The obstacle problem for conformal metrics on compact Riemannian manifolds

Abstract

We prove a priori estimates up to their second order derivatives for solutions to the obstacle problem of curvature equations on Riemannian manifolds $(M^{n}, g)$ arising from conformal deformation. With the a priori estimates the existence of a $C^{1,1}$ solution to the obstacle problem with Dirichlet boundary value is obtained by approximation.

1 Introduction

Let $(M^{n}, g)$ be a compact Riemannian manifold of dimension $n \geq3$ with smooth boundary ∂M, $\bar{M}:= M \cup \partial M$. In conformal geometry, it is interesting to find a complete metric $\tilde{g} \in[g]$, the conformal class of g, with which the manifold has prescribed curvature. In general, such conformal deformation can be interpreted by certain partial differential equations. See [8, 13, 22, 25, 26] for more details.

In [8], Guan studied the existence of a complete conformal metric g̃ of negative Ricci curvature on M satisfying

$$ f \bigl( - \lambda\bigl(\tilde{g}^{-1} \mathrm{Ric}_{\tilde{g}}\bigr)\bigr) = \psi\quad \mbox{in } M, $$

(1.1)

where $\mathrm{Ric}_{\tilde{g}}$ is the Ricci tensor of g̃, and $\lambda(\tilde{g}^{-1} \mathrm{Ric}_{\tilde{g}}) = (\lambda_{1}, \ldots , \lambda_{n})$ are the eigenvalues of $\tilde{g}^{-1} \mathrm {Ric}_{\tilde{g}}$. The transformation formula for the Ricci tensor under conformal deformation $\tilde{g} = e^{2 u} g$ is given by

$$\frac{1}{n - 2} \mathrm{Ric}_{\tilde{g}} = \frac{1}{n - 2} \mathrm{Ric}_{g} - \nabla^{2} u - \biggl(\frac {\Delta u}{n - 2} + \vert \nabla u \vert ^{2} \biggr) g + du \otimes du, $$

where ∇u, $\nabla^{2} u$, and Δu denote the gradient, Hessian, and Laplacian of u with respect to the metric g, respectively. When f is homogenous of degree one, it is easy to verify that equation (1.1) is equivalent to the following form:

$$ f \biggl(\lambda \biggl(g^{-1} \biggl[ \nabla^{2} u + \frac{\Delta u}{n - 2} g + \vert \nabla u \vert ^{2} g - du \otimes du - \frac{\mathrm{Ric}_{g}}{n - 2} \biggr] \biggr) \biggr) = \frac{\psi(x)}{n - 2} e^{2u}. $$

(1.2)

In this paper, we study the obstacle problem of equation (1.2). More generally, let

$$T [u] := \nabla^{2} u + s \, du \otimes du + \biggl( \gamma\Delta u - \frac{t}{2} \vert \nabla u \vert ^{2} \biggr) g + \chi, $$

where χ is a smooth $(0,2)$ tensor, $\gamma> 0$ is a constant, and $s, t \in\mathbb{R}$. We consider the following equation:

$$ \max \bigl\{ u -h, - \bigl(f \bigl(\lambda\bigl(g^{-1} T [u]\bigr)\bigr) - \psi[u]\bigr) \bigr\} = 0 \quad\mbox{in } M $$

(1.3)

with the Dirichlet boundary condition

$$ u = \varphi\quad\mbox{on } \partial M, $$

(1.4)

where $h \in C^{3} (\bar{M})$, $\varphi\in C^{4} (\partial M)$, $h > \varphi$ on ∂M, $\psi[u] = \psi(x, u)$ is a positive function in $C^{3} (\bar{M} \times\mathbb{R})$.

