On a class of obstacle problem for Hessian equations on Riemannian manifolds

Abstract

In this paper, we establish the a priori $C^{2}$ estimates for solutions of a class of obstacle problem for Hessian equations on Riemannian manifolds. Some applications are also discussed. The main contribution of this paper is the boundary estimates for second-order derivatives.

1 Introduction

Let $(\overline{M}, g)$ be a compact manifold with smooth boundary ∂M. In this paper, we are concerned with the obstacle problem

$$\begin{aligned} \max \bigl\{ u - \phi, - \bigl(f \bigl(\lambda \bigl(\nabla ^{2} u + \chi \bigr)\bigr) - \psi (x, u, \nabla u)\bigr)\bigr\} = 0\quad \text{in } M \end{aligned}$$

(1.1)

with the boundary condition

$$\begin{aligned} u = \varphi \quad\text{on } \partial M, \end{aligned}$$

(1.2)

where f is a smooth, symmetric function defined in an open convex cone $\Gamma \subset \mathbb{R}^{n}$ with a vertex at the origin and

$$\begin{aligned} \Gamma _{n} = \bigl\{ \lambda = (\lambda _{1}, \ldots, \lambda _{n}) \in \mathbb{R}^{n}: \text{ each } \lambda _{i} > 0\bigr\} \subseteq \Gamma \neq \mathbb{R}^{n}, \end{aligned}$$

$\nabla ^{2} u$ denotes the Hessian of u, χ is a $(0, 2)$-tensor field, $\lambda (h)$ denotes the eigenvalues of a $(0, 2)$-tensor field h with respect to the metric g and $\varphi \in C^{4} (\partial M)$. In this work, we assume the obstacle function $\phi \in C^{3} (\overline{M})$ satisfies $\phi = \varphi$ on ∂M.

We shall use a penalization technique to establish the a priori $C^{2}$ estimates for a singular perturbation problem (see (2.1)). A similar problem was studied in [14] and [1], where the obstacle function ϕ is assumed to satisfy $\phi > \varphi$ on ∂M so that near the boundary ∂M, the solution of (2.1) satisfies the Hessian-type equation

$$\begin{aligned} f \bigl(\lambda \bigl(\nabla ^{2} u + \chi \bigr)\bigr) = \psi (x, u, \nabla u) \end{aligned}$$

(1.3)

and the second-order boundary estimates follow from studies on Hessian-type equations (see [6], [9], and [10] for examples). In the current paper the obstacle function ϕ is allowed to equal φ on the boundary so that the main difficulty is from the boundary estimates for second-order derivatives.

As in [3], we suppose the function $f \in C^{2} (\Gamma ) \cap C^{0} (\overline{\Gamma})$ satisfies the structure conditions:

$$\begin{aligned} &f_{i} = f_{\lambda _{i}} \equiv \frac{\partial f}{\partial \lambda _{i}} > 0 \quad\text{in $\Gamma $}, 1 \leq i \leq n, \end{aligned}$$

(1.4)

$$\begin{aligned} &\text{$f$ is concave in $\Gamma $}, \end{aligned}$$

(1.5)

and

$$\begin{aligned} \textstyle\begin{cases} f>0& \text{in $\Gamma $}, \\ f=0& \text{on $\partial \Gamma $}. \end{cases}\displaystyle \end{aligned}$$

(1.6)

In addition, f is also assumed to satisfy that for any positive constants $\mu _{1}, \mu _{2}$ with $0 < \mu _{1} < \mu _{2} < \sup_{\Gamma }f$ there exists a positive constant $c_{0}$ depending on $\mu _{1}$ and $\mu _{2}$ such that

$$\begin{aligned} \bigl(f_{1} (\lambda ) \cdot \cdots \cdot f_{n} (\lambda )\bigr)^{1/n} \geq c_{0} \end{aligned}$$

(1.7)

for any $\lambda \in \Gamma _{\mu _{1}, \mu _{2}}:= \{\lambda \in \Gamma: \mu _{1} \leq f (\lambda ) \leq \mu _{2}\}$ and

$$\begin{aligned} f_{i} (\lambda ) \geq c_{1} \biggl(1 + \sum _{j} f_{j} \biggr) \quad\text{for any } \lambda \in \Gamma \text{ with } \lambda _{i} < 0. \end{aligned}$$

(1.8)

Furthermore, f is supposed to satisfy that for any $A > 0$ and any compact set $K \subset \Gamma$, there exists $R = R (A, K) > 0$ such that

$$\begin{aligned} f (\lambda _{1}, \ldots, \lambda _{n-1}, \lambda _{n} + R) \geq A, \quad\text{for all } \lambda \in K \end{aligned}$$

(1.9)

and

$$\begin{aligned} f (R \lambda ) \geq A, \quad\text{for all } \lambda \in K. \end{aligned}$$

(1.10)

Following [3], we assume that there exists a large number $R > 0$ such that at each $x \in \partial M$,

$$\begin{aligned} \bigl(\kappa _{1} (x), \ldots, \kappa _{n-1} (x), R\bigr) \in \Gamma, \end{aligned}$$

(1.11)

where $(\kappa _{1} (x), \ldots, \kappa _{n-1} (x))$ are the principal curvatures of ∂M at x (relative to the interior normal). Since the function ψ may depend on ∇u, we assume there exists an admissible subsolution $\underline{u} \in C^{2} (\overline{M})$ satisfying

$$\begin{aligned} \textstyle\begin{cases} f (\lambda (\nabla ^{2} \underline{u} + \chi )) \geq \psi (x, \underline{u}, \nabla \underline{u}) &\text{in } M, \\ \underline{u} = \varphi &\text{on } \partial M , \\ \underline{u} \leq \phi &\text{in } M. \end{cases}\displaystyle \end{aligned}$$

