A linear time approximation scheme for scheduling unbounded batch machines with delivery times and inclusive processing set restrictions

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

1.   Introduction

One classical problem in convex geometry is the Minkowski problem, which is to find convex hypersurfaces in $\mathbb{R}^{n+1}$ whose Gaussian curvature is prescribed as a function defined on $\mathbb{S}^n$ 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

where $u$ is the support function of the convex hypersurface, $\nabla^2u+ug_{\mathbb{S}^n}$ the spherical Hessian matrix of the function $u$. If we take an orthonormal frame on ${\mathbb{S}^n}$, the spherical Hessian of $u$ is $W_u(x): = u_{ij}(x)+u(x)\delta_{ij}$, 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 $1\le k < n$, is often called the Christoffel-Minkowski problem. It corresponds to finding convex solutions of the nonlinear Hessian equation

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

where $\alpha(x), \alpha_l(x)\, (0\le l\le k-1)$ are some functions on $\mathbb{S}^n$. 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 ${\mathbb R}^{n+1}$ and $M = {{\rm{graph}}}\, u$ in a neighbourhood of some point at which we calculate. Let $A$ be the second fundamental form of $M$, $\lambda(A) = (\lambda_1, \cdots, \lambda_n)\in {\mathbb R}^n$ the eigenvalues of $A$ with respect to the induced metric of $M\subset {\mathbb R}^{n+1}$, i.e., the principle curvatures of $M$, and $\sigma_k(\lambda)$ the $k$-th elementary symmetric function, $\sigma_0(\lambda) = 1$. 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

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

where $\sigma_k(A): = \sigma_k(\lambda(A))$, and the coefficients satisfy $\alpha_{k-2} > 0$ and $\alpha_l\geq 0$ for $0\leq l\leq k-3$.

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

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

This equation is equivalent to

where $\lambda_i$'s are the eigenvalues of $D^2u$. It is called supercritical if $\theta\in(\frac{(n-2)\pi}{2}, \frac{n\pi}{2})$ and hypercritical if $\theta\in(\frac{(n-1)\pi}{2}, \frac{n\pi}{2})$. 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., $0\leq \theta < \frac{(n-2)\pi}{2}$, 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 $(M, \omega)$:

where $\chi$ is a closed real $(1, 1)$-form, $\chi_u = \chi+\sqrt{-1}\partial\bar\partial u$, and $\theta_0$ is the angles of the complex number $\int_{M}(\chi+\sqrt{-1}\omega)^n$, $u$ 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 $\theta\in(\pi, \frac{3\pi}{2})$, the special Lagrangian equation (1.3) is

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 $\Omega$ be a bounded domain in $\mathbb{R}^n$, and let $u\in C^4(\Omega)\cap C^{0, 1}(\bar\Omega)$ be an admissible solution of

where $g(x, u)$ and $\alpha_l(x, u) > 0$, $l = 0, 1, \cdots, k-2$, are given smooth functions on $\bar\Omega\times\mathbb{R}$ and $\phi$ is affine, $\lambda = (\lambda_1, \cdots, \lambda_n)$ is the vector of the principal curvatures of graph $u$. $u$ is the admissible solution in the sense that $\lambda\in \Gamma_{k}$ for points on the graph of $u$, with

For simplicity we denote $F = G_k-\sum_{l = 0}^{k-2}\alpha_lG_l$ and $G_l = \sigma_l(\lambda)/\sigma_{k-1}(\lambda)$ for $l = 0, 1, \cdots, k-2, k$. 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 $l$ $(0\leq l\leq k-2)$, $\alpha_l, g\in C^{1, 1}(\bar\Omega\times\mathbb{R})$, $\alpha_l > 0$, and $g > 0$ or $g < 0$. $\phi$ is affine in $(1.4)$. For any fixed $\beta > 0$, if $u\in C^4(\Omega)\cap C^{0, 1}(\bar\Omega)$ is an admissible solution of $(1.4)$, then there exists a constant $C$, depending only on $n, k, \beta, ||u||_{C^1(\bar\Omega)}, \alpha_l, g$ and their first and second derivatives, such that the second fundamental form ${\bf{A}}$ of graph $u$ satisfies

Remark 1.1. Comparing with ^[16], here we require $g > 0$ or $g < 0$ additionally. Also our curvature estimates still hold if $\alpha_l\equiv 0$ for some $0\leq l\leq k-2$. More over, if $\alpha_l\equiv 0$ for all $l = 0, 1, \cdots, k-2$, 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 $\phi-u > 0$ on $\Omega$. We apply the strong maximum principle for the minimal graph equation. Since $\phi$ is affine, it satisfies the following minimal graph equation

Since $u$ is $k$-admissible solution, and $n\geq k\geq 2$, graph of $u$ is mean-convex and $Qu > Q\phi = 0$. By the comparison principle for quasilinear equations (Theorem 10.1 in ^[10]), we then have $\phi > u$ on $\Omega$.

