)C0,1(ˉΩ) is an admissible solution of (1.4), then there exists a constant C, depending only on n,k,β,||u||C1(ˉΩ),αl,g and their first and second derivatives, such that the second fundamental form A of graph u satisfies

    |A|C(ϕu)β.

    Remark 1.1. Comparing with [16], here we require g>0 or g<0 additionally. Also our curvature estimates still hold if αl0 for some 0lk2. More over, if αl0 for all l=0,1,,k2, Eq (1.4) becomes the Hessian quotient equation and the results can be followed from [29].

    To see that this is an interior curvature estimate, we need to verify that ϕu>0 on Ω. We apply the strong maximum principle for the minimal graph equation. Since ϕ is affine, it satisfies the following minimal graph equation

    Qu:=(1+|Du|2)uuiujuij=nH(1+|Du|2)32=0onΩ.

    Since u is k-admissible solution, and nk2, graph of u is mean-convex and Qu>Qϕ=0. By the comparison principle for quasilinear equations (Theorem 10.1 in [10]), we then have ϕ>u on Ω.

    The main application of the curvature bound of Theorem 1.1 is to extend various existence results for the Dirichlet problem for curvature equations of mixed Hessian type.

    Theorem 1.2. Let Ω be a bounded domain in Rn, let αl,gC1,1(ˉΩ×R) satisfying inf|g|>0, ug(x,u)0, αl>0 and uαl(x,u)0. Suppose there is an admissible function u_C2(Ω)C0,1(ˉΩ) satisfying

    F[u_]g(x,u_)inΩ,u_=0onΩ. (1.5)

    Then the problem

    F[u]=g(x,u)inΩ,u=0onΩ. (1.6)

    has a unique admissible solution uC3,α(Ω)C0,1(ˉΩ) for all α(0,1).

    Remark 1.2. ug0, uαl(x,u)0 and the existence of sub-solutions are required in the C0 estimate. The C1 interior estimate is a slightly modification of the result in Theorem 5.1.1 [36] since the coefficients g, αl of (1.2) are independent of u. We use conditions ug0 and uαl(x,u)0 again to eliminate extra terms in the C1 estimate.

    As a further application of the a priori curvature estimate we also consider a Plateau-type problem for locally convex Weingarten hypersurfaces. Let Σ be a finite collection of disjoint, smooth, closed, codimension 2 submanifolds of Rn+1. Suppose Σ bounds a locally uniformly convex hypersurface M0 with

    f(n)(λ0):=σnσn1(λ0)n2l=0αlσlσn1(λ0)c,

    where λ0=(λ01,,λ0n) are the principal curvatures of M0 and αl's are positive constants, c0 is a constant. Is there a locally convex hypersurface M with boundary Σ and f(n)(λ)=c, where λ=(λ1,,λn) are the principal curvatures of M?

    Theorem 1.3. Let Σ, f(n)(λ) be as above. If Σ bounds a locally uniformly convex hypersurface M0 with f(n)(λ0)c at each point of M0. Then Σ bounds a smooth, locally convex hypersurface M with f(n)(λ)=c at each point of M.

    We compute using a local orthonormal frame field ˆe1,,ˆen defined on M= graph u in a neighbourhood of the point at which we are computing. The standard basis of Rn+1 is denoted by e1,,en+1. Covariant differentiation on M in the direction ˆei is denoted by i. The components of the second fundamental form A of M in the basis ˆe1,,ˆen are denoted by (hij). Thus

    hij=Dˆeiˆej,ν,

    where D and , denote the usual connection and inner product on Rn+1, and ν denotes the upward unit normal

    ν=(Du,1)1+|Du|2.

    The differential equation in (1.4) can then be expressed as

    F(A,X)=g(X). (2.1)

    As usual we denote first and second partial derivatives of F with respect to hij by Fij and Fij,rs. We assume summation from 1 to n over repeated Latin indices unless otherwise indicated. Following two lemmas are similar to the ones in [29] with minor changes, so we omit the proof.

    Lemma 2.1. The second fundamental form hab satisfies

    Fijijhab=Fij,rsahijbhrs+FijhijhaphpbFijhiphpjhababg+k2l=0(aαlbGl+bαlaGl)+k2l=0abαlGl.

    Lemma 2.2. For any α=1,,n+1, we have

    Fijijνα+Fijhiphpjνα=g,eαk2l=0αl,eαGl.

    Lemma 2.3. There is a constant C>0, depending only on n,k,infαl,|g|C0, so that for any l=0,1,,k2,

    |Gl|C.

    Proof. Proof by contradiction. If the result is not true, then for any integer i, there is an admissible solution u(i), a point x(i)Ω and an index 0l(i)k2, so that

    σl(i)σk1(λ[u(i)])>iatx(i).

    By passing to a subsequence, we may assume l(i)l and x(i)xˉΩ as i+. Therefore

    limi+σlσk1(λ[u(i)])(x(i))=+,

    or we may simply write σlσk1+ if no ambiguilty arises. Since αl>0, and g is bounded, by (1.4) we have σkσk1+. For i large enough, σk>0. By Newton-MacLaurin inequalities, we have

    σlσk1=σlσl+1σk2σk1C(σk1σk)k1l0.

    We therefore get a contradiction.

    Proof of Theorem 1.1. Here the argument comes from [29]. Let η=ϕu. η>0 in Ω. For a function Φ to be chosen and a constant β>0 fixed, we consider the function

    ˜W(X,ξ)=ηβ(expΦ(νn+1))hξξ

    for all XM and all unit vector ξTXM. Then ˜W attains its maximum at an interior point X0M, in a direction ξ0TX0M which we may take to be ˆe1. We may assume that (hij) is diagonal at X0 with eigenvalues λ1λ2λn. Without loss of generality we may assume that the ˆe1,,ˆen has been chosen so that iˆej=0 at X0 for all i,j=1,,n. Let τ=ˆe1. Then W(X)=˜W(X,τ) is defined near X0 and has an interior maximum at X0. Let Z:=habτaτb. By the special choice of frame and the fact that hij is diagonal at X0 in this frame, we can see that

    iZ=ih11andijZ=ijh11atX0

    Therefore the scalar function Z satisfies the same equation as the component h11 of the tensor hij. Thus at X0, we have

    iWW=βiηη+Φiνn+1+ih11h11=0 (2.2)

    and

    ijWWiWjWW2=β(ijηηiηjηη2)+Φiνn+1jνn+1+Φijνn+1+ijh11h11ih11jh11h211 (2.3)

    is nonpositive in the sense of matrices at X0. By Lemmas 2.1 and 2.2, we have, at X0,

    0βFij(ijηηiηjηη2)+ΦFijiνn+1jνn+1(Φνn+1+1)Fijhiphpj+Fijhijh1111gh11+Φg,en+11h11Fij,rs1hij1hrsFijih11jh11h211k2l=0Φαl,en+1σlσk1+k2l=01h11(21αl1σlσk1+11αlσlσk1). (2.4)

    Using Gauss's formula

    ijXα=hijνα,

    we have

    11g(X)=n+1α=1gXα11Xα+n+1α,β=12gXαXβ1Xα1Xβ=n+1α=1gXαναh11+n+1α,β=12gXαXβ1Xα1Xβ.

    Consequently,

    |11gh11|C.

    For the same reason, we have for all l=0,,k2,

    |11αlh11|C.

    Taking Lemma 2.3 into count, we estimate the two terms in the last line of (2.4) as

    k2l=0Φαl,en+1σlσk1+k2l=01h1111αlσlσk1C|Φ|C.

    Recall that F=GkαlGl and it is well-known that the operator (σk1σl)1k1l is concave for 0lk2. It follows that

    (1Gl)1k1lisaconcaveoperatorforl=0,1,,k2.

    For any symmetric matrix (Bij)Rn×n, we have

    {(1Gl)1k1l}ij,rsBijBrs0.

    Direct computation shows that

    Gij,rslBijBrs1Glklk1l(GijlBij)2.

    Note that Gk is also a concave operator.

    1h11Fij,rs1hij1hrs+k2l=02h111αl1σlσk1=1h11Gij,rsk1hij1hrs+k2l=0αlh11Gij,rsl1hij1hrs+k2l=02h111αl1σlσk11h11k2l=0G1lαlCl(1Gl+1αlClαlGl)21h11k2l=0(1αl)2ClαlGlCh11

    where Cl=klk1l. By the homogeneity of Gl's, we see that

    Fijhij=Gk+k2l=0αl(k1l)GlGk+k2l=0αlσlσk1inf|g|>0.

    Using Lemma 2.3 again, we have

    FijhijC.

    Next we assume that ϕ has been extended to be constant in the en+1 direction.

    ijη=nα,β=12ϕXαXβiXαjXβ+nα=1ϕXαijXαijXn+1=nα=1ϕXαναhijhijνn+1.

    Consequently,

    Fijijη=(nα=1ϕXανανn+1)Fijhij.

    Using above estimates in (2.4), we have, at X0,

    0CβηβFijiηjηη2+ΦFijiνn+1jνn+1Fijih11jh11h211(Φνn+1+1)Fijhiphpj+inf|g|h11C(1+|Φ|). (2.5)

    Next, using (2.2), we have

    Fijih11jh11h211=Fij(βiηη+Φiνn+1)(βjηη+Φjνn+1)(1+γ1)β2Fijiηjηη2+(1+γ)(Φ)2Fijiνn+1jνn+1

    for any γ>0. Therefore at X0 we have, since |η|C,

    0CβηC[β+(1+γ1)β2]ni=1Fiiη2+[Φ(1+γ)(Φ)2]Fijiνn+1jνn+1[Φνn+1+1]Fijhiphpj+inf|g|h11C(1+|Φ|). (2.6)

    We choose a positive constant a, so that

    a12νn+1=121+|Du|2

    which depends only on supΩ|Du|. Therefore

    1νn+1a1aC.