(1.12)

As in [6], the function $\psi (x, z, p) \in C^{2} (T^{*}\overline{M} \times \mathbb{R}) > 0$ satisfies

$$\begin{aligned} & \psi (x, z, p) \text{ is convex in } p, \end{aligned}$$

(1.13)

$$\begin{aligned} &\sup_{(x, z, p) \in T^{*} \overline{M} \times \mathbb{R}} \frac{- \psi _{z} (x, z, p)}{\psi (x, z, p)} < \infty \end{aligned}$$

(1.14)

and the growth condition

$$\begin{aligned} \begin{aligned}& p \cdot \nabla _{p} \psi (x, z, p) \leq \bar{\psi} (x, z) \bigl(1 + \vert p \vert ^{ \gamma _{1}} \bigr), \\ &p \cdot \nabla _{x} \psi (x, z, p) + \vert p \vert ^{2} \psi _{z} (x, z, p) \geq \bar{\psi} (x, z) \bigl(1 + \vert p \vert ^{\gamma _{2}}\bigr), \end{aligned} \end{aligned}$$

(1.15)

when $|p|$ is sufficiently large, where $\gamma _{1} < 2$, $\gamma _{2} < 4$ are positive constants and ψ̄ is a positive-continuous function of $(x, z) \in \overline{\Omega} \times \mathbb{R}$.

Definition 1.1

A function $u \in C^{2} (M)$ is called admissible if $\lambda (\nabla ^{2} u + \chi ) \in \Gamma$ in Ω.

Our main results are stated as follows.

Theorem 1.2

Suppose f satisfies (1.4)–(1.11) and there exists an admissible subsolution $\underline{u} \in C^{2} (\overline{M})$ satisfying (1.12). Assume that $\psi > 0$ satisfies (1.13)–(1.15), $\varphi \in C^{4} (\partial M)$, ϕ is admissible in M and $\phi = \varphi$ on ∂M. Then, there exists an admissible solution $u \in C^{1,1} (\overline{M})$ of (1.1) and (1.2).

Furthermore, $u \in C^{3,\alpha} (E)$ for any $\alpha \in (0, 1)$ and the Hessian equation (1.3) holds in E, where $E:= \{x \in M: u(x) < \phi (x)\}$.

Note that in Theorem 1.2, the function ϕ is assumed to be admissible. Under the homogeneous boundary condition, i.e., $\varphi \equiv 0$, and that $\chi \equiv 0$, we can remove this assumption.

Theorem 1.3

Assume that $\chi \equiv 0$ in (1.1). Suppose (1.4)–(1.11) and there exists an admissible subsolution $\underline{u} \in C^{2} (\overline{M})$ satisfying (1.12) with $\varphi \equiv 0$. Assume that $\psi > 0$ satisfies (1.13)–(1.15), $\varphi \equiv 0$ and $\phi \equiv 0$ on ∂M. Then, there exists an admissible solution $u \in C^{1,1} (\overline{M})$ of (1.1) and (1.2) and $u \in C^{3,\alpha} (E)$ for any $\alpha \in (0, 1)$ and satisfies (1.3) in E.

Typical examples are given by $f = \sigma ^{1/k}_{k}$, $1 \leq k \leq n$, defined on the cone $\Gamma _{k} = \{\lambda \in \mathbb{R}^{n}: \sigma _{j} (\lambda ) > 0, j = 1, \ldots, k\}$, where $\sigma _{k} (\lambda )$ are the elementary symmetric functions

$$\begin{aligned} \sigma _{k} (\lambda ) = \sum _{i_{1} < \cdots < i_{k}} \lambda _{i_{1}} \ldots \lambda _{i_{k}},\quad k = 1, \ldots, n. \end{aligned}$$

(1.16)

Other interesting examples satisfying (1.4)–(1.11) (see [13]) are

$$\begin{aligned} f (\lambda ) = \sigma _{k}^{1/k} (\mu _{1}, \ldots, \mu _{n}), \end{aligned}$$

(1.17)

defined on the cone $\Gamma = \{\lambda \in \mathbb{R}^{n}: (\mu _{1}, \ldots, \mu _{n}) \in \Gamma _{k}\}$, where $\mu _{i}$ are defined by

$$\begin{aligned} \mu _{i} = \sum_{j \neq i} \lambda _{j},\quad i = 1, \ldots, n. \end{aligned}$$

It is an interesting question whether we can establish the a priori second-order estimates without the condition (1.13). We note that such a condition is necessary in general (see [11]). It is a longstanding problem of the global $C^{2}$ estimates for the k-Hessian equation

$$\begin{aligned} \sigma _{k} \bigl(\lambda \bigl(D^{2} u\bigr)\bigr) = \psi (x, u, Du) \end{aligned}$$

dropping the condition (1.13). The cases $k=2$, $k=n-1$, and $k=n-2$ were resolved by Guan–Ren–Wang [11], Ren–Wang [20], and Ren–Wang [21], respectively. It is still open for general k. Chu–Jiao [5] considered the case (1.17) and established the curvature estimates without the condition (1.13). Jiao–Liu [13] studied the corresponding Dirichlet problem. It is of interest to ask if the above methods can be applied to the related obstacle problem (1.1).