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 $\Omega$ be a bounded domain in $\mathbb{R}^n$, let $\alpha_l, g\in C^{1, 1}(\bar\Omega\times\mathbb{R})$ satisfying $\inf |g| > 0$, $\partial_u g(x, u)\leq 0$, $\alpha_l > 0$ and $\partial_u \alpha_l(x, u)\geq 0$. Suppose there is an admissible function $\underline{u}\in C^2(\Omega)\cap C^{0, 1}(\bar\Omega)$ satisfying

Then the problem

has a unique admissible solution $u\in C^{3, \alpha}(\Omega)\cap C^{0, 1}(\bar\Omega)$ for all $\alpha\in (0, 1)$.

Remark 1.2. $\partial_u g\leq 0$, $\partial_u \alpha_l(x, u)\geq 0$ and the existence of sub-solutions are required in the $C^0$ estimate. The $C^1$ interior estimate is a slightly modification of the result in Theorem 5.1.1 ^[36] since the coefficients $g$, $\alpha_l$ of (1.2) are independent of $u$. We use conditions $\partial_u g\leq 0$ and $\partial_u \alpha_l(x, u)\geq 0$ again to eliminate extra terms in the $C^1$ estimate.

As a further application of the a priori curvature estimate we also consider a Plateau-type problem for locally convex Weingarten hypersurfaces. Let $\Sigma$ be a finite collection of disjoint, smooth, closed, codimension 2 submanifolds of $\mathbb{R}^{n+1}$. Suppose $\Sigma$ bounds a locally uniformly convex hypersurface $\mathcal{M}_0$ with

where $\lambda^0 = (\lambda^0_1, \cdots, \lambda^0_n)$ are the principal curvatures of $\mathcal{M}_0$ and $\alpha_l$'s are positive constants, $c\neq 0$ is a constant. Is there a locally convex hypersurface $\mathcal{M}$ with boundary $\Sigma$ and $f_{(n)}(\lambda) = c$, where $\lambda = (\lambda_1, \cdots, \lambda_n)$ are the principal curvatures of $\mathcal{M}$?

Theorem 1.3. Let $\Sigma$, $f_{(n)}(\lambda)$ be as above. If $\Sigma$ bounds a locally uniformly convex hypersurface $\mathcal{M}_0$ with $f_{(n)}(\lambda^0)\geq c$ at each point of $\mathcal{M}_0$. Then $\Sigma$ bounds a smooth, locally convex hypersurface $\mathcal{M}$ with $f_{(n)}(\lambda) = c$ at each point of $\mathcal{M}$.

2.   Proof of the curvature bound

We compute using a local orthonormal frame field $\hat{{\bf{e}}}_1, \cdots, \hat{{\bf{e}}}_n$ defined on $\mathcal{M} =$ graph $u$ in a neighbourhood of the point at which we are computing. The standard basis of $\mathbb{R}^{n+1}$ is denoted by ${\bf{e}}_1, \cdots, {\bf{e}}_{n+1}$. Covariant differentiation on $\mathcal{M}$ in the direction $\hat{{\bf{e}}}_i$ is denoted by $\nabla_i$. The components of the second fundamental form ${\bf{A}}$ of $\mathcal{M}$ in the basis $\hat{{\bf{e}}}_1, \cdots, \hat{{\bf{e}}}_n$ are denoted by $(h_{ij})$. Thus

where $D$ and $\langle\cdot, \cdot\rangle$ denote the usual connection and inner product on $\mathbb{R}^{n+1}$, and $\nu$ denotes the upward unit normal

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

As usual we denote first and second partial derivatives of $F$ with respect to $h_{ij}$ by $F^{ij}$ and $F^{ij, rs}$. 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 $h_{ab}$ satisfies

Lemma 2.2. For any $\alpha = 1, \cdots, n+1$, we have

Lemma 2.3. There is a constant $C > 0$, depending only on $n, k, \inf \alpha_l, |g|_{C^0}$, so that for any $l = 0, 1, \cdots, k-2$,

Proof. Proof by contradiction. If the result is not true, then for any integer $i$, there is an admissible solution $u_{(i)}$, a point $x_{(i)}\in \Omega$ and an index $0\leq l_{(i)}\leq k-2$, so that

By passing to a subsequence, we may assume $l_{(i)}\rightarrow l_{\infty}$ and $x_{(i)}\rightarrow x_{\infty}\in\bar\Omega$ as $i\rightarrow +\infty$. Therefore

or we may simply write $\frac{\sigma_{l_{\infty}}}{\sigma_{k-1}}\rightarrow +\infty$ if no ambiguilty arises. Since $\alpha_{l_{\infty}} > 0$, and $g$ is bounded, by (1.4) we have $\frac{\sigma_k}{\sigma_{k-1}}\rightarrow +\infty$. For $i$ large enough, $\sigma_k > 0$. By Newton-MacLaurin inequalities, we have

We therefore get a contradiction.