    We now choose

    Φ(t)=log(ta).

    Then

    Φ(t)=1ta,Φ(t)=1(ta)2,

    and

    (Φt+1)=ata,Φ(1+γ)(Φ)2=γ(ta)2.

    By direct computation, we have iνn+1=hipˆep,en+1, and therefore

    Fijiνn+1jνn+1=Fijhiphjqˆep,en+1ˆeq,en+1Fijhiphpj.

    Next we choose 0<γa22, then we have

    (Φt+1)+[Φ(1+γ)(Φ)2]=ataγ(ta)212a2(ta)2>0.

    Thus we have

    0CβηC(β,a)η2(ni=1Fii)+inf|g|h11C(a). (2.7)

    In the following we show that ni=1FiiC. By the definition of operator F and Lemma 2.3, we have

    ni=1Fii=ni=1(σkσk1)iini=1k2l=0αl(σlσk1)ii=ni=1σk1(λ|i)σk1σkσ2k1ni=1σk2(λ|i)+α0σ2k1ni=1σk2(λ|i)+k2l=1αliσlσk2(λ|i)iσk1σl1(λ|i)σ2k1=nk+1(nk+2)σkσk2σ2k1+(nk+2)α0σk2σ2k1+k2l=1αl(nk+2)σlσk2(nl+1)σk1σl1σ2k1nk+1+(nk+2)|σkσk1Gk2|+(nk+2)|α0|C0|Gk2G0|+(nk+2)|Gk2|k2l=1|αl|C0|Gl|.

    From Eq (1.4) we have |Gk|C, therefore iFiiC. At X0, we get an upper bound

    λ1C(β,a)η2.

    Consequently, W(X0) satisfies an upper bound. Since W(X)W(X0), we get the required upper bound for the maximum principle curvature. Since λΓk and nk2, u is at least mean-convex and

    ni=1λi>0.

    Therefore λn(n1)λ1 and

    |A|=ni=1λ2iC(n)λ1C(ϕu)β.

    In this section we prove Theorem 1.2. By comparison principle, we have 0uu_. For any Ω, \inf_{\Omega'}\underline{u}\leq u\leq c(\Omega') < 0 . First we show the gradient bound of admissible solutions of (1.6). We need following lemmas to prove the gradient estimate.

    Lemma 3.1. Suppose A = \{a_{ij}\}_{n\times n} satisfies \lambda(A)\in\Gamma_{k-1} , a_{11} < 0 and \{a_{ij}\}_{2\leq i, j\leq n} is diagonal, then

    \begin{equation} \sum\limits_{i = 2}^n\frac{\partial F}{\partial a_{1i}}a_{1i}\leq 0. \end{equation} (3.1)

    Proof. Let

    \begin{equation*} B = \left( \begin{array}{cccc} a_{11} & 0 & \cdots & 0\\ 0 & a_{22} & \ldots & 0\\ \vdots & \vdots & \ddots & \vdots\\ 0 & 0 &\cdots &a_{nn} \end{array} \right), \quad C = \left( \begin{array}{cccc} 0 & a_{12} & \cdots & a_{1n}\\ a_{21} & 0 & \cdots & 0\\ \vdots & \vdots & \ddots & \vdots\\ a_{n1} & 0 &\cdots & 0 \end{array} \right). \end{equation*}

    A(t): = B+tC , f(t): = F(A(t)) . Suppose a_{1i} = a_{i1} for all 2\leq i\leq n . Directly we have

    \sigma_k(A(t)) = \sigma_k(B)-t^2\sum\limits_{i = 2}^n a_{1i}^2\sigma_{k-2}(B|1i),

    where (B|ij) is the submatrix of B formed by deleting i -th, j -th rows and columns. Easily we see that for t\in[-1, 1] , \lambda(A(t))\in\Gamma_{k-1} and f is concave on [-1, 1] . f(-1) = f(1) = F(A) . So f'(1)\leq 0 . While

    f'(1) = 2\sum\limits_{i = 2}^n\frac{\partial F}{\partial a_{1i}}a_{1i}.

    Remark 3.1. By the concavity of \frac{\sigma_k}{\sigma_{k-1}} , we can prove following inequality with \lambda(B)\in\Gamma_{k-1}

    \begin{equation} \sigma_{k-2}(B|1i)\sigma_{k-1}(B)-\sigma_{k-3}(B|1i)\sigma_k(B)\geq 0\quad \forall 2\leq i\leq n. \end{equation} (3.2)

    We let f(t) = \frac{\sigma_{k}}{\sigma_{k-1}}(A(t)) .

    f'(1) = \frac{-2(\sum_{i = 2}^na_{1i}\sigma_{k-2}(B|1i))}{\sigma_{k-1}(A)}-\frac{\sigma_k(A)(\sum_{i = 2}^n a_{1i}^2\sigma_{k-3}(B|1i))}{\sigma_{k-1}^2(A)}\leq 0.

    Equivalently,

    \begin{equation} \sigma_{k-1}(B)\big(\sum\limits_{i = 2}^na_{1i}^2\sigma_{k-2}(B|1i)\big)-\sigma_{k}(B)\big(\sum\limits_{i = 2}^na_{1i}^2\sigma_{k-3}(B|1i)\big)\geq 0. \end{equation} (3.3)

    We can choose a_{1i} > 0 small enough and a_{1j} = 0 for j\ne i and 2\leq j\leq n , so that \lambda(A)\in \Gamma_{k-1} . Then (3.3) implies (3.2).

    Lemma 3.2. Let \alpha_{k-2} > 0 and \alpha_l\geq 0 for 0\leq l\leq k-3 . Suppose symmetric matrix A = \{a_{ij}\}_{n\times n} satisfying

    \lambda(A)\in\Gamma_{k-1}, a_{11} < 0, \;{{{and}}}\, \{a_{ij}\}_{2\leq i,j\leq n}\, {{is\; diagonal}}.

    Then

    \begin{equation} \frac{\partial F}{\partial a_{11}}\geq C_0\big(\sum\limits_{i = 1}^n\frac{\partial F}{\partial a_{ii}}\big) \end{equation} (3.4)

    where C_0 depends on n, k, |u|_{C^0}, |g|_{C^0}, \inf \alpha_{k-2} .

    Proof. Note that

    \begin{align*} \frac{\partial}{\partial a_{11}}(\frac{\sigma_l}{\sigma_{k-1}}(A)) = &\frac{\sigma_{l-1}(A|1)\sigma_{k-1}(A)-\sigma_l(A)\sigma_{k-2}(A|1)}{\sigma_{k-1}^2(A)}\\ = &\sum\limits_{i = 2}^n\frac{a_{1i}^2}{\sigma_{k-1}^2(A)}[\sigma_{l-2}(A|1i)\sigma_{k-2}(A|1)-\sigma_{l-1}(A|1)\sigma_{k-3}(A|1i)]\\ &+\sigma_{k-1}^{-2}(A)[\sigma_{l-1}(A|1)\sigma_{k-1}(A|1)-\sigma_l(A|1)\sigma_{k-2}(A|1)]. \end{align*}

    For 0\leq l\leq k-2 ,

    \frac{\partial}{\partial a_{11}}(\frac{\sigma_l}{\sigma_{k-1}}(A))\leq -C_{n,l}\frac{\sigma_l(A|1)\sigma_{k-2}(A|1)}{\sigma_{k-1}^{2}(A)}.

    As for l = k ,

    \frac{\partial}{\partial a_{11}}(\frac{\sigma_k}{\sigma_{k-1}}(A))\geq C_{n,k}\frac{\sigma_{k-1}^2(A|1)}{\sigma_{k-1}^{2}(A)}\geq C_{n,k}.

    Therefore

    \frac{\partial F}{\partial a_{11}}\geq C_{n,k}+C_{n,k}\inf\alpha_{k-2}\frac{\sigma_{k-2}^2(A|1)}{\sigma_{k-1}^{2}(A)}.

    Next we compute \sum_{i = 1}^n \frac{\partial F}{\partial a_{ii}} as

    \begin{align*} \sum\limits_{i = 1}^n \frac{\partial F}{\partial a_{ii}} = &n-k+1-(n-k+2)\frac{\sigma_k\sigma_{k-2}}{\sigma_{k-1}^2}(A)+(n-k+2)\alpha_0\frac{\sigma_{k-2}}{\sigma_{k-1}^2}(A)\\ &+\sum\limits_{l = 1}^{k-2}\alpha_l\frac{(n-k+2)\sigma_l(A)\sigma_{k-2}(A)-(n-l+1)\sigma_{k-1}(A)\sigma_{l-1}(A)}{\sigma_{k-1}^2(A)}\\ &\leq n-k+1-(n-k+2)\frac{\sigma_{k-2}(A)}{\sigma_{k-1}(A)}(\frac{\sigma_k}{\sigma_{k-1}}(A)-\sum\limits_{l = 0}^{k-2}\alpha_l\frac{\sigma_{l}}{\sigma_{k-1}}(A))\\ &\leq C_{n,k}+C_{n,k}|g|_{C^0}\frac{\sigma_{k-2}}{\sigma_{k-1}}(A)\\ &\leq C(n,k,|g|_{C^0})+C(n,k,|g|_{C^0})(\frac{\sigma_{k-2}}{\sigma_{k-1}}(A))^2\\ &\leq C(n,k,|g|_{C^0})+C(n,k,|g|_{C^0})\frac{\sigma_{k-2}^2(A|1)}{\sigma_{k-1}^2(A)}\\ &\leq C(n,k,|g|_{C^0},\inf\alpha_{k-2})\frac{\partial F}{\partial a_{11}}. \end{align*}

    Lemma 3.3. For any \Omega'\Subset\Omega , there is a constant C depending only on \Omega', n, k, \alpha_l, g and their first derivatives, such that if u is an admissible solution of (1.6), then

    |Du|\leq C

    on \Omega' .