Given a function $v: \Omega \rightarrow \mathbb{R}$, denote $M_{v}:= \{(x, v(x)): x \in \Omega \}$ to be the graphic hypersurface defined by v. Then, the Gauss curvature of $M_{v}$ is

$$\begin{aligned} K (M_{v}) = \frac{\det D^{2} v}{(1+ \vert Dv \vert ^{2})^{(n+2)/2}}. \end{aligned}$$

A classic problem in differential geometry is to find a convex graphic hypersurface with prescribed Gauss curvature K that is equivalent to solving a Monge–Ampère equation

$$\begin{aligned} \det D^{2} u = K (x, u) \bigl(1+ \vert Du \vert ^{2}\bigr)^{(n+2)/2}. \end{aligned}$$

(1.18)

It is also of interest to find hypersurfaces having prescribed Gauss curvature under an obstacle. Such a problem is also equivalent to an obstacle for Monge–Ampère equations. Xiong–Bao [25] proved the $C^{1,1}$ regularity under the condition that the obstacle function is strictly larger than the boundary data. A similar question can be asked if the Gauss curvature is replaced with other kinds of curvatures, such as the mean curvature [4]. The following two theorems can be regarded as applications of Theorem 1.2 and Theorem 1.3.

Theorem 1.4

Let Ω be a uniformly convex bounded domain in $\mathbb{R}^{n}$. Given a function $K (x, z) \in C^{2} (\overline{\Omega}\times \mathbb{R}) > 0$ satisfying that there exists a positive constant A such that

$$\begin{aligned} K_{z} (x, z) \geq - A K (x, z), \quad\textit{for all } (x, z) \in \overline{\Omega}\times \mathbb{R} \end{aligned}$$

(1.19)

and a piece of uniformly convex graphic hypersurface $M_{\phi}$, suppose there exists a uniformly convex graphic hypersurface $M_{\underline{u}}$ under $M_{\phi}$ satisfying the Gauss curvature of $M_{\underline{u}}$,

$$\begin{aligned} K (M_{\underline{u}}) \bigl(x, \underline{u} (x)\bigr) \geq K \bigl(x, \underline{u} (x)\bigr)\quad \textit{for } x \in \overline{\Omega} \end{aligned}$$

(1.20)

and $\underline{u} = \phi$ on ∂Ω. Then, there exists a $C^{1,1}$ graphic hypersurface $M_{u}$ under $M_{\phi}$ with the same boundary such that $K (M_{u}) \geq K (x, u)$ in Ω and $K (M_{u}) = K (x, u)$ in $E:= \{x \in \Omega: u(x) < \phi (x)\}$.

Theorem 1.5

Suppose $K (x, z) \in C^{2} (\overline{\Omega}\times \mathbb{R}) > 0$ satisfying (1.19). The graphic hypersurface $M_{\phi}$ is of constant boundary, suppose there exist a uniformly convex graphic hypersurface $M_{\underline{u}}$ under $M_{\phi}$ satisfying (1.20) and $\underline{u} = \phi$ on ∂Ω. Then, there exists a $C^{1,1}$ graphic hypersurface $M_{u}$ under $M_{\phi}$ with the same constant boundary such that $K (M_{u}) \geq K (x, u)$ in Ω and $K (M_{u}) = K (x, u)$ in E.

Other applications of the obstacle problem for Hessian equations can be found in [2], [4], [15], [19], [22], and so on. The reader is referred to [1] for more applications and background of (1.1).

Similar problems were studied in [14], [1], and [12] under various conditions. In this work, we are mainly concerned with the boundary estimates for second-order estimates. The main difficulty is from the existence of a disturbance term $\beta _{\epsilon}$ in (2.1). It is also why the conditions (1.9)–(1.11) are needed.

The obstacle problem for Monge–Ampère equations (when $f = \sigma _{n}^{1/n}$) was studied extensively, see [2], [16], [17], [22], and [25] for examples. For the obstacle problem of Hessian equations on Riemannian manifolds, the reader is referred to [1], [12], and [14]. We refer the reader to [6], [8], [10], [18], and [24] for the study of Hessian-type equations on Riemannian manifolds.

In Sect. 2, we provide the general idea to prove Theorems 1.2 and 1.3 for which we introduce an approximating problem using a penalization technique. Section 3 is devoted to the boundary estimates for second-order estimates for the solution of the approximating problem.

2 Preliminaries

As in [14] and [25], we consider the singular perturbation problem

$$\begin{aligned} \textstyle\begin{cases} f (\lambda (\nabla ^{2} u + \chi ) = \psi (x, u, \nabla u) + \beta _{\epsilon }(u - \phi ) & \text{in } M, \\ u = \varphi & \text{on } \partial M, \end{cases}\displaystyle \end{aligned}$$

(2.1)

where the penalty function $\beta _{\epsilon}$ is defined by

$$\begin{aligned} \beta _{\epsilon }(z) = \textstyle\begin{cases} 0, & z \leq 0, \\ z^{3} / \epsilon, & z > 0, \end{cases}\displaystyle \end{aligned}$$

for $\epsilon \in (0, 1)$. Obviously, $\beta _{\epsilon }\in C^{2} (\mathbb{R})$ satisfies

$$\begin{aligned} \begin{aligned} & \beta _{\epsilon}, \beta '_{\epsilon}, \beta ''_{\epsilon } \geq 0; \\ & \beta _{\epsilon }(z) \rightarrow \infty \quad\text{as } \epsilon \rightarrow 0^{+}, \text{ whenever } z > 0; \\ & \beta _{\epsilon }(z) = 0, \quad\text{whenever } z \leq 0. \end{aligned} \end{aligned}$$

(2.2)

Since $\underline{u} \leq \phi$, $\underline{u}$ is also a subsolution to (2.1). Let $u_{\epsilon }\in C^{3} (\overline{M}) \cap C^{4} (M)$ be an admissible solution of (2.1) with $u_{\epsilon }\geq \underline{u}$. We shall show that there exists a constant C independent of ϵ such that

$$\begin{aligned} \Vert u_{\epsilon} \Vert _{C^{2} (\overline{M})} \leq C \end{aligned}$$

(2.3)

for small ϵ.