Equations as (1.1) and (1.3) are the Hessian equations, which were well studied by many authors such as [2, 7, 9–12, 23, 24]. Generally, $f \in C^{2} (\Gamma) \cap C^{0} (\bar{\Gamma})$ is a symmetric function of $\lambda\in\mathbb{R}^{n}$, defined in an open, convex, and symmetric cone $\Gamma\subsetneqq\mathbb{R}^{n}$, with vertex at the origin, which contains the positive cone: $\Gamma_{n}^{+} : =\{\lambda\in\mathbb{R}^{n}: \mbox{each component } \lambda_{i}>0\}$ and satisfies the following fundamental structure conditions:

$$ f_{i} \equiv\frac{\partial f}{\partial\lambda_{i}} > 0\quad \mbox{in } \Gamma, 1 \leq i \leq n, $$

(1.5)

$$ \mbox{$f$ is a concave function}, $$

(1.6)

and

$$ f > 0 \quad\mbox{in } \Gamma, \qquad f = 0 \quad\mbox{on } \partial\Gamma. $$

(1.7)

Here, for convenience, we also assume that

$$ \mbox{$f$ is homogeneous of degree one}. $$

(1.8)

We observe that by the concavity and homogeneity of f,

$$ \sum f_{i} (\lambda) = f (\lambda) + \sum f_{i} (\lambda) (1 - \lambda _{i}) \geq f (1, \ldots, 1) > 0 \quad\mbox{in } \Gamma. $$

(1.9)

Important classes of f are the elementary symmetric functions and their quotients, i.e.,

$$f (\lambda) = (\sigma_{k})^{\frac{1}{k}} (\lambda) : = \biggl(\sum _{1 \leq i_{1} < \cdots< i_{k} \leq n} \lambda_{i_{1}} \cdots \lambda_{i_{k}} \biggr)^{\frac {1}{k}}, \quad 1 \leq k \leq n, $$

and

$$f (\lambda) = \biggl(\frac{\sigma_{k}}{\sigma_{l}} \biggr)^{\frac {1}{k-l}}, \quad0 \leq l < k \leq n. $$

Let F be defined by $F (r) = f (\lambda(r))$ for $r = \{r_{ij}\} \in\mathcal{S}^{n \times n}$ with $\lambda(r) \in\Gamma$, where $\mathcal{S}^{n \times n}$ is the set of $n \times n$ symmetric matrices. It is shown in [2] that (1.5) implies F is an elliptic operator and (1.6) ensures that F is concave.

A function $u \in C^{2} (M)$ is called admissible at $x \in M$ if $\lambda(g^{-1} T [u]) (x) \in\Gamma$, and we call it admissible in M when it is admissible at each x in M. In this paper, we prove the existence of an admissible viscosity solution of (1.3) and (1.4) in $C^{1, 1} (\bar{M})$ (see [1, 3] for the definition of viscosity solutions).

Many authors have studied various obstacle problems. In [6], Gerhardt considered a hypersurface bounded from below by an obstacle with prescribed mean curvature in $\mathbb{R}^{n}$. Lee [17] considered the obstacle problem for the Monge–Ampère equation (i.e., $f = (\sigma_{n})^{\frac{1}{n}}$) for the case that $T [u] = D^{2} u$, $\psi\equiv1$, and $\varphi\equiv0$, and proved the $C^{1,1}$ regularity of the viscosity solution in a strictly convex domain in $\mathbb{R}^{n}$. Xiong and Bao [27] extended the work of Lee to a nonconvex domain in $\mathbb{R}^{n}$ with general ψ and φ under additional assumptions. Bao, Dong, and Jiao treated a class of obstacle problems in [1] assuming that $T[u] = \nabla^{2} u + A(x, u, \nabla u)$, under a certain technical assumption. Because of the term $\gamma\Delta u$ ($\gamma> 0$), here we only need a minimal amount of assumptions. For other works, see [4, 14, 15, 18–21].

Our main result is the following theorem.

Theorem 1.1

Assume that (1.5)–(1.8) and either the following condition

$$ \lim_{z \rightarrow+ \infty} \psi(x, z) = +\infty, \quad\forall x \in\bar{M}, $$

(1.10)

or

$$ \frac{2s - n t}{1 + n\gamma} < 2 \lambda_{1} $$

(1.11)

hold, where $\lambda_{1}$ is the first eigenvalue of the problem

$$ \textstyle\begin{cases} \Delta u + \lambda(\operatorname{tr} \chi)^{+} u = 0 & \textit{on } \bar{M},\\ u = 0 & \textit{on } \partial M \end{cases} $$

(1.12)