    Proof. Since we require that \partial_u g\leq 0 and \partial_u\alpha_l\geq 0 , we only need to modify the equation (5.42) in [36] (i.e., (A.6)), where extra terms \sum_{l = 0}^{k-2}\frac{(\alpha_l)_u}{\log u_1}\frac{\sigma_l(A)}{\sigma_{k-1}(A)}-\frac{g_u}{\log u_1} should be included. These terms are all good terms and Zhou's proof will also hold in our case. For reader's convenience, we sketch the proof in the appendix below.

    Now we give the proof of Theorem 1.2.

    Proof of Thorem 1.2. The theorem can be proved by solving uniformlly elliptic approximating problems.

    F_{\epsilon}[u_{\epsilon}] = -g_{\epsilon}(x,u_{\epsilon})\quad {\rm{in}}\,\Omega,\quad\quad u_{\epsilon} = 0\quad {\rm{on}}\,\partial \Omega,

    for \epsilon > 0 small, and \underline{u} is an admissible subsolution for each of the approximating problems. By the comparison principle and Theorem 1.1, the interior gradient estimates in [36](modified), we have uniform C^2 interior estimates for u_{\epsilon} . Then Evans-Krylov's theory, together with Schauder theory, imply uniform estimates for ||u_{\epsilon}||_{C^{3, \alpha}(\Omega')} for any \Omega'\Subset\Omega . Theorem 2 then follows by extracting a suitable subsequence as \epsilon\rightarrow 0 .

    In this section we prove Theorem 1.3. The notion of locally convex hypersurface we use is the same as that in [29].

    Definition 4.1. A compact, connected, locally convex hypersurface \mathcal{M} (possibly with boundary) in \mathbb{R}^{n+1} is an immersion of an n -dimensional, compact, oriented and connected manifold \mathcal{N} (possibly with boundary) in \mathbb{R}^{n+1} , that is, a mapping T:\mathcal{N}\rightarrow \mathcal{M}\subset\mathbb{R}^{n+1} , such that for any p\in\mathcal{N} there is a neighbourhood \omega_p\subset\mathcal{N} such that

    T is a homeomorphism from \omega_p to T(\omega_p) ;

    T(\omega_p) is a convex graph;

    the convexity of T(\omega_p) agrees with the orientation.

    Since \mathcal{M} is immersed, a point x\in\mathcal{M} may be the image of several points in \mathcal{N} . Since \mathcal{M} and \mathcal{N} are compact, T^{-1}(x) consists of only finitely many points. Let r > 0 and x\in\mathcal{M} . For small enough r , T^{-1}(\mathcal{M}\cap B_r^{n+1}(x)) consists of several disjoint open sets U_1, \cdots, U_s of \mathcal{N} such that T|_{U_i} is a homeomorphism of U_i onto T(U_i) for each i = 1, \cdots, s . By an r -neighbourhood \omega_r(x) of x in \mathcal{M} we mean any one of the sets T(U_i) . We say that \omega_r(x) is convex if \omega_r(x) lies on the boundary of its convex hull.

    We shall use following lemma (see [32] Theorem A) to prove Theorem 1.3.

    Lemma 4.1. Let \mathcal{M}_0\subset B_R(0) be a locally convex hypersurface with C^2 -boundary \partial \mathcal{M}_0 . Suppose that on \partial \mathcal{M}_0 , the principal curvatures \lambda_1^0, \cdots, \lambda_n^0 of \mathcal{M}_0 satisfy

    C_0^{-1}\leq \lambda_i^0\leq C_0,\quad i = 1,2,\cdots,n,

    for some C_0 > 0 . Then there exist positive constants r and \alpha , depending only on n, C_0, R and \partial \mathcal{M}_0 , such that for any point p\in\mathcal{M}_0 , each r -neighbourhood \omega_r(p) of p is convex, and there is a closed cone C_{p, \alpha} with vertex p and angle \alpha such that \omega_r(p)\cap C_{p, \alpha} = \{p\} .

    Note that for any point p\in\mathcal{M}_0 , if one chooses the axial direction of the cone C_{p, \alpha} as the x_{n+1} -axis, then each \delta -neighbourhood of p can be represented as a graph,

    x_{n+1} = u(x),\quad |x|\leq \delta,

    for any \delta < r\sin(\alpha/2) . The cone condition also implies

    |Du(x)|\leq C,\quad |x| < \delta,

    where C > 0 only depends on \alpha . Lemma 4.1 holds not just for \mathcal{M}_0 , but also for a family of locally convex hypersurfaces, with uniform r and \alpha .

    For 2\leq k\leq n , denote

    f_{(k)}(\lambda) = \frac{\sigma_k}{\sigma_{k-1}}(\lambda)-\sum\limits_{l = 0}^{n-2}\alpha_l\frac{\sigma_l}{\sigma_{k-1}}(\lambda).

    \alpha_l 's are positive constants. With the aid of Lemma 4.1, we use the Perron method to obtain a viscosity solution of the Plateau problem for the curvature function f_{(n)} , using the following lemma.

    Lemma 4.2. Let \Omega be a bounded domain in \mathbb{R}^n with Lipschitz boundary. Let \phi\in C^{0, 1}(\bar\Omega) be a k -convex viscosity subsolution of

    \begin{equation} f_{(k)}(\lambda) = \frac{\sigma_k}{\sigma_{k-1}}(\lambda)-\sum\limits_{l = 0}^{k-2}\alpha_l\frac{\sigma_l}{\sigma_{k-1}}(\lambda) = c\quad {{{in}}}\,\Omega, \end{equation} (4.1)

    where \alpha_l > 0 and c\neq 0 are all constants. Then there is a viscosity solution u of (4.1) such that u = \phi on \partial \Omega .

    Proof. The proof uses the well-known Perron method. Let \Psi denote the set of k -convex subsolutions v of (4.1) with v = \phi on \partial \Omega . Then \Psi is not empty and the required solution u is given by

    u(x) = \sup\{v(x):v\in\Psi\}.

    It is a standard argument. The key ingredient that needs to be mentioned is the solvability of the Dirichlet problem

    \begin{equation} f_{(k)}(\lambda) = c\quad {\rm{in}}\, \, B_r,\quad\quad u = u_0\quad {\rm{on}}\, \, \partial B_r, \end{equation} (4.2)

    in small enough balls B_r\subset\mathbb{R}^n , if u_0 is any Lipschitz viscosity subsolution of (4.2). This is a consequence of [31] Theorem 6.2 with slight modification.

    Using Lemma 4.2 and the argument of [32], we conclude that there is a locally convex hypersurface \mathcal{M} with boundary \Sigma which satisfies the equation f_{(n)}(\lambda) = c in the viscosity sense; that is, for any point p\in\mathcal{M} , if \mathcal{M} is locally represented as the graph of a convex function u (by Lemma 4.1), then u is a viscosity solution of f_{(n)}(\lambda) = c .

    Following we discuss the regularity of \mathcal{M} . The interior regularity follows in the same way as [29].

    Boundary regularity

    The boundary regularity of \mathcal{M} is a local property. The boundary estimates we need are contained in [19,21]. However, they can not be applied directly to \mathcal{M} . Since we are working in a neighbourhood of a boundary point p_0\in\mathcal{M} , which we may take to be the origin, we may assume that for a smooth bounded domain \Omega\subset\mathbb{R}^n with 0\in\partial \Omega and small enough \rho > 0 we have

    \mathcal{M}\cap(B_{\rho}\times\mathbb{R}) = {\rm{graph}}\,u,\quad\mathcal{M}_0\cap(B_{\rho}\times\mathbb{R}) = {\rm{graph}}\,u_0,

    where u\in C^{\infty}(\Omega_{\rho})\cap C^{0, 1}(\bar\Omega_{\rho}) , and u_0\in C^{\infty}(\bar\Omega_{\rho}) are k -convex solutions of

    f_{(k)}[u] = c\quad {\rm{in}}\,\Omega_{\rho},\quad\quad f_{(k)}[u_0]\geq c\quad {\rm{in}}\,\Omega_{\rho},

    with

    u\geq u_0\quad{\rm{in}}\,\Omega_{\rho},\quad\quad u = u_0\quad{\rm{on}}\,\partial\Omega\cap B_{\rho}.

    We may choose the coordinate system in \mathbb{R}^n in such a way that \Omega is uniformly convex, and moreover, so that for some \epsilon_0 > 0 we have

    \begin{equation} \frac{\sigma_{k-1}(\kappa')}{\sigma_{k-2}(\kappa')}\geq \epsilon_0 > 0 \end{equation} (4.3)

    on \partial \Omega\cap B_{\rho} , where \kappa' = (\kappa'_1, \cdots, \kappa'_{n-1}) denotes the vector of principal curvatures of \partial\Omega . We recall that the principal curvatures of graph( u ) are the eigenvalues of the matrix

    (I-\frac{Du\otimes Du}{1+|Du|^2})(\frac{D^2u}{\sqrt{1+|Du|^2}}).

    We denote \sigma_k(p, r) as the k -th elementary symmetric function of the eigenvalues of the matrix

    (I-\frac{p\otimes p}{1+|p|^2})r,\,p = (p_1,\cdots,p_n),\,r = (r_{ij})_{n\times n}.