The $C^{0}$ estimates can be easily derived from the fact that $\Gamma \subset \Gamma _{1}$ and $u \geq \underline{u}$. The following lemma is crucial for our estimates, and its proof can be found in [1] (see [25] for the case of the Monge–Ampère equation). For completeness, we provide a proof here.

Lemma 2.1

There exists a positive constant $c_{2}$ independent of ϵ such that

$$\begin{aligned} 0 \leq \beta _{\epsilon }(u_{\epsilon }- \phi ) \leq c_{2} \quad\textit{on } \overline{M}. \end{aligned}$$

(2.4)

Proof

We consider the maximal value of $u_{\epsilon }- \phi$ on M̅. We may assume it is achieved at an interior point $x_{0} \in M$ since $u_{\epsilon }- \phi = \varphi - \phi = 0$ on ∂M. We have, at $x_{0}$,

$$\begin{aligned} \nabla (u_{\epsilon }- \phi ) = 0 \end{aligned}$$

(2.5)

and

$$\begin{aligned} \begin{aligned} \nabla ^{2} u_{\epsilon } \leq \nabla ^{2} \phi . \end{aligned} \end{aligned}$$

(2.6)

It follows that, at $x_{0}$,

$$\begin{aligned} \begin{aligned} 0 \leq {}& \beta _{\epsilon }(u_{\epsilon }- \phi ) = f \bigl(\lambda \bigl( \nabla ^{2} u_{\epsilon }+ \chi \bigr) \bigr) - \psi (x, u, \nabla \phi ) \\ \leq {}& f \bigl(\lambda \bigl(\nabla ^{2} \phi + \chi \bigr)\bigr) - \psi (x, u, \nabla \phi ) \leq c_{2} \end{aligned} \end{aligned}$$

for some positive constant $c_{2}$ depending only on $\|\phi \|_{C^{2} (\overline{M})}$ and (2.4) holds. □

After establishing the estimate (2.3), we can find a subsequence $u_{\epsilon _{k}}$ and a function $u \in C^{1,1} (\overline{\Omega})$ such that

$$\begin{aligned} u_{\epsilon _{k}} \rightarrow u \quad\text{in } C^{1,\alpha} (\bar{M}), \forall \alpha \in (0, 1), \text{ as } \epsilon _{k} \rightarrow 0. \end{aligned}$$

Then, we see u is an admissible solution of (1.1) and (1.2) as in [25]. The fact that $u \in C^{3, \alpha} (E)$ and satisfies (1.3) in E follows from the Evans–Krylov theory.

The $C^{1}$ bound under conditions (1.8) and (1.15) was derived in [14]. It was also shown in [14] how to establish the estimates for second-order derivatives from their bound on the boundary. This paper will focus on the estimates for second-order estimates on the boundary.

Let $u \in C^{4} (\overline{M})$ be an admissible function. For simplicity we shall use the notation $U = \chi + \nabla ^{2} u$ and, under an orthonormal local frame $e_{1}, \ldots, e_{n}$,

$$\begin{aligned} &U_{ij} \equiv U (e_{i}, e_{j}) = \chi _{ij} + \nabla _{ij} u, \\ & \nabla _{k} U_{ij} \equiv \nabla U (e_{i}, e_{j}, e_{k}) = \nabla _{k} \chi _{ij} + \nabla _{kij} u. \end{aligned}$$

(2.7)

Let F be the function defined by

$$\begin{aligned} F (h) = f \bigl(\lambda (h)\bigr) \end{aligned}$$

for a $(0,2)$-tensor h on M. Equation (2.1) is therefore written in the form

$$\begin{aligned} F (U) = \psi (x, u, \nabla u) + \beta _{\epsilon }(u-\phi ). \end{aligned}$$

(2.8)

Following the literature we denote throughout this paper

$$\begin{aligned} F^{ij} = \frac{\partial F}{\partial h_{ij}} (U), \qquad F^{ij, kl} = \frac{\partial ^{2} F}{\partial h_{ij} \partial h_{kl}} (U) \end{aligned}$$

under an orthonormal local frame $e_{1}, \ldots, e_{n}$. The matrix $\{F^{ij}\}$ has eigenvalues $f_{1}, \ldots, f_{n}$ and is positive-definite by assumption (1.4), while (1.5) implies that F is a concave function of $U_{ij}$ (see [3]). Moreover, when $U_{ij}$ is diagonal so is $\{F^{ij}\}$. We can derive from (1.4)–(1.6) that

$$\begin{aligned} \sum_{i} f_{i} (\lambda ) \lambda _{i} \geq 0\quad \text{for any } \lambda \in \Gamma. \end{aligned}$$

(2.9)

We need the following lemmas that were proved in [7].

Lemma 2.2

Let $A = \{A_{ij}\} \in \mathcal{S}^{n\times n}$ with $\lambda (A) = (\lambda _{1}, \ldots, \lambda _{n}) \in \Gamma$ and $F^{ij} = \frac{\partial F (A)}{\partial A_{ij}}$ with eigenvalues $f_{1}, \ldots, f_{n}$, where $\mathcal{S}^{n\times n}$ is the space of all symmetric matrices. There exists an index r such that

$$\begin{aligned} \sum_{\beta \leq n-1} F^{ij} A_{i \beta} A_{\beta j} \geq \frac{1}{2} \sum _{i \neq r} f_{i} \lambda _{i}^{2}. \end{aligned}$$

(2.10)

Lemma 2.3

For any index r and $\epsilon > 0$, there exists a positive constant C depending only on n such that

$$\begin{aligned} \sum f_{i} \vert \lambda _{i} \vert \leq \epsilon \sum_{i \neq r} f_{i} \lambda _{i}^{2} + \frac{C}{\epsilon} \sum f_{i} + Q (r), \end{aligned}$$

(2.11)

where $Q (r) = f (\lambda ) - f (1, \ldots, 1)$ if $\lambda _{r} \geq 0$ and $Q (r) = 0$ if $\lambda _{r} < 0$.