Proof of Theorem 1.1. Here the argument comes from ^[29]. Let $\eta = \phi-u$. $\eta > 0$ in $\Omega$. For a function $\Phi$ to be chosen and a constant $\beta > 0$ fixed, we consider the function

for all $X\in\mathcal{M}$ and all unit vector $\xi\in \rm{T}_X\mathcal{M}$. Then $\tilde W$ attains its maximum at an interior point $X_0\in\mathcal{M}$, in a direction $\xi_0\in \rm{T}_{X_0}\mathcal{M}$ which we may take to be $\hat{{\bf{e}}}_1$. We may assume that $(h_{ij})$ is diagonal at $X_0$ with eigenvalues $\lambda_1\geq \lambda_2\geq\cdots\geq\lambda_n$. Without loss of generality we may assume that the $\hat{{\bf{e}}}_1, \cdots, \hat{{\bf{e}}}_n$ has been chosen so that $\nabla_i\hat{{\bf{e}}}_j = 0$ at $X_0$ for all $i, j = 1, \cdots, n$. Let $\tau = \hat{{\bf{e}}}_1$. Then $W(X) = \tilde W(X, \tau)$ is defined near $X_0$ and has an interior maximum at $X_0$. Let $Z: = h_{ab}\tau_a\tau_b$. By the special choice of frame and the fact that $h_{ij}$ is diagonal at $X_0$ in this frame, we can see that

Therefore the scalar function $Z$ satisfies the same equation as the component $h_{11}$ of the tensor $h_{ij}$. Thus at $X_0$, we have

and

is nonpositive in the sense of matrices at $X_0$. By Lemmas 2.1 and 2.2, we have, at $X_0$,

Using Gauss's formula

we have

Consequently,

For the same reason, we have for all $l = 0, \cdots, k-2$,

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

Recall that $F = G_k-\sum \alpha_lG_l$ and it is well-known that the operator $(\frac{\sigma_{k-1}}{\sigma_{l}})^{\frac{1}{k-1-l}}$ is concave for $0\leq l\leq k-2$. It follows that

For any symmetric matrix $(B_{ij})\in\mathbb{R}^{n\times n}$, we have

Direct computation shows that

Note that $G_k$ is also a concave operator.

where $C_l = \frac{k-l}{k-1-l}$. By the homogeneity of $G_l$'s, we see that

Using Lemma 2.3 again, we have

Next we assume that $\phi$ has been extended to be constant in the ${\bf{e}}_{n+1}$ direction.

Consequently,

Using above estimates in (2.4), we have, at $X_0$,

Next, using (2.2), we have

for any $\gamma > 0$. Therefore at $X_0$ we have, since $|\nabla\eta|\leq C$,

We choose a positive constant $a$, so that

which depends only on $\sup_{\Omega}|Du|$. Therefore

We now choose

Then

and

By direct computation, we have $\nabla_i\nu_{n+1} = -h_{ip}\langle \hat{{\bf{e}}}_p, {\bf{e}}_{n+1}\rangle$, and therefore

Next we choose $0 < \gamma\leq\frac{a^2}{2}$, then we have

Thus we have

In the following we show that $\sum_{i = 1}^nF^{ii}\leq C$. By the definition of operator $F$ and Lemma 2.3, we have

From Eq (1.4) we have $|G_k|\leq C$, therefore $\sum_i F^{ii}\leq C$. At $X_0$, we get an upper bound

Consequently, $W(X_0)$ satisfies an upper bound. Since $W(X)\leq W(X_0)$, we get the required upper bound for the maximum principle curvature. Since $\lambda\in\Gamma_{k}$ and $n\geq k\geq 2$, $u$ is at least mean-convex and

Therefore $\lambda_n\geq -(n-1)\lambda_1$ and

3.   The Dirichlet problem

In this section we prove Theorem 1.2. By comparison principle, we have $0\geq u\geq \underline{u}$. For any $\Omega'\Subset\Omega$, $\inf_{\Omega'}\underline{u}\leq u\leq c(\Omega') < 0$. 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

Proof. Let

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

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

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

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

Equivalently,

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

Then

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

Proof. Note that

For $0\leq l\leq k-2$,

As for $l = k$,

Therefore

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

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

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.

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 $\epsilon\rightarrow 0$.

4.   The Plateau problem

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

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,

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

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

$\alpha_l$'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

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

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

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

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

with

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

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

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

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

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

and the development

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

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

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

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

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

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

Our lower barrier function $v$ will be of the form

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

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

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

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

is valid for small $\delta$. Therefore

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

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

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

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

By the definition of $\tilde w$, we have

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

Directly we have

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

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

Then we obtain

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

Thus

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

Furthermore,

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

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

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

for some $\gamma$, which implies

for all $\alpha\neq \gamma$. Hence

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

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

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

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

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

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

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

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

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 $G(y) > C^{-1} > 0$.

Therefore $G(0)$ is strictly positive and we have

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

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

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

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

with

and

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.

Acknowledgments

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

Conflict of interest

The authors declare no conflict of interest.

A.   Appendix. Proof of Lemma 3.3

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

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

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

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

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

Direct computation shows that

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

where we have used (3.1). Therefore

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

for $i\geq 2$,

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

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

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

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

Therefore

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

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

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

We claim that

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

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 $\lambda(A)\in\Gamma_k$.

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