    Let f_{(k)}(p, r) = \frac{\sigma_k}{\sigma_{k-1}}(p, r)-\sum_{l = 0}^{k-2}\alpha_l(1+|p|^2)^{\frac{k-l}{2}}\frac{\sigma_l}{\sigma_{k-1}}(p, r) . \lambda(r) is the vector formed by eigenvalues of r . For any p\in\mathbb{R}^n and symmetric matrices r, s with \lambda(r), \lambda(s)\in \Gamma_k , we have

    \begin{equation} \sum\limits_{i,j}\frac{\partial f_{(k)}}{\partial r_{ij}}(p,r)s_{ij}\geq f_{(k)}(p,s)+\sum\limits_{l = 0}^{k-2}(k-l)\alpha_l(1+|p|^2)^{\frac{k-l}{2}}\frac{\sigma_l}{\sigma_{k-1}}(p,r). \end{equation} (4.4)

    For later purposes we note the simple estimate, if r\geq 0 ,

    \frac{1}{1+|p|^2}\sigma_k(0,r)\leq\sigma_k(p,r)\leq \sigma_k(0,r),

    and the development

    \sigma_k(p,r) = \frac{1+|\tilde p|^2}{1+|p|^2}r_{nn}\sigma_{k-1}(\tilde p,\tilde r)+O((|r_{st}|^k)_{(s,t)\neq (n,n)}),

    where p = (p_1, \cdots, p_n)\in\mathbb{R}^n , r = (r_{ij})_{n\times n} , \tilde p = (p_1, \cdots, p_{n-1})\in\mathbb{R}^{n-1} , \tilde r = (r_{ij})_{i, j = 1, \cdots, n-1} .

    We suppose that \partial \Omega is the graph of \omega:B_{\rho}^{n-1}(0)\subset\mathbb{R}^n\rightarrow \mathbb{R} and u(\tilde x, \omega(\tilde x)) = \varphi(\tilde x) . Furthermore, \omega(0) = 0 , D\omega(0) = 0 , D\varphi(0) = 0 and \omega is a strictly convex function of \tilde x . The curvature equation is equivalent to

    \begin{equation} f_{(k)}(Du,D^2u) = c\sqrt{1+|Du|^2} \end{equation} (4.5)

    defined in some domain \Omega\subset\mathbb{R}^n . We have following boundary estimates for second derivatives of u .

    Lemma 4.3. Let u\in C^3(\bar\Omega) be a k -convex solution of (4.5). We assume (4.3) with \epsilon > 0 . Then the estimate

    \begin{equation} |D^2u(0)|\leq C(n,k,\alpha_l,c,\epsilon,||\omega||_{C^3},||\varphi||_{C^4},||u||_{C^1},\lambda_{\min}(D^2\omega(0))) \end{equation} (4.6)

    holds true where \lambda_{\min} denotes the smallest eigenvalue.

    Remark 4.1. On \partial \Omega , we have for i, j = 1, \cdots, n-1 ,

    \begin{align*} &u_i+u_n\omega_i = \varphi_i,\\ &u_{ij}+u_{in}\omega_j+u_{nj}\omega_i+u_{nn}\omega_i\omega_j+u_n\omega_{ij} = \varphi_{ij}. \end{align*}

    Therefore |u_{ij}(0)| = |\varphi_{ij}(0)-u_n(0)\omega_{ij}(0)|\leq C . It remains to show that |u_{in}(0)|\leq C and |u_{nn}(0)|\leq C . We follow [19,21] to obtain mixed second derivative boundary estimates and double normal second derivative boundary estimate.

    Proof. Let

    \Omega_{d,\kappa} = \{x(\tilde x,x_n)\in\Omega||\tilde x| < d,\omega(\tilde x) < x_n < \tilde\omega(\tilde x)+\frac{\kappa}{2}d^2\}

    where 0 < d < \rho , \tilde\omega(\tilde x): = \omega(\tilde x)-\frac{\kappa}{2}|\tilde x|^2 , and \kappa > 0 is chosen small enough such that \tilde\omega is still strictly convex. We decompose \partial\Omega_{d, \kappa} = \partial_1\Omega_{d, \kappa}\cup\partial_2\Omega_{d, \kappa}\cup\partial_3\Omega_{d, \kappa} with

    \begin{align*} &\partial_1\Omega_{d,\kappa} = \{x\in\partial\Omega_{d,\kappa}|x_n = \omega(\tilde x)\},\\ &\partial_2\Omega_{d,\kappa} = \{x\in\partial\Omega_{d,\kappa}|x_n = \omega(\tilde x)+\frac{\kappa}{2}d^2\},\\ &\partial_3\Omega_{d,\kappa} = \{x\in\partial\Omega_{d,\kappa}||\tilde x| = d\}. \end{align*}

    Our lower barrier function v will be of the form

    \begin{equation} v(x) = \theta(\tilde x)+h(\rho(x)) \end{equation} (4.7)

    where \theta(\tilde x) is an arbitrary C^2 -function, h(\rho) = \exp\{B\rho\}-\exp\{\kappa Bd^2\} and \rho(x) = \kappa d^2+\tilde\omega(\tilde x)-x_n . Denote F^{ij} = \frac{\partial f_{(k)}(Du, D^2u)}{\partial u_{ij}} .

    Mixed second derivative boundary estimates

    By (4.4) and Lemma 2.3, we have

    F^{ij}v_{ij}\geq f_{(k)}(Du,D^2v)+C

    where C depends only on n, k, \alpha_l 's, c, ||Du||_{C^0} . We choose an orthonormal frame \{b_i\}_{i = 1}^n with b_n = -\frac{D\rho}{|D\rho|} and denote v_{(s)} = \frac{\partial v}{\partial b_s} . Directly, we have

    \begin{align*} &v_{(s)} = \theta_{(s)}+h'\rho_{(s)},\,(1\leq s\leq n-1);\quad v_{(n)} = \theta_{(n)}-h'\sqrt{1+|D\tilde \omega|^2};\\ &v_{(st)} = \theta_{(st)}+h'\tilde\omega_{(st)},\,(s,t)\neq (n,n);\\ &v_{(nn)} = \theta_{(nn)}+h'\tilde \omega_{(nn)}+h''(1+|D\tilde\omega|^2). \end{align*}

    We may choose d small so that |Du| is also small. Note that |D\tilde\omega| is small since we can choose d, \kappa small. By choosing large enough B , we caculate

    \begin{align*} f_{(k)}(Du,D^2v) = &\frac{\sigma_k}{\sigma_{k-1}}(Du,D^2v)-\sum\limits_{l = 0}^{k-2}\alpha_l(1+|Du|^2)^{\frac{k-l}{2}}\frac{\sigma_l}{\sigma_{k-1}}(Du,D^2v)\\ \geq & (1-\epsilon)\frac{\sigma_k}{\sigma_{k-1}}(0,D^2v)-2\sum\limits_{l = 0}^{k-2}\alpha_l\frac{\sigma_l}{\sigma_{k-1}}(0,D^2v)\\ \geq & (1-\epsilon)^2h'\frac{\sigma_{k-1}}{\sigma_{k-2}}(0,\tilde \omega_{(st)})-2\sum\limits_{l = 1}^{k-2}\alpha_l (h')^{l-k+1}\frac{\sigma_{l-1}}{\sigma_{k-2}}(0,\tilde \omega_{(st)})-o(B^{-1}) \end{align*}

    where in the last line, 1\leq s, t\leq n-1 . Finally, we see that for large enough B and small enough d and \kappa the estimate

    (1-\delta)h'\leq |Dv|\leq (1+\delta)h'

    is valid for small \delta . Therefore

    \begin{equation} F^{ij}v_{ij}\geq (1-\epsilon)\frac{\sigma_{k-1}}{\sigma_{k-2}}(0,\tilde \omega_{(st)})|Dv|+C. \end{equation} (4.8)

    Let \tau be a C^2 -smooth vector field which is tangential along \partial \Omega . Following [19,21] we then introduce the function

    w = 1-\exp(-a\tilde w)-b|x|^2

    where \tilde w = u_{\tau}-\frac{1}{2}\sum_{i = 1}^{n-1}u_s^2 and a, b are positive constants. Since on \partial_1\Omega_{d, \kappa} , u = \varphi , and

    w|_{\partial_1\Omega_{d,\kappa}}\geq a\varphi_{\tau}-c|\tilde x|^2,\,w(0) = 0,\,w|_{\partial_2\Omega_{d,\kappa}\cup\partial_3\Omega_{d,\kappa}}\geq-M

    for suitable constants c, M depending on a, b, ||u||_{C^1} and ||\varphi||_{C^1} . By differentiation of Eq (4.5), we obtain

    F^{ij}u_{ijp}+F^iu_{ip} = c\bar v_p

    where F^i: = \frac{\partial f_{(k)}}{\partial u_i} and \bar v: = \sqrt{1+|Du|^2} .

    \begin{align} F^{ij}\tilde w_{ij} = &F^{ij}u_{ijp}\tau_p+F^{ij}(u_{pj}\tau_{pi}+u_{pi}\tau_{pj})+F^{ij}\tau_{ijp}u_p-\\&\sum\limits_{s = 1}^{n-1}F^{ij}(u_{is}u_{js}+u_{sij}u_s)\\ = &c(\bar v_p\tau_p-\sum\limits_{s = 1}^{n-1}\bar v_su_s)-F^iu_{ip}\tau_p+\\&\sum\limits_{s = 1}^{n-1}F^iu_{is}u_s+F^{ij}(u_{pj}\tau_{pi}+u_{pi}\tau_{pj})\\ &+F^{ij}\tau_{ijp}u_p-\sum\limits_{s = 1}^{n-1}F^{ij}u_{is}u_{js}. \end{align} (4.9)

    By the definition of \tilde w , we have

    \begin{equation} c(\bar v_p\tau_p-\sum\limits_{s = 1}^{n-1}\bar v_su_s) = \frac{c}{\bar v}\big(\langle D\tilde w,Du\rangle-\rm{Hess}(\tau)(Du,Du)\big). \end{equation} (4.10)

    Then we compute F^i . Denote b_{ij} = \delta_{ij}-\frac{u_iu_j}{\bar v^2} and c_{ij} = b_{ip}u_{pj} . f_{(k)} can be rewritten as

    f_{(k)} = f_{(k)}(c_{ij},\bar v) = \frac{\sigma_k}{\sigma_{k-1}}(c_{ij})-\sum\limits_{l = 0}^{k-2}\alpha_l\bar v^{k-l}\frac{\sigma_l}{\sigma_{k-1}}(c_{ij}).