In the following section, we will drop the subscript ϵ for convenience.

3 Estimates for second-order derivatives on the boundary

In this section, we establish the boundary estimates for second-order derivatives of the solution of (2.1). Fix an arbitrary point $x_{0} \in \partial M$. We choose smooth orthonormal local frames $e_{1}, \ldots, e_{n}$ around $x_{0}$ such that when restricted to ∂M, $e_{n}$ is normal to ∂M.

Let $\rho (x)$ denote the distance from x to $x_{0}$,

$$\begin{aligned} \rho (x) \equiv \operatorname{dist}_{M^{n}} (x, x_{0}), \end{aligned}$$

and $M_{\delta} = \{x \in M: \rho (x) < \delta \}$. Since ∂M is smooth we may assume the distance function to ∂M

$$\begin{aligned} d (x) \equiv \operatorname{dist} (x, \partial M) \end{aligned}$$

is smooth in $M_{\delta _{0}}$ for fixed $\delta _{0} > 0$ sufficiently small (depending only on the curvature of M and the principal curvatures of ∂M). Since $\nabla _{ij} \rho ^{2} (x_{0}) = 2 \delta _{ij}$, we may assume ρ is smooth in $M_{\delta _{0}}$ and

$$\begin{aligned} \{\delta _{ij}\} \leq \bigl\{ \nabla _{ij} \rho ^{2}\bigr\} \leq 3 \{\delta _{ij} \} \quad\text{in } M_{\delta _{0}}. \end{aligned}$$

(3.1)

Since $u - \underline{u} = 0$ on ∂M we have

$$\begin{aligned} \nabla _{\alpha \beta} (u - \underline{u}) = - \nabla _{n} (u - \underline{u}) \varPi (e_{\alpha}, e_{\beta}),\quad \forall 1 \leq \alpha, \beta < n \text{ on $\partial M$} , \end{aligned}$$

(3.2)

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

$$\begin{aligned} \bigl\vert \nabla _{\alpha \beta} u (x_{0}) \bigr\vert \leq C, \quad\text{for } 1 \leq \alpha, \beta \leq n-1. \end{aligned}$$

(3.3)

Next, we establish the estimate

$$\begin{aligned} \bigl\vert \nabla _{\alpha n} u (x_{0}) \bigr\vert \leq C\quad \text{for } \alpha \leq n - 1. \end{aligned}$$

(3.4)

Define the linear operator L by

$$\begin{aligned} L w:= F^{ij} \nabla _{ij} w - \psi _{p_{k}} \nabla _{k} w - \beta _{ \epsilon}' (u - \phi ) w, \quad\text{for } w \in C^{2} (M). \end{aligned}$$

We first need to construct a barrier as Lemma 6.2 of [6].

Lemma 3.1

Let

$$\begin{aligned} v:= u - \underline{u} + t d - N d^{2}. \end{aligned}$$

Then, there exist positive constants t, δ sufficiently small and N sufficiently large such that

$$\begin{aligned} L v \leq - \epsilon _{0} \biggl(1 + \sum _{i} F^{ii} \biggr) \end{aligned}$$

(3.5)

and

$$\begin{aligned} v \geq 0 \end{aligned}$$

(3.6)

in $M_{\delta}$ for some uniform constat $\epsilon _{0} > 0$.

Proof

First, there exists a positive constant $\theta _{0}$ such that $\underline{u} - \theta _{0} \rho ^{2}$ is also admissible. By (2.4) and the concavity of F, we have

$$\begin{aligned} F^{ij} \nabla _{ij} (u - \underline{u}) &\leq - \theta _{0} \sum_{i} F^{ii} - F \bigl(\nabla ^{2} \underline{u} - \theta _{0} g + \chi \bigr) + F \bigl( \nabla ^{2} u + \chi \bigr) \\ &= - \theta _{0} \sum_{i} F^{ii} - F \bigl(\nabla ^{2} \underline{u} - \theta _{0} g + \chi \bigr) + \psi + \beta _{ \epsilon} \\ &\leq - \theta _{0} \sum_{i} F^{ii} + C, \end{aligned}$$

where the constant C depends on $\|u\|_{C^{1} (\overline{M})}$ and the constant $c_{2}$ in (2.4). Recall that $f_{i} = \frac{\partial f}{\partial \lambda _{i}}$, where $\lambda = \lambda (\nabla ^{2} u + \chi )$ for $i = 1, \ldots, n$. Without loss of generality, we may assume $f_{n} = \min_{i} \{f_{i}\}$. Next, since $\nabla d \equiv 1$ on the boundary, we have

$$\begin{aligned} F^{ij} \nabla _{ij} \bigl(d^{2}\bigr) \geq f_{n} + 2 d F^{ij} \nabla _{ij} d \geq f_{n} - C \delta \sum_{i} F^{ii} \quad\text{in } M_{\delta}, \end{aligned}$$

for δ sufficiently small. It follows that

$$\begin{aligned} L v + \beta _{\epsilon }v \leq - \theta _{0} \sum _{i} F^{ii} - N f_{n} + C (\delta N + t) \biggl(1 + \sum_{i} F^{ii} \biggr) \end{aligned}$$

in $M_{\delta}$. By (1.7), we have

$$\begin{aligned} \frac{\theta _{0}}{4} \sum_{i} F^{ii} + N f_{n} \geq \frac{n \theta _{0}}{4} (N f_{1} \cdot \cdots \cdot f_{n})^{1/n} \geq \frac{n c_{0} \theta _{0} N^{1/n}}{4}. \end{aligned}$$