($\lambda_{1} = + \infty$ if $\operatorname{tr} \chi\leq0$). Then there exists a viscosity solution $u \in C^{1,1} (\bar{M})$ to (1.3) and (1.4), if there exists a subsolution $\underline{u} \in C^{0} (\bar{M}) \cap C^{1} (\bar{M} _{\delta})$ for some $\delta> 0$ such that

$$ \textstyle\begin{cases} f (\lambda(g^{-1} T [\underline{u}])) \geq\psi[\underline{u}], & \textit{in } M,\\ \underline{u} = \varphi, & \textit{on } \partial M, \\ \underline{u} \leq h, & \textit{in } M, \end{cases} $$

(1.13)

where $M_{\delta}= \{x \in M: \operatorname{dist} (x, \partial M) \leq \delta\}$. Moreover, we have that $u \in C^{3, \alpha} (E)$ for any $\alpha\in (0, 1)$, and $f(\lambda(g^{-1} T [u])) = \psi[u]$ in E, where $E:= \{x \in M: u (x) < h (x)\}$.

Remark 1.2

(1.10), as well as (1.11), is used in Lemma 3.2 to derive an upper bound for u. Assumption (1.13) is just applied to derive a lower bound for u on M and $\nabla_{\nu} u$ on ∂M, where ν is the interior unit normal to ∂M.

Remark 1.3

We can construct some subsolutions of (1.2) satisfying (1.13) as in [15] following ideas from [2] and [7] since

$$\vert \nabla u \vert ^{2} g - du \otimes du $$

is positive definite and that we can obtain a priori upper bound of any admissible function (Lemma 3.2) under additional conditions that there exists a sufficiently large number $R > 0$ such that at each point $x \in\partial M$,

$$ (\kappa_{1}, \ldots, \kappa_{n-1}, R) \in \Gamma, $$

(1.14)

where $\kappa_{1}, \ldots, \kappa_{n-1}$ are the principal curvatures of ∂M with respect to the interior normal, and that for every $C > 0$ and every compact set K in Γ there is a number $R = R (C, K)$ such that

$$ f (R \lambda) \geq C \quad\mbox{for all } \lambda\in K. $$

(1.15)

We use a penalization technique to prove the existence of viscosity solutions to (1.3) and (1.4). We shall consider the following singular perturbation problem:

$$ \textstyle\begin{cases} f (\lambda( g^{-1} T [u] )) = \psi[u] + \beta_{\varepsilon}(u-h) & \mbox{in } M,\\ u = \varphi& \mbox{on } \partial M, \end{cases} $$

(1.16)

where the penalty function $\beta_{\varepsilon}\in C^{2} (\mathbb{R})$ satisfies

$$ \begin{aligned} & \beta_{\varepsilon}, \beta'_{\varepsilon}, \beta''_{\varepsilon}\geq 0 \quad\mbox{on } \mathbb{R}, \beta_{\varepsilon}(z) = 0, \mbox{ whenever } z \leq0; \\ & \beta_{\varepsilon}(z) \rightarrow\infty\quad\mbox{as } \varepsilon \rightarrow0^{+}, \mbox{ whenever } z > 0. \end{aligned} $$

(1.17)

An example given in [27] is

$$ \beta_{\varepsilon}(z) = \textstyle\begin{cases} 0, & z \leq0,\\ z^{3} / \varepsilon, & z > 0, \end{cases} $$

(1.18)

for $\varepsilon\in(0, 1)$. Observe that $\underline{u}$ is also a subsolution to (1.16).

Let

$$\mathscr{U} = \bigl\{ u_{\varepsilon} | u_{\varepsilon} \in C^{4}( \bar{M}) \mbox{ is an admissible solution of } (1.16) \mbox{ with } u_{\varepsilon} \geq\underline{u} \mbox{ on } \bar{M} \bigr\} . $$

We aim to derive the uniform bound

$$ \vert u_{\varepsilon} \vert _{C^{2} (\bar{M})} \leq C $$

(1.19)

for $u_{\varepsilon}\in\mathscr{U}$, where C is independent of ε. After establishing (1.19), the equation (1.16) becomes uniformly elliptic by (1.7). By Evans–Krylov [5], [16] theorem, we can derive the $C^{2, \alpha}$ estimates (which may depend on ε) of $u_{\varepsilon}$. Higher estimates can be derived by Schauder theory. Following the proof as in [8] or [1], we can prove there exists an admissible solution $u_{\varepsilon}$ to (1.16). Then we can conclude by (1.19) that there exists a viscosity solution $u \in C^{1, 1} (\bar{M})$ to (1.3) and (1.4), see [1, 27].