    Directly we have

    \begin{align*} F^i = &\frac{\partial f_{(k)}}{\partial u_i} = \frac{\partial f_{(k)}}{\partial c_{pq}}\frac{\partial c_{pq}}{\partial u_i}+\frac{\partial f_{(k)}}{\partial \bar v}\frac{\partial \bar v}{\partial u_i}\\ = &-\frac{1}{\bar v^2}f_{(k)}^{iq}u_{ql}u_l-\frac{1}{\bar v^2}f_{(k)}^{pq}u_{iq}u_p+\frac{2}{\bar v^3}f_{(k)}^{pq}u_pu_lu_{lq}u_i \\&-\sum\limits_{l = 0}^{k-2}\alpha_l(k-l)\bar v^{k-l-2}\frac{\sigma_l}{\sigma_{k-1}}(c_{ij})u_i \end{align*}

    where f_{(k)}^{pq}: = \frac{\partial f_{(k)}}{\partial c_{pq}} . Therefore

    \begin{eqnarray} -F^iu_{ip}\tau_p+\sum\limits_{s = 1}^{n-1}F^iu_{is}u_s = \Big(-\frac{1}{\bar v^2}f_{(k)}^{iq}u_{ql}u_l-\frac{1}{\bar v^2}f_{(k)}^{pq}u_{iq}u_p+\frac{2}{\bar v^3}f_{(k)}^{pq}u_pu_lu_{lq}u_i\\ -\sum\limits_{l = 0}^{k-2}\alpha_l(k-l)\bar v^{k-l-2}\frac{\sigma_l}{\sigma_{k-1}}(c_{ij})u_i\Big)(-\tilde w_i+u_p\tau_{pi}). \end{eqnarray} (4.11)

    In order to derive the right hand side of (4.11), we use the same coordinate system as [21], which corresponds to the projection of principal curvature directions of the graph of u onto \mathbb{R}^n\supset\Omega . Fixing a point y\in\Omega , we choose a basis of eigenvectors \hat e_1, \cdots, \hat e_n of the matrix (c_{ij}) at y , corresponding to the eigenvalues \lambda_1, \cdots, \lambda_n and orthonormal with respect to the inner product given by the matrix I+Du\otimes Du . Using a subscript \alpha to denote differentiation with respect to \hat e_{\alpha} , \alpha = 1, \cdots, n , so that

    u_{\alpha} = \hat e_{\alpha}^iu_i = \langle Du,\hat e_{\alpha}\rangle,\quad u_{\alpha\alpha} = \lambda_{\alpha} = \hat e_{\alpha}^i\hat e_{\alpha}^ju_{ij}.

    Then we obtain

    \begin{align*} \frac{1}{\bar v^2}f_{(k)}^{iq}u_{ql}u_l(\tilde w_i-u_p\tau_{pi}) = &\frac{1}{\bar v^2}\frac{\partial f_{(k)}}{\partial \lambda_{\alpha}}\lambda_{\alpha}u_{\alpha}(\tilde \omega_{\alpha}-\rm{Hess}(\tau)(Du,\hat e_{\alpha}))\\ \leq& \delta \frac{\partial f_{(k)}}{\partial \lambda_{\alpha}}\lambda_{\alpha}^2+C(\delta)\frac{\partial f_{(k)}}{\partial \lambda_{\alpha}}\tilde w_{\alpha}^2+C(\delta)\sum\limits_{\alpha = 1}^n\frac{\partial f_{(k)}}{\partial \lambda_{\alpha}}. \end{align*}

    The second term of (4.11) can be estimated in the same way as above. As for the third term of (4.11), we calculate as

    |f_{(k)}^{pq}u_pu_lu_{lq}| = |f_{(k)}^{pq}(u_{pq}-c_{pq})|\leq C|Du|^2.

    Thus

    \begin{equation} -F^iu_{ip}\tau_p+\sum\limits_{s = 1}^{n-1}F^iu_{is}u_s\leq 2\delta \frac{\partial f_{(k)}}{\partial \lambda_{\alpha}}\lambda_{\alpha}^2 \\+C(\delta)\frac{\partial f_{(k)}}{\partial \lambda_{\alpha}}\tilde w_{\alpha}^2+C(\delta)\sum\limits_{\alpha = 1}^n\frac{\partial f_{(k)}}{\partial \lambda_{\alpha}}+C|\tilde w_iu_i-\tau_{ij}u_iu_j|. \end{equation} (4.12)

    Let (\eta_i^{\alpha}) denote the inverse matrix to (\hat e^i_{\alpha}) , we write

    u_{s\alpha} = \hat e^{i}_{\alpha}u_{is} = \lambda_{\alpha}\eta^{\alpha}_s.

    Furthermore,

    \sum\limits_{s = 1}^{n-1}\frac{\partial f_{(k)}}{\partial \lambda_{\alpha}}u_{s\alpha}^2 = \frac{\partial f_{(k)}}{\partial \lambda_{\alpha}}\lambda_{\alpha}^2\sum\limits_{s = 1}^{n-1}(\eta^{\alpha}_s)^2.

    Now we reason similarly to [21]. If for all \alpha = 1, \cdots, n , we have

    \begin{equation} \sum\limits_{s = 1}^{n-1}(\eta^{\alpha}_s)^2\geq \epsilon > 0 \end{equation} (4.13)

    where \epsilon is a small postive number. Then we clearly have

    \begin{equation} \sum\limits_{s = 1}^{n-1}\frac{\partial f_{(k)}}{\partial \lambda_{\alpha}}u_{s\alpha}^2\geq \epsilon \frac{\partial f_{(k)}}{\partial \lambda_{\alpha}}\lambda_{\alpha}^2. \end{equation} (4.14)

    On the other hand, if (4.13) is not true, then

    \sum\limits_{s = 1}^{n-1}(\eta^{\gamma}_s)^2 < \epsilon

    for some \gamma , which implies

    \sum\limits_{s = 1}^{n-1}(\eta^{\alpha}_s)^2\geq \delta_0 > 0

    for all \alpha\neq \gamma . Hence

    \begin{equation} \sum\limits_{s = 1}^{n-1}\frac{\partial f_{(k)}}{\partial \lambda_{\alpha}}u_{s\alpha}^2\geq \delta_0\sum\limits_{\alpha\neq \gamma} \frac{\partial f_{(k)}}{\partial \lambda_{\alpha}}\lambda_{\alpha}^2. \end{equation} (4.15)

    Then we use Theorem 3, 4 in [22] to deduce that

    \begin{align*} &\sum\limits_{\alpha\neq \gamma}(\frac{\sigma_k}{\sigma_{k-1}})_{,\alpha}\lambda_{\alpha}^2\geq\frac{1}{C(n,k)}(\frac{\sigma_k}{\sigma_{k-1}})_{,\alpha}\lambda_{\alpha}^2,\\ &\sum\limits_{\alpha\neq \gamma}(-\frac{\sigma_l}{\sigma_{k-1}})_{,\alpha}\lambda_{\alpha}^2\geq\frac{1}{C(n,k,l)}(-\frac{\sigma_l}{\sigma_{k-1}})_{,\alpha}\lambda_{\alpha}^2,\\ &\sum\limits_{\alpha\neq \gamma}(-\frac{1}{\sigma_{k-1}})_{,\alpha}\lambda_{\alpha}^2\geq\frac{1}{C(n,k,0)}(-\frac{1}{\sigma_{k-1}})_{,\alpha}\lambda_{\alpha}^2-\frac{1}{C(n,k,0)}\frac{\sigma_1}{\sigma_{k-1}} \end{align*}

    where subscript ', \alpha ' denotes differentiation with respect to \lambda_{\alpha} . Therefore,

    \begin{equation} \sum\limits_{s = 1}^{n-1}\frac{\partial f_{(k)}}{\partial \lambda_{\alpha}}u_{s\alpha}^2\geq \delta'\frac{\partial f_{(k)}}{\partial \lambda_{\alpha}}\lambda_{\alpha}^2-C. \end{equation} (4.16)

    Combing (4.9), (4.10), (4.12), (4.16), we have

    \begin{equation} F^{ij}\tilde w_{ij}\leq C|\langle D\tilde w,Du\rangle|+CF^{ij}\tilde w_i\tilde w_j+C\sum\limits_{i = 1}^nF^{ii} \end{equation} (4.17)

    where we have chosen \delta < < \delta' , so that \frac{\partial f_{(k)}}{\partial \lambda_{\alpha}}\lambda_{\alpha}^2 can be discarded. Note that in (4.17), we also have used the fact that \sum_{i = 1}^nF^{ii}\geq C_0 > 0 . By choosing a, b large, we conclude that

    \begin{equation} F^{ij}w_{ij}\leq C|\langle Dw,Du\rangle|. \end{equation} (4.18)

    From (4.8), (4.18), by comparison principle, we have at 0 ,

    u_{\tau n}(0) = \frac{1}{a}w_n(0)\geq \frac{1}{a}v_n(0).