Thus, we can choose N sufficiently large and t, δ sufficiently small such that

$$\begin{aligned} L v + \beta _{\epsilon }v \leq - \frac{\theta _{0}}{2} \sum _{i} F^{ii} - c_{3} N^{1/n}. \end{aligned}$$

We may further make δ sufficiently small such that $v \geq 0$ in $M_{\delta}$. Since $\beta '_{\epsilon }\geq 0$ we obtain (3.5). □

From formula $(4.7)$ in [7] and differentiating the equation (2.1), we have

$$\begin{aligned} \bigl\vert L \nabla _{k} (u - \phi ) \bigr\vert \leq C \biggl(1 + \sum_{i} F^{ii} + \sum _{i} f_{i} \vert \lambda _{i} \vert \biggr), \quad\text{for } 1 \leq k \leq n, \end{aligned}$$

(3.7)

where C is a positive constant depending only on $\|u\|_{C^{1} (\overline{M})}$, $\|\phi \|_{C^{3} (\overline{M})}$ and $\|\psi \|_{C^{1}}$. Similar to formula $(4.9)$ in [7], by Lemma 2.2, we find that

$$\begin{aligned} \begin{aligned} L \biggl(\sum _{\beta \leq n-1} \bigl(\nabla _{\beta }(u - \phi ) \bigr)^{2} \biggr) \geq {}& \sum_{\beta \leq n-1} F^{ij} U_{\beta i} U_{\beta j} - C \biggl(1 + \sum _{i} F^{ii} + \sum_{i} f_{i} \vert \lambda _{i} \vert \biggr) \\ & {}+ \beta '_{\epsilon }\sum_{\beta \leq n-1} \bigl(\nabla _{\beta }(u - \phi )\bigr)^{2} \\ \geq {}& \frac{1}{2} \sum_{i \neq r} f_{i} \lambda _{i}^{2} - C \biggl(1 + \sum _{i} F^{ii} + \sum _{i} f_{i} \vert \lambda _{i} \vert \biggr) \end{aligned} \end{aligned}$$

(3.8)

for some index $1 \leq r \leq n$. Let

$$\begin{aligned} \varPsi = A_{1} v + A_{2} \rho ^{2} - A_{3} \sum_{\beta \leq n-1} \bigl\vert \nabla _{\beta }(u - \phi ) \bigr\vert ^{2} \end{aligned}$$

(3.9)

as in [7]. Combining (2.11), (3.7), and (3.8), we can choose $A_{1} \gg A_{2} \gg A_{3} \gg 1$ such that

$$\begin{aligned} L \bigl(\varPsi \pm \nabla _{\alpha }(u - \phi )\bigr) \leq 0\quad \text{in } M_{ \delta } \end{aligned}$$

and

$$\begin{aligned} \varPsi \pm \nabla _{\alpha }(u - \phi ) \geq 0 \quad\text{on } \partial M_{ \delta } \end{aligned}$$

for any index $1 \leq \alpha \leq n-1$. Then, by the maximum principle, we have

$$\begin{aligned} \varPsi \pm \nabla _{\alpha }(u - \phi ) \geq 0 \quad\text{on } \overline{M}_{\delta}. \end{aligned}$$

Since

$$\begin{aligned} \varPsi \pm \nabla _{\alpha }(u - \phi ) = 0 \quad\text{at } x_{0} \end{aligned}$$

we obtain (3.4).

Since $\Delta u + \mathrm{tr} (\chi ) > 0$ in M, it suffices to establish the upper bound

$$\begin{aligned} \nabla _{nn} u (x_{0}) \leq C. \end{aligned}$$

(3.10)

We first suppose ϕ is admissible in M. As in [7], following an idea of Trudinger [23] we prove that there are uniform constants $c_{0}, R_{0}$ such that for all $R > R_{0}$, $(\lambda ' [\{U_{\alpha \beta} (x_{0})\}], R) \in \Gamma$ and

$$\begin{aligned} f \bigl(\lambda ' \bigl[\bigl\{ U_{\alpha \beta} (x_{0})\bigr\} \bigr], R\bigr) \geq \psi [u] (x_{0}) + \beta _{\epsilon }(x_{0}) + c_{0}, \end{aligned}$$

(3.11)

which implies (3.10) by Lemma 1.2 in [3], where $\lambda ' [\{U_{\alpha \beta}\}] = (\lambda '_{1}, \ldots, \lambda '_{n-1})$ denote the eigenvalues of the $(n-1) \times (n-1)$ matrix $\{U_{\alpha \beta}\}$ ($1 \leq \alpha, \beta \leq n-1$). Denote

$$\begin{aligned} \tilde{m}_{R}:= \min_{x \in \partial M} f \bigl(\lambda ' \bigl[\bigl\{ U_{\alpha \beta} (x)\bigr\} \bigr], R\bigr). \end{aligned}$$

Suppose $\tilde{m}_{R}$ is achieved at a point $x_{0} \in \partial M$. Choose local orthonormal frames ${e_{1}},{e_{2}}, \ldots,{e_{n}}$ around $x_{0}$ as before and assume $\nabla _{n n} u (x_{0}) \geq \nabla _{n n} \phi (x_{0})$. Let $\Phi _{ij}:= \nabla _{ij} \phi + \chi _{ij}$ and

$$\begin{aligned} \tilde{c}_{R}:= \min_{x \in \overline{M}_{\delta _{0}}} f \bigl(\lambda ' \bigl[ \bigl\{ \Phi _{\alpha \beta} (x) \bigr\} \bigr], R\bigr) \end{aligned}$$

for $\delta _{0}$ sufficiently small such that $e_{1}, \ldots, e_{n}$ are well defined in $\overline{M}_{\delta _{0}}$. By (1.9) and the fact that ϕ is admissible, we see that

$$\begin{aligned} \lim_{R \rightarrow + \infty} \tilde{c}_{R} = + \infty. \end{aligned}$$

(3.12)

We wish to show $\tilde{m}_{R} \rightarrow + \infty$ as $R \rightarrow + \infty$. Without loss of generality we assume $\tilde{m}_{R} < \tilde{c}_{R}/2$ (otherwise we are done by (3.12)).