Thus, our main work is focused on the a priori estimates for admissible solutions up to their second order derivatives. In Sect. 2, we achieve the estimates for second order derivatives. Finally, we end this paper with gradient and $C^{0}$ estimates in Sect. 3.

2 Estimates for second order derivatives

In this section, we prove a priori estimates of second order derivatives for admissible solutions. From now on, we drop the subscript ε when there is no possible confusion.

Theorem 2.1

Assume that f satisfies (1.5)–(1.8) and $u \in C^{4} (\bar{M})$ is an admissible solution to (1.16). Then

$$ \sup_{M} \bigl\vert \nabla^{2} u \bigr\vert \leq C \Bigl(1 + \sup_{\partial M} \bigl\vert \nabla^{2} u \bigr\vert \Bigr), $$

(2.1)

where C depends on $|u|_{C^{1}(\bar{M})}$ and other known data.

Proof

Set

$$W(x) = \max_{ \xi\in T_{x} M, \vert \xi \vert = 1} \bigl( \nabla_{\xi\xi} u + s \vert \nabla_{\xi} u \vert ^{2} \bigr) e^{\phi}, \quad x \in\bar{M}, $$

where ϕ is a function to be determined. Assume that W is achieved at an interior point $x_{0} \in M$ and a unit direction $\xi\in T_{x_{0}} M$. Choose a smooth orthonormal local frame $e_{1}, \ldots, e_{n}$ about $x_{0}$ such that $\xi= e_{1}$, $\nabla_{i} e_{j} (x_{0}) = 0$ and that $T_{ij} (x_{0})$ is diagonal. We write $G = \nabla_{11} u + s |\nabla _{1} u|^{2}$. Assume $G (x_{0}) > 0$ (otherwise we are done).

At the point $x_{0}$, where the function $\log G + \phi$ (defined near $x_{0}$) attains its maximum, we have

$$ \frac{\nabla_{i} G}{G} + \nabla_{i} \phi= 0, \quad i = 1, \ldots, n, $$

(2.2)

and

$$ \frac{\nabla_{ii} G}{G} - \biggl(\frac{\nabla_{i} G}{G} \biggr)^{2} + \nabla_{ii} \phi\leq0. $$

(2.3)

By (2.3) we have

$$ F^{ii} \bigl(\nabla_{ii} G + G \nabla_{ii} \phi- G \vert \nabla_{i} \phi \vert ^{2} \bigr) \leq0 $$

(2.4)

and

$$ \Delta G + G \Delta\phi- G \vert \nabla\phi \vert ^{2} \leq0. $$

(2.5)

Since $\gamma> 0$, we obtain

$$ F^{ii} \bigl(\nabla_{ii} G + \gamma\Delta G + G \nabla_{ii} \phi+ \gamma G \Delta\phi- G \vert \nabla_{i} \phi \vert ^{2} - \gamma G \vert \nabla\phi \vert ^{2} \bigr) \leq0. $$

(2.6)

By calculation, we get

$$ \nabla_{i} G = \nabla_{i11} u + 2 s \nabla_{1} u \nabla_{i1} u, $$

(2.7)

and

$$ \nabla_{ii} G = \nabla_{ii11} u + 2 s \bigl( \vert \nabla_{i1} u \vert ^{2} + \nabla_{1} u \nabla_{ii1} u\bigr). $$

(2.8)

Recall the formula for interchanging order of covariant derivatives

$$ \nabla_{ijk} v - \nabla_{kij} v = R^{l}_{kij} \nabla_{l} v, $$

(2.9)

and

$$ \begin{aligned}[b] \nabla_{ijkl} v - \nabla_{klij} v = {}&R^{m}_{ljk} \nabla_{im} v + \nabla_{i} R^{m}_{ljk} \nabla_{m} v + R^{m}_{lik} \nabla_{jm} v \\ &{}+ R^{m}_{jik} \nabla_{lm} v + R^{m}_{jil} \nabla_{km} v + \nabla_{k} R^{m}_{jil} \nabla_{m} v. \end{aligned} $$