    Since \tau is an arbitrary tangential direction at 0\in\partial\Omega , if we replace \tau by -\tau , we get an upper bound for u_{\tau n}(0) .

    Double normal second derivative boundary estimate

    We turn to estimate |u_{nn}(0)| . The idea is to estimate u_{nn} in a first step at some optimally chosen point y and in a second step conclude from this the estimate in the given point. We introduce a smooth moving orthonormal frame \{b_1, \cdots, b_n\} with b_n = (-\omega_{\tilde x}, 1)/\sqrt{1+|\omega_{\tilde x}|^2} being the upward normal to \partial\Omega . Here \omega_{\tilde x} is the gradient of \omega(\tilde x) . Let

    G = \frac{\sigma_{k-1}}{\sigma_{k-2}}(u_{(\tilde x)},u_{(\tilde x\tilde x)})-\sum\limits_{l = 1}^{k-2}\alpha_l\sqrt{1+|Du|^2}^{k-l}\frac{\sigma_{l-1}}{\sigma_{k-2}}(u_{(\tilde x)},u_{(\tilde x\tilde x)})-c\sqrt{1+|Du|^2}

    on \partial\Omega , where u_{(\tilde x)} = (\frac{\partial u}{\partial b_1}, \cdots, \frac{\partial u}{\partial b_{n-1}}) and u_{(\tilde x\tilde x)} = (\frac{\partial^2 u}{\partial b_i\partial b_j})_{1\leq i, j\leq n-1} . For simplicity, we denote \tilde p = u_{(\tilde x)} , \tilde r = u_{(\tilde x\tilde x)} , \bar v = \sqrt{1+|Du|^2} . First we observe that

    f_{(k)}(p,r) < \lim\limits_{r_{nn}\rightarrow +\infty}f_{(k)}(p,r) = \frac{\sigma_{k-1}}{\sigma_{k-2}}(\tilde p,\tilde r)-\sum\limits_{l = 1}^{k-2}\alpha_l\bar v^{k-l}\frac{\sigma_{l-1}}{\sigma_{k-2}}(\tilde p,\tilde r)

    from what we see that G > 0 . Hence the function

    \tilde G = G(x)+\frac{4|\tilde x|^2}{\bar \rho^2}\bar G

    with \bar G = \max\{G(x)|x\in\partial\Omega, \, |\tilde x| < \rho\} and 0 < \bar\rho < \rho attains its minimum over \partial\Omega\cap B_{\rho}(0) at some point y\in\partial\Omega\cap B_{\bar\rho/2}(0) . If |u_{nn}(y)| < C , then G(y) > C^{-1} > 0 .

    G(0) = \tilde G(0)\geq \tilde G(y) > G(y) > C^{-1} > 0.

    Therefore G(0) is strictly positive and we have

    |u_{nn}(0)| < +\infty.

    To check that |u_{nn}(y)| < +\infty , we proceed in essentially the same way as in mixed second derivative estimates. The point y plays the role of the origin and the function \tilde w is defined as

    \tilde w(x) = -(u_n(x)-u_n(y))-K|Du(x)-Du(y)|^2

    where K is a sufficiently big constant. In order to apply the comparison principle, we need to obtain that

    w(x)\geq \tilde \theta(\tilde x)-C|\tilde x-\tilde y|^2(x\in\partial\Omega\cap B_{\rho}(0))

    where \tilde \theta is some C^2 -smooth function. We reason similarly to Lemma 2.5 in [19]. The choice of the moving frame gives

    u_{(s)} = \varphi_{(s)},\quad u_{(st)} = \varphi_{(st)}-u_n\omega_{(st)}(s,t = 1,\cdots,n-1).

    By the concavity of \frac{\sigma_{k-1}}{\sigma_{k-2}}(\tilde p, \tilde r) , -\frac{\sigma_{l}}{\sigma_{k-2}}(\tilde p, \tilde r)(l = 0, \cdots, k-3) in \tilde r and the convexity of \sqrt{1+|\tilde p|^2} in \tilde p , we compute

    \begin{equation} 0\leq \tilde G(x)-\tilde G(y)\leq g(y,x)(u_n(y)-u_n(x))+h(y,x) \end{equation} (4.19)

    with

    \begin{align*} g(y,x) = &(\frac{\sigma_{k-1}}{\sigma_{k-2}})^{st}(\tilde p(x),\tilde r(y))\omega_{(st)}(x)-\\ &\sum\limits_{l = 1}^{k-2}\alpha_l\bar v^{k-l}(x)(\frac{\sigma_{l-1}}{\sigma_{k-2}})^{st}(\tilde p(x),\tilde r(y))\omega_{(st)}(x)\\ &+\Big(\sum\limits_{l = 1}^{k-2}\alpha_l(k-l)\bar v^{k-l-2}(y)\frac{\sigma_{l-1}}{\sigma_{k-2}}(\tilde p(y),\tilde r(y))+c\bar v^{-1}(y)\Big)\\ &(u_n(y)-u_i(y)\omega_i(x)) \end{align*}

    and

    \begin{align*} h(y,x) = &\frac{\sigma_{k-1}}{\sigma_{k-2}}(\varphi_{\tilde x}(x),\tilde r(y))-\frac{\sigma_{k-1}}{\sigma_{k-2}}(\varphi_{\tilde x}(y),\tilde r(y))\\ &+\big(\frac{\sigma_{k-1}}{\sigma_{k-2}}\big)^{st}(\varphi_{\tilde x}(x),\tilde r(y))\Psi_{st}(y,x)\\ &-\sum\limits_{l = 1}^{k-2}\alpha_l\bar v^{k-l}\big(\frac{\sigma_{l-1}}{\sigma_{k-2}}\big)^{st}(\varphi_{\tilde x}(x),\tilde r(y))\Psi_{st}(y,x)\\ &+\sum\limits_{l = 1}^{k-2} \alpha_l\bar v^{k-l}\Big(\frac{\sigma_{l-1}}{\sigma_{k-2}}(\varphi_{\tilde x}(y),\tilde r(y))-\frac{\sigma_{l-1}}{\sigma_{k-2}}(\varphi_{\tilde x}(x),\tilde r(y))\Big)\\ &+[c\bar v^{-1}-\sum\limits_{l = 1}^{k-2}\alpha_l(k-l)\bar v^{k-l-2}\frac{\sigma_{l-1}}{\sigma_{k-2}}(\tilde p(y),\tilde r(y))]\\ &\cdot A+\frac{4\bar G}{\bar \rho^2}(|\tilde x|^2-|\tilde y|^2) \end{align*}

    where \Psi_{st}(y, x) = \varphi_{(st)}(x)-\varphi_{(st)}(y)-u_n(y)(\omega_{(st)}(x)-\omega_{(st)}(y)) , A = [\varphi_i(y)-\varphi_i(x)-u_n(y)(\omega_i(y)-\omega_i(x))]u_i(y) . We may take \tilde \theta(\tilde x) = -\frac{h}{g}(y, x) if we can show that g(y, x) > 0 . This is true since |Du| is small and -(\frac{\sigma_{l-1}}{\sigma_{k-1}})^{st} is semi-positive definite, together with condition (4.3). This completes the proof of the boundary regularity.

    The authors were supported by NSFC, grant nos. 12031017 and 11971424.

    The authors declare no conflict of interest.

    In this appendix, we sketch the proof of Lemma 3.3 for reader's convenience. For the original proof, see [36].

    Without loss of generality, we assume \Omega = B_r(0) . Let \rho = r^2-|x|^2 , M = osc_{B_r}u , \tilde g(u) = \frac{1}{M}(M+u-\inf_{B_r} u) , \phi(x, \xi) = \rho(x)\tilde g(u)\log(u_{\xi}(x)) . This auxiliary function \phi comes from [34]. Suppose \phi attains its maximum at (x_0, e_1) . Furthermore, by rotating e_2, \cdots, e_n , we can assume that \{u_{ij}(x_0)\}_{2\leq i, j\leq n} is diagonal. Thus \varphi(x) = \log\rho(x)+\log\tilde g(u(x))+\log\log u_1 also attains a local maximum at x_0\in B_r(0) . At x_0 , we have

    \begin{equation} 0 = \varphi_i = \frac{\rho_i}{\rho}+\frac{\tilde g_i}{\tilde g}+\frac{u_{1i}}{u_1\log u_1}, \end{equation} (A.1)
    \begin{equation} 0\geq \varphi_{ij} = \frac{\rho_{ij}}{\rho}-\frac{\rho_i\rho_j}{\rho^2}+\frac{\tilde g_{ij}}{\tilde g}-\frac{\tilde g_i\tilde g_j}{\tilde g^2}+\frac{u_{1ij}}{u_1\log u_1}-(1+\frac{1}{\log u_1})\frac{u_{1i}u_{1j}}{u_1^2\log u_1}. \end{equation} (A.2)

    Only in this proof we denote that F^{ij}: = \frac{\partial F}{\partial u_{ij}} . F^{ij} is positive definite. Taking trace with \varphi_{ij} and using (A.1), we have

    \begin{align} 0&\geq F^{ij}\varphi_{ij}\\ & = F^{ij}\Big(\frac{\rho_{ij}}{\rho}+2\frac{\rho_i}{\rho}\frac{\tilde g_j}{\tilde g}+\frac{\tilde g_{ij}}{\tilde g}\Big)+F^{ij}\Big(\frac{u_{1ij}}{u_1\log u_1}-(1+\frac{2}{\log u_1})\frac{u_{1i}u_{1j}}{u_1^2\log u_1}\Big)\\ &: = \mathcal{A}+\mathcal{B}. \end{align} (A.3)