For a symmetric $(n - 1) \times (n - 1)$ matrix $\{r_{\alpha {\beta}}\}$ such that $(\lambda ' [\{r_{\alpha \beta}\}], R) \in \Gamma$, define

$$\begin{aligned} \tilde{F}[r_{\alpha \beta}]:= f \bigl(\lambda ' \bigl[ \{r_{\alpha \beta}\}\bigr], R\bigr) . \end{aligned}$$

Note that F̃ is concave by (1.5). Let

$$\begin{aligned} \tilde{F}^{\alpha {\beta}}_{0} = \frac{\partial \tilde{F}}{\partial r_{\alpha \beta}} \bigl[U_{\alpha \beta} (x_{0})\bigr]. \end{aligned}$$

We find

$$\begin{aligned} \tilde{F}^{\alpha {\beta}}_{0} U_{\alpha {\beta}} - \tilde{F}^{ \alpha {\beta}}_{0} U_{\alpha {\beta}} (x_{0}) \geq \tilde{F}[U_{ \alpha {\beta}}] - \tilde{m}_{R} \geq 0\quad \text{on $\partial M$ near } x_{0}. \end{aligned}$$

(3.13)

By (3.2) we have on ∂M near $x_{0}$,

$$\begin{aligned} U_{\alpha {\beta}} = \Phi _{\alpha {\beta}} - \nabla _{n} (u - \phi ) \sigma _{\alpha {\beta}}, \end{aligned}$$

(3.14)

where $\sigma _{\alpha {\beta}} = \langle \nabla _{\alpha} e_{\beta}, e_{n} \rangle$; note that $\sigma _{\alpha \beta} = \varPi (e_{\alpha}, e_{\beta})$ on ∂M. Define

$$\begin{aligned} Q = - \eta \nabla _{n} (u - \phi ) + \tilde{F}^{\alpha {\beta}}_{0} \Phi _{\alpha {\beta}} - \tilde{F}^{\alpha {\beta}}_{0} U_{\alpha { \beta}} (x_{0}), \end{aligned}$$

where $\eta = \tilde{F}^{\alpha {\beta}}_{0} \sigma _{\alpha {\beta}}$. From (3.13) and (3.14) we see that $Q (x_{0}) = 0$ and $Q \geq 0$ on ∂M near $x_{0}$. Furthermore, we have

$$\begin{aligned} \begin{aligned} Q &\geq - \eta \nabla _{n} (u - \phi ) + \tilde{F}[\Phi _{\alpha \beta}] - \tilde{F}\bigl[U_{\alpha \beta} (x_{0})\bigr] \\ &\geq - \eta \nabla _{n} (u - \phi ) + \tilde{c}_{R} - \tilde{m}_{R} \\ &\geq - \eta \nabla _{n} (u - \phi ) + \frac{\tilde{c}_{R}}{2} \quad\text{in } \overline{M}_{\delta _{0}}. \end{aligned} \end{aligned}$$

(3.15)

By (3.7) and (3.15), we have

$$\begin{aligned} \begin{aligned} L Q &\leq C \mathcal{F} \Bigl(1 + \sum F^{ii} + \sum f_{i} \vert \lambda _{i} \vert \Bigr) - \frac{\tilde{c}_{R}}{2} \beta '_{\epsilon} \\ &\leq C \mathcal{F} \Bigl(1 + \sum F^{ii} + \sum f_{i} \vert \lambda _{i} \vert \Bigr), \end{aligned} \end{aligned}$$

(3.16)

where

$$\begin{aligned} \mathcal{F}:= \sum_{\alpha \leq n-1} \tilde{F}^{\alpha \alpha}_{0}. \end{aligned}$$

Recall that Ψ is defined in (3.9). Choosing $A_{1} \gg A_{2} \gg A_{3} \gg 1$ as before, we derive

$$\begin{aligned} \textstyle\begin{cases} L (\mathcal{F} \varPsi + Q) \leq 0 &\text{in $M_{\delta}$}, \\ \mathcal{F} \varPsi + Q \geq 0 &\text{on $\partial M_{\delta}$}. \end{cases}\displaystyle \end{aligned}$$

(3.17)

By the maximum principle, $\mathcal{F} \varPsi + Q \geq 0$ in $M_{\delta}$. Thus,

$$\begin{aligned} \nabla _{n} Q (x_{0}) \geq - \mathcal{F} \nabla _{n} \varPsi (x_{0}) \geq -C \mathcal{F}. \end{aligned}$$

(3.18)