(2.10)

It follows from (2.10)

$$ \nabla_{ii} G \geq\nabla_{11ii} u + 2 s \bigl( \vert \nabla_{i1} u \vert ^{2} + \nabla _{1} u \nabla_{1ii} u\bigr) - C ( 1 + G), $$

(2.11)

and

$$ \begin{aligned}[b] \nabla_{ii} G + \gamma \Delta G \geq{}& \nabla_{11ii} u + 2 s \bigl( \vert \nabla_{i1} u \vert ^{2} + \nabla_{1} u \nabla_{1ii} u\bigr) + \gamma\nabla _{11} (\Delta u) \\ & {} + 2 s \gamma \bigl( \vert \nabla_{i1} u \vert ^{2} + \nabla_{1} u \nabla_{1} (\Delta u) \bigr) - C (1 + G). \end{aligned} $$

(2.12)

Differentiating equation (1.16) once at $x_{0}$, we obtain for $1 \leq k \leq n$,

$$ \nabla_{k} F = F^{ii} \nabla_{k} T_{ii} = \psi_{x_{k}} + \psi_{z} \nabla_{k} u + \nabla_{k} \beta_{\varepsilon}(u - h). $$

(2.13)

It is easy to see that

$$ \begin{aligned}[b] F^{ii} \nabla_{1} (\nabla_{ii} u + \gamma\Delta u) &= F^{ii} \nabla_{1} \biggl(T_{ii} [u] - s \vert \nabla_{i} u \vert ^{2} + \frac{t}{2} \vert \nabla u \vert ^{2} - \chi_{ii} \biggr) \\ &\geq\nabla_{1} F - 2s F^{ii} \nabla_{i} u \nabla_{1i} u + t \nabla_{k} u \nabla_{1k} u\sum _{i} F^{ii} - \sum _{i} F^{ii} \end{aligned} $$

(2.14)

and that

$$ \begin{aligned}[b] F^{ii} \nabla_{11} (\nabla_{ii} u + \gamma\Delta u) = {}& F^{ii} \nabla_{11} \biggl(T_{ii} [u] - s \vert \nabla_{i} u \vert ^{2} + \frac {t}{2} \vert \nabla u \vert ^{2} - \chi_{ii} \biggr) \\ \geq{}& F^{ii} \nabla_{11} T_{ii} [u] - 2s F^{ii} \bigl( \nabla_{i} u \nabla_{11i} u + \vert \nabla_{1i} u \vert ^{2} \bigr) \\ &{} + t \sum_{k} \bigl(\nabla_{k} u \nabla_{11k} u + \vert \nabla_{1k} u \vert ^{2} \bigr) \sum_{i} F^{ii} - C \sum _{i} F^{ii}. \end{aligned} $$

(2.15)

With (2.9) we see

$$ \begin{aligned}[b] 2s \nabla_{i} u \nabla_{11i} u & \leq2s \nabla_{i} u (\nabla_{i} G - 2s \nabla_{1} u \nabla_{i1} u) + C \\ & \leq- 4s^{2} \nabla_{i} u \nabla_{1} u \nabla_{i1} u + C \bigl( 1 + G \vert \nabla\phi \vert \bigr), \end{aligned} $$

(2.16)

and similarly

$$ t \nabla_{k} u \nabla_{11k} u \geq- 2st \nabla_{k} u \nabla_{1} u \nabla _{k1} u - C \bigl( 1 + G \vert \nabla\phi \vert \bigr). $$

(2.17)

With (2.12), (2.14)–(2.17), and the concavity of F, we derive

$$ \begin{aligned} F^{ii} ( \nabla_{ii} G + \gamma\Delta G) \geq{}& \nabla_{11} F + 2s \nabla_{1} u \nabla_{1} F + (2s \gamma+ t) \sum _{k} \vert \nabla_{1k} u \vert ^{2} \sum F^{ii} \\ & {} - C \biggl(G + G \vert \nabla\phi \vert + \sum _{j,k} \vert \nabla_{jk} u \vert \biggr) \\ \geq{} & \nabla_{11} F + 2s \nabla_{1} u \nabla_{1} F - C \bigl(G^{2} + G \vert \nabla\phi \vert \bigr). \end{aligned} $$