    It is well-known that the principal curvatures of graph u are the eigenvalues of matrix A = (a_{ij})_{n\times n} :

    a_{ij} = \frac{1}{W}\Big(u_{ij}-\frac{u_iu_lu_{lj}}{W(W+1)}-\frac{u_ju_lu_{li}}{W(W+1)}+\frac{u_iu_ju_pu_qu_{pq}}{W^2(W+1)^2}\Big)

    where W = \sqrt{1+|Du|^2} . Next we compute F^{ij} at x_0 .

    \begin{equation*} \frac{\partial a_{ij}}{\partial u_{ij}} = \left\{ \begin{array}{cl} \frac{1}{W^3} & i = j = 1,\\ \frac{1}{W^2} & i = 1,j\geq 2\,\rm{or}\,i\geq 2,j = 1,\\ \frac{1}{W} & i\geq 2,j\geq 2. \end{array} \right. \end{equation*}

    For two different sets \{p, q\}\neq\{i, j\} , \frac{\partial a_{pq}}{\partial u_{ij}} = 0 . Therefore

    \begin{equation*} F^{ij} = \frac{\partial F}{\partial a_{ij}}\frac{\partial a_{ij}}{\partial u_{ij}} = \left\{ \begin{array}{cl} \frac{1}{W^3}\frac{\partial F}{\partial a_{11}} & i = j = 1,\\ \frac{1}{W^2}\frac{\partial F}{\partial a_{ij}} & i = 1,j\geq 2\,\rm{or}\,i\geq 2,j = 1,\\ \frac{1}{W}\frac{\partial F}{\partial a_{ij}} & i\geq 2,j\geq 2. \end{array} \right. \end{equation*}

    Direct computation shows that

    \mathcal{A} = \frac{-2}{\rho}(\sum\limits_{i = 1}^n\frac{\partial F}{\partial u_{ii}})+\frac{1}{M\tilde g}(\sum\limits_{i,j = 1}^n\frac{\partial F}{\partial u_{ij}}\cdot u_{ij})+\frac{2u_1}{M\rho\tilde g}\sum\limits_{i = 1}^n\frac{\partial F}{\partial u_{1i}}\cdot \rho_i,
    \sum\limits_{i = 1}^n\frac{\partial F}{\partial u_{ii}} = \frac{\partial F}{\partial a_{11}}\frac{1}{W^3}+\sum\limits_{i = 2}^n \frac{\partial F}{\partial a_{ii}}\frac{1}{W}\leq \frac{1}{W}\sum\limits_{i = 1}^n\frac{\partial F}{\partial a_{ii}},
    \sum\limits_{i,j = 1}^n\frac{\partial F}{\partial u_{ij}}\cdot u_{ij} = \sum\limits_{i,j = 1}^n\frac{\partial F}{\partial a_{ij}}\cdot a_{ij} = \frac{\sigma_k}{\sigma_{k-1}}(A)-\sum\limits_{l = 0}^{k-2}\alpha_l(l-k+1)\frac{\sigma_l}{\sigma_{k-1}}(A).

    By (A.1), suppose that u_1\gg 1 , then we have u_{11} < 0 and

    \begin{align*} \frac{2u_1}{M\rho\tilde g}\sum\limits_{i = 1}^n\frac{\partial F}{\partial u_{1i}}\cdot \rho_i& = \frac{2u_1}{M\rho\tilde g}(\frac{\partial F}{\partial u_{11}}\rho_1+\sum\limits_{i = 2}^n\frac{\partial F}{\partial u_{1i}}\rho_i)\\ & = \frac{2u_1}{M\rho\tilde g}(\frac{\partial F}{\partial a_{11}}\frac{\rho_1}{W^3}-\sum\limits_{i = 2}^n\frac{\partial F}{\partial a_{1i}}\frac{u_{1i}\rho}{W^2u_1\log u_{1}})\\ &\geq -\frac{4ru_1}{MW^3\rho\tilde g}\frac{\partial F}{\partial a_{11}}-\frac{2}{M\tilde g\log u_1}\sum\limits_{i = 2}^n\frac{\partial F}{\partial a_{1i}}a_{1i}\\ &\geq -\frac{4ru_1}{MW^3\rho\tilde g}\sum\limits_{i = 1}^n\frac{\partial F}{\partial a_{ii}} \end{align*}

    where we have used (3.1). Therefore

    \begin{equation} \mathcal{A}\geq (-\frac{2}{W\rho}-C\frac{u_1}{W^3})(\sum\limits_{i = 1}^n\frac{\partial F}{\partial a_{ii}})+\frac{1}{M\tilde g}(-g+\sum\limits_{l = 0}^{k-2}\alpha_l(k-l)\frac{\sigma_l}{\sigma_{k-1}}(A)). \end{equation} (A.4)

    In the following we turn to estimate \mathcal{B} . By the definition of a_{ij} , we have at x_0 ,

    \frac{\partial a_{11}}{\partial x_1} = \frac{1}{W^3}u_{111}-\frac{3u_1}{W^5}u_{11}^2-\frac{2u_1}{W^3(W+1)}\sum\limits_{k = 2}^nu_{k1}^2,

    for i\geq 2 ,

    \frac{\partial a_{1i}}{\partial x_1} = \frac{1}{W^2}u_{1i1}-\frac{2u_1}{W^4}u_{11}u_{1i}-\frac{u_1}{W^2(W+1)}u_{1i}u_{ii}-\frac{u_1}{W^3(W+1)}u_{11}u_{1i},
    \frac{\partial a_{ii}}{\partial x_1} = \frac{1}{W}u_{ii1}-\frac{u_1}{W^3}u_{11}u_{ii}-\frac{2u_1}{W^2(W+1)}u_{1i}^2,

    for i\geq 2, j\geq 2, i\neq j ,

    \frac{\partial a_{ij}}{\partial x_1} = \frac{1}{W}u_{ij1}-2\frac{u_{i1}u_{j1}u_1}{W^2(W+1)}.

    Taking derivatives with respect to x_1 on both sides of (1.4), we have

    \sum\limits_{i,j = 1}^n\frac{\partial F}{\partial a_{ij}}\frac{\partial a_{ij}}{\partial x_1}-\sum\limits_{l = 0}^{k-2}(\alpha_l)_{,1}\frac{\sigma_{l}}{\sigma_{k-2}} = -g_{,1}.

    For the first term of \mathcal{B} , we calculate as

    \sum\limits_{i,j = 1}^n\frac{\partial F}{\partial u_{ij}}\frac{u_{ij1}}{u_1\log u_1} = \frac{1}{u_1\log u_1}\Big(\frac{\partial F}{\partial a_{11}}\frac{u_{111}}{W^3}+2\sum\limits_{i\geq 2}\frac{\partial F}{\partial a_{1i}}\frac{u_{1i1}}{W^2}+\sum\limits_{i,j\geq 2}\frac{\partial F}{\partial a_{ij}}\frac{u_{ij1}}{W}\Big),
    \begin{align*} F^{ij}u_{ij1} = &\frac{\partial F}{\partial a_{11}}\Big(\frac{\partial a_{11}}{\partial x_1}+\frac{3u_1}{W^5}u_{11}^2+\frac{2u_1}{W^3(W+1)}\sum\limits_{k\geq 2} u_{k1}^2\Big)\\ &+2\sum\limits_{i\geq 2}\frac{\partial F}{\partial a_{1i}}\Big(\frac{\partial a_{1i}}{\partial x_1}+\frac{2u_1}{W^4}u_{11}u_{1i}\\ &+\frac{u_1u_{1i}u_{ii}}{W^2(W+1)}+\frac{u_1u_{11}u_{1i}}{W^3(W+1)}\Big)\\ &+\sum\limits_{i\neq j\geq 2}\frac{\partial F}{\partial a_{ij}}\Big(\frac{\partial a_{ij}}{\partial x_1}+2\frac{u_1u_{i1}u_{j1}}{W^2(W+1)}\Big)\\ &+\sum\limits_{i = 2}^n\frac{\partial F}{\partial a_{ii}}\Big(\frac{\partial a_{ii}}{\partial x_1}+\frac{u_1u_{11}u_{ii}}{W^3}+\frac{2u_1u_{1i}^2}{W^2(W+1)}\Big)\\ = &-g_{,1}+\sum\limits_{l = 0}^{k-2}(\alpha_l)_{,1}\frac{\sigma_{l}}{\sigma_{k-1}}(A)+\frac{u_1u_{11}}{W^2}\Big(-g+\sum\limits_{l = 0}^{k-2}(k-l)\alpha_l\frac{\sigma_l}{\sigma_{k-1}}(A)\Big)\\ &+\frac{\partial F}{\partial a_{11}}\Big(\frac{2u_1}{W^5}u_{11}^2+\frac{2u_1}{W^3(W+1)}\sum\limits_{k\geq 2} u_{k1}^2\Big)\\ &+2\sum\limits_{i\geq 2}\frac{\partial F}{\partial a_{1i}}\Big(\frac{u_1}{W^4}u_{11}u_{1i}+\frac{u_1u_{1i}u_{ii}}{W^2(W+1)}\\ &+\frac{u_1u_{11}u_{1i}}{W^3(W+1)}\Big)+2\sum\limits_{i\neq j\geq 2}\frac{\partial F}{\partial a_{ij}}\frac{u_1u_{i1}u_{j1}}{W^2(W+1)}\\ &+2\sum\limits_{i = 2}^n\frac{\partial F}{\partial a_{ii}}\frac{u_1u_{1i}^2}{W^2(W+1)}. \end{align*}

    For the second term of \mathcal{B} , we calculate

    \sum\limits_{i,j = 1}^n\frac{\partial F}{\partial u_{ij}}u_{1i}u_{1j} = \frac{\partial F}{\partial a_{11}}\frac{u_{11}^2}{W^3}+2\sum\limits_{i = 2}^n\frac{\partial F}{\partial a_{1i}}\frac{u_{11}u_{1i}}{W^2}+\sum\limits_{2\leq i,j\leq n}\frac{\partial F}{\partial a_{ij}}\frac{u_{1i}u_{1j}}{W}.