By (1.11), we see, at $x_{0}$, $(\lambda ' (\sigma _{\alpha \beta}), \sqrt{R_{0}}) \in \Gamma$ for some $R_{0}$ sufficiently large. Thus, there exists a uniform constant $\epsilon _{0} > 0$ such that $(\lambda '({\sigma _{\alpha \beta }} - {\epsilon _{0}}{\delta _{\alpha \beta }}),\sqrt {R} ) \in \Gamma$ for all $R \geq R_{0}$. From the concavity of F̃ and (1.10) we find, at $x_{0}$,

$$\begin{aligned} \begin{aligned} \sqrt{R} \tilde{F}_{0}^{\alpha \beta} \sigma _{\alpha \beta}& = \sqrt{R} \tilde{F}_{0}^{\alpha \beta} (\sigma _{\alpha \beta} - \epsilon _{0} \delta _{\alpha \beta}) - \tilde{F}_{0}^{\alpha \beta} U_{ \alpha \beta} (x_{0}) + \tilde{F}_{0}^{\alpha \beta} U_{\alpha \beta} (x_{0}) + \sqrt{R} \epsilon _{0} \mathcal{F} \\ &\geq \tilde{F}\bigl[\sqrt{R} (\sigma _{\alpha \beta} - \epsilon _{0} \delta _{\alpha \beta})\bigr] - \tilde{F}\bigl[U_{\alpha \beta} (x_{0}) \bigr] + \tilde{F}_{0}^{\alpha \beta} U_{\alpha \beta} (x_{0}) + \sqrt{R} \epsilon _{0} \mathcal{F} \\ &\geq f \bigl(\sqrt{R} \bigl(\lambda ' (\sigma _{\alpha \beta} - \epsilon _{0} \delta _{\alpha \beta}), \sqrt{R}\bigr)\bigr) - \tilde{F} \bigl[U_{\alpha \beta} (x_{0})\bigr] + \sqrt{R} \epsilon _{0} \mathcal{F} - C \mathcal{F} \\ &\geq f \bigl(\sqrt{R} \bigl(\lambda ' (\sigma _{\alpha \beta} - \epsilon _{0} \delta _{\alpha \beta}), \sqrt{R_{0}}\bigr) \bigr) - \tilde{m}_{R} + \frac{\sqrt{R}}{2} \epsilon _{0} \mathcal{F} \\ &\geq C (R) - \tilde{m}_{R} + \frac{\sqrt{R}}{2} \epsilon _{0} \mathcal{F}, \end{aligned} \end{aligned}$$

provided R is sufficiently large, where $\lim_{R \rightarrow + \infty} C (R) = + \infty$. We may assume $\tilde{m}_{R} \leq C(R)$ for otherwise we are done. It follows that, at $x_{0}$,

$$\begin{aligned} \eta = \tilde{F}_{0}^{\alpha \beta} \sigma _{\alpha \beta} \geq \frac{\epsilon _{0}}{2} \mathcal{F}. \end{aligned}$$

(3.19)

Combining (3.18) and (3.19) we obtain

$$\begin{aligned} \nabla _{nn} u (x_{0}) \leq C. \end{aligned}$$

We have established an a priori upper bound for all eigenvalues of $\{U_{ij} (x_{0})\}$. Consequently, $\lambda [\{U_{ij} (x_{0})\}]$ is contained in a compact subset of Γ by (1.6), and therefore

$$\begin{aligned} \lim_{R \rightarrow + \infty} \tilde{m}_{R} = + \infty \end{aligned}$$

by (1.9). This proves (3.11) and the proof of (3.10) is complete.

We now consider the case $\chi \equiv 0$ and $\varphi \equiv 0$ on ∂M to prove Theorem 1.3. By [3] we have

$$\begin{aligned} \Delta u \geq \delta _{0} > 0 \end{aligned}$$

(3.20)

for some positive constant $\delta _{0}$ depending only on $\psi _{0} = \inf \psi > 0$. Let $u_{0}$ be defined by the equation

$$\begin{aligned} \Delta u_{0} = \delta _{0} \quad\text{in } M \end{aligned}$$

with $u_{0} = 0$ on ∂M. By the maximum principle and Hopf’s lemma, we see $u_{0} < 0$ in M and $(u_{0})_{\nu }< 0$ on ∂M, where ν is the unit interior normal to ∂M. Since ∂M is compact, there exists a uniform constant $\gamma _{1} > 0$ such that $(u_{0})_{\nu }\leq - \gamma _{1}$ on ∂M. By (3.20) and the maximum principle, we find that

$$\begin{aligned} u \leq u_{0} \quad\text{in } M \text{ and } u = u_{0} = 0 \text{ on } \partial M. \end{aligned}$$

It follows that

$$\begin{aligned} \nabla _{n} u (x_{0}) \leq \nabla _{n} (u_{0}) (x_{0}) \leq - \gamma _{1}. \end{aligned}$$

(3.21)

We find, at $x_{0} \in \partial M$,

$$\begin{aligned} \nabla _{\alpha \beta} u = - \nabla _{n} u \varPi (e_{\alpha}, e_{ \beta}), \quad\text{for } 1 \leq \alpha, \beta \leq n-1. \end{aligned}$$

Since $\underline{u} = 0$, we have, at $x_{0}$,

$$\begin{aligned} \nabla _{\alpha \beta} \underline{u} = - \nabla _{n} \underline{u} \varPi (e_{\alpha}, e_{\beta}), \quad\text{for } 1 \leq \alpha, \beta \leq n-1. \end{aligned}$$

Therefore,

$$\begin{aligned} \nabla _{\alpha \beta} u = \frac{\nabla _{n} u}{\nabla _{n} \underline{u}} \nabla _{\alpha \beta} \underline{u}. \end{aligned}$$

By (3.21), we then find the eigenvalues of the $(n-1)\times (n-1)$ matrix $\{\nabla _{\alpha \beta} u (x_{0})\}_{\alpha, \beta \leq n-1}$ $\lambda ' \{\nabla _{\alpha \beta} u (x_{0})\}$ belong to a compact subset of $\Gamma '$, where $\Gamma '$ denotes the projection of Γ to $\lambda ' = (\lambda _{1}, \ldots, \lambda _{n-1})$ of Γ. By (1.9) and Lemma 1.2 of [3], we can prove (3.10).