    Therefore

    \begin{align*} \mathcal{B} = &\frac{1}{u_1\log u_1}\Big(-g_{,1}+\sum\limits_{l = 0}^{k-2}(\alpha_l)_{,1}\frac{\sigma_{l}}{\sigma_{k-1}}(A)\Big)\\ &+\frac{u_{11}}{W^2\log u_1}\Big(-g+\sum\limits_{l = 0}^{k-2}(k-l)\alpha_l\frac{\sigma_l}{\sigma_{k-1}}(A)\Big)\\ &+\Big(\frac{2}{W^5\log u_1}-(1+\frac{2}{\log u_1})\frac{1}{u_1^2\log u_1W^3}\Big)\\ &\frac{\partial F}{\partial a_{11}}u_{11}^2+\frac{2}{W^3(W+1)\log u_1}\sum\limits_{k\geq 2} \frac{\partial F}{\partial a_{11}}u_{k1}^2\\ &+\Big(\frac{2}{W^4\log u_1}+\frac{2}{W^3(W+1)\log u_1}\\ &-(1+\frac{2}{\log u_1})\frac{2}{W^2u_1^2\log u_1}\Big)\sum\limits_{i = 2}^n\frac{\partial F}{\partial a_{1i}}u_{11}u_{1i}\\ &+\frac{2}{W^2(W+1)\log u_1}\sum\limits_{i = 2}^n\frac{\partial F}{\partial a_{1i}}u_{1i}u_{ii}\\ &+\Big(\frac{2}{W^2(W+1)\log u_1}-\frac{1+2/\log u_1}{Wu_1^2\log u_1}\Big)\times\\ &\sum\limits_{2\leq i,j\leq n}\frac{\partial F}{\partial a_{ij}}u_{1i}u_{1j}. \end{align*}

    Since \{\frac{\partial F}{\partial a_{ij}}\}_{1\leq i, j\leq n} is positive definite, so is \{\frac{\partial F}{\partial a_{ij}}\}_{2\leq i, j\leq n} . W = \sqrt{1+u_1^2}\approx u_1 . Therefore

    \begin{align} \mathcal{B}\geq&\frac{1}{u_1\log u_1}\Big(-g_{,1}+\sum\limits_{l = 0}^{k-2}(\alpha_l)_{,1}\frac{\sigma_{l}}{\sigma_{k-1}}(A)\Big)\\&+\frac{u_{11}}{W^2\log u_1}\Big(-g+\sum\limits_{l = 0}^{k-2}(k-l)\alpha_l\frac{\sigma_l}{\sigma_{k-1}}(A)\Big)\\ &+\frac{1-\delta}{W^5\log u_1}\frac{\partial F}{\partial a_{11}}u_{11}^2+\frac{2}{W^2(W+1)\log u_1}\big(\sum\limits_{i = 2}^n\frac{\partial F}{\partial a_{1i}}u_{1i}u_{ii}\big) \end{align} (A.5)

    where \delta > 0 is a small constant, depending only on u_1 . By (A.3), (A.4), (A.5), we have

    \begin{align} 0\geq& (-\frac{2}{W\rho}-\frac{Cu_1}{W^3})(\sum\limits_{i = 1}^n\frac{\partial F}{\partial a_{ii}})+(\frac{1}{M\tilde g}+\frac{u_{11}}{W^2\log u_1})(-g+\sum\limits_{l = 0}^n\alpha_l(k-l)\frac{\sigma_l}{\sigma_{k-1}})\\ &+\frac{1}{u_1\log u_1}(-g_{,1}+\sum\limits_{l = 0}^{k-2}(\alpha_l)_{,1}\frac{\sigma_l}{\sigma_{k-1}})+\frac{1-\delta}{W^5\log u_1}\frac{\partial F}{\partial a_{11}}u_{11}^2\\ &+\frac{2}{W^2(W+1)\log u_1}\big(\sum\limits_{i = 2}^n\frac{\partial F}{\partial a_{1i}}u_{1i}u_{ii}\big). \end{align} (A.6)

    Since we require that g_u\leq 0 and (\alpha_l)_u\geq 0 ,

    -g_{,1}+\sum\limits_{l = 0}^{k-2}(\alpha_l)_{,1}\frac{\sigma_l}{\sigma_{k-1}}\geq-\frac{\partial g}{\partial x_1}+\sum\limits_{l = 0}^{k-2}\frac{\partial \alpha_l}{\partial x_1}\frac{\sigma_l}{\sigma_{k-1}}.

    We claim that

    \begin{equation} \sum\limits_{i = 2}^n\frac{\partial F}{\partial a_{1i}}u_{1i}u_{ii}\geq -C\frac{u_1^2\log^2 u_1}{W}\frac{|D\rho|^2}{\rho^2}\frac{\partial F}{\partial a_{11}}. \end{equation} (A.7)

    We deter the proof of (A.7). By (A.1), we see that the leading term in (A.6) is \frac{1-\delta}{W^5\log u_1}\frac{\partial F}{\partial a_{11}}u_{11}^2\approx\frac{\log u_1}{W}\frac{\partial F}{\partial a_{11}} > 0 . Other terms have order at most O(W^{-1}) , therefore

    \log u_1\leq C.

    The interior gradient estimate is proved after we check (A.7). Let \Upsilon = \{2\leq j\leq n|a_{jj}\geq 0\} . Note that a_{11} < 0 and \lambda(A)\in\Gamma_k .

    \begin{align*} \sum\limits_{i = 2}^{n} \frac{\partial F}{\partial a_{1 i}} u_{1 i} u_{i i} = &-\sum\limits_{i = 2}^{n}\left[\frac{\sigma_{k-2}(A|1i) \sigma_{k-1}(A)-\sigma_{k-3}(A|1 i) \sigma_{k}(A)}{\sigma_{k-1}^{2}(A)}\right.\\ &\left.+\sum\limits_{l = 1}^{k-2} \alpha_{l}\frac{\sigma_{k-3}(A|1 i) \sigma_{l}(A)-\sigma_{l-2}(A|1 i) \sigma_{k-1}(A)}{\sigma_{k-1}^{2}(A)}\right] a_{i 1} u_{1 i} u_{i i} \\ \geq &-\sum\limits_{i \in \Upsilon}^{n}\left[\frac{\sigma_{k-2}(A|1 i) \sigma_{k-1}(A)-\sigma_{k-3}(A|1 i) \sigma_{k}(A)}{\sigma_{k-1}^{2}(A)}\right.\\ &\left.+\sum\limits_{l = 1}^{k-2} \alpha_{l}\frac{\sigma_{k-3}(A|1 i) \sigma_{l}(A)-\sigma_{l-2}(A|1 i) \sigma_{k-1}(A)}{\sigma_{k-1}^{2}(A)}\right] a_{i i} \frac{u_{1 i}^{2}}{W} \\ \geq &-\sum\limits_{i \in \Upsilon}\left[\frac{a_{i i} \sigma_{k-2}(A|1 i) \sigma_{k-1}(A)}{\sigma_{k-1}^{2}(A)}+\sum\limits_{l = 1}^{k-2} \alpha_{l}\frac{a_{i i} \sigma_{k-3}(A|1 i) \sigma_{l}(A)}{\sigma_{k-1}^{2}(A)}\right] \frac{u_{1 i}^{2}}{W} \\ \geq &-\sum\limits_{i \in \Upsilon}\left[\frac{C_{n,k}\sigma_{k-1}(A|1) \sigma_{k-1}(A)}{\sigma_{k-1}^{2}(A)}+\sum\limits_{l = 1}^{k-2} \alpha_{l}\frac{C_{n,k}\sigma_{k-2}(A|1) \sigma_{l}(A)}{\sigma_{k-1}^{2}(A)}\right] \frac{u_{1 i}^{2}}{W} \\ \geq &-\left[\frac{C_{n,k}\sigma_{k-1}(A|1) \sigma_{k-1}(A)}{\sigma_{k-1}^{2}(A)}+\sum\limits_{l = 1}^{k-2} \alpha_{l}\frac{C_{n,k}\sigma_{k-2}(A|1) \sigma_{l}(A)}{\sigma_{k-1}^{2}(A)}\right] \sum\limits_{i = 2}^{n} \frac{u_{1 i}^{2}}{W} \\ \geq &-\left[C_{n,k}\frac{\sigma_{k-1}(A|1) \sigma_{k-1}(A)-\sigma_{k-2}(A|1) \sigma_{k}(A)}{\sigma_{k-1}^{2}(A)}\right.\\ &+\left.\sum\limits_{l = 0}^{k-2}\alpha_lC_{n,k}\frac{\sigma_{k-2}(A|1)\sigma_l(A)-\sigma_{l-1}(A)\sigma_{k-1}(A)}{\sigma_{k-1}^2(A)}\right]\sum\limits_{i = 2}^n\frac{u_{1i}^2}{W}\\ \geq &-C(n,k)\sum\limits_{i = 2}^n\frac{u_{1i}^2}{W}\frac{\partial F}{\partial a_{11}}\\ \geq &-C(n,k)\frac{u_1^2\log^2u_1}{W}\frac{|D\rho|^2}{\rho^2}\frac{\partial F}{\partial a_{11}}. \end{align*}

    Thus (A.7) holds and the gradient estimate is proved.



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