(2.18)

By (1.9) and $\beta_{\varepsilon}'' > 0$ it follows from (2.6) and (2.18) that

$$ \begin{aligned}[b] &F^{ii} \bigl( \nabla_{ii} \phi- \vert \nabla_{i} \phi \vert ^{2} \bigr) + \gamma \bigl( \Delta\phi- \vert \nabla\phi \vert ^{2} \bigr) \sum F^{ii} \\ &\quad\leq C \bigl(G + \vert \nabla\phi \vert \bigr) \sum F^{ii} + \biggl(\frac {C}{G} - 1 \biggr)\beta'_{\varepsilon} (u - h). \end{aligned} $$

(2.19)

Let

$$\phi:= \eta(w) = \biggl(1 - \frac{w}{2 a} \biggr)^{-1/2} , \quad w = \frac{ \vert \nabla u \vert ^{2}}{2}, $$

where $a > \sup_{M} w$ is a constant to be determined. We have

$$1 \leq\eta< \sqrt{2} , \quad\eta' = \frac{\eta^{3}}{4 a}, \qquad \eta'' = \frac{3 \eta^{\prime2}}{\eta} $$

and

$$ \nabla_{ii} \phi- \vert \nabla_{i} \phi \vert ^{2} = \eta' \nabla_{ii} w + \bigl( \eta'' - \eta^{\prime2} \bigr) \vert \nabla_{i} w \vert ^{2} \geq\eta' \nabla_{ii} w. $$

(2.20)

Next, by (2.14)

$$ \begin{aligned}[b] &F^{ii} ( \nabla_{ii} w + \gamma\Delta w ) \\ &\quad= F^{ii} \biggl(\sum_{l} \vert \nabla_{il} u \vert ^{2} + \gamma\sum _{k,l} \vert \nabla_{kl} u \vert ^{2} \biggr) + F^{ii} \nabla_{l} u \bigl( \nabla_{iil} u + \gamma\Delta(\nabla_{l} u) \bigr) \\ &\quad\geq F^{ii} \nabla_{l} u \biggl(\nabla_{lii} u + \gamma\sum_{k}\nabla_{lkk} u \biggr) + \bigl(\gamma G^{2} - C G \bigr) \sum F^{ii} \\ &\quad\geq- C \beta'_{\varepsilon} (u - h) + \bigl(\gamma G^{2} - C G \bigr) \sum F^{ii}. \end{aligned} $$

(2.21)

Combining (2.19), (2.20), (2.21), and $|\nabla \phi| \leq C \eta' G$, we have

$$ \eta' \bigl(\gamma G^{2} - C G \bigr) \sum F^{ii} \leq C \bigl(G + \eta' G \bigr) \sum F^{ii} + \biggl(\frac{C}{G} - 1 + C \eta' \biggr)\beta '_{\varepsilon} (u - h). $$

(2.22)

We could assume that $G \geq2 C$. When $a > 2 C$, the coefficient of $\beta'_{\varepsilon}(u - h)$ is negative. Then we can derive $G \leq\frac{4 a C}{\gamma}$. □

To derive the boundary estimates for $\nabla^{2} u$, we note that $\operatorname{tr} (s du \otimes du - \frac{t}{2} |\nabla u|^{2} g + \chi ) \leq C$ on M̄, where C is independent of ε, though it may depend on $|u|_{C^{1}(\bar{M})}$. As in [1, 4], let H be the solution to

$$\textstyle\begin{cases} (1 + n \gamma )\Delta H + C = 0 & \mbox{in } M, \\ H = \varphi& \mbox{on } \partial M. \end{cases} $$

Then we have $u \leq H$ in M by the maximum principle and $\beta_{\varepsilon}(u - h) \equiv0$ in $M_{\delta}= \{x \in M: \operatorname{dist} (x, \partial M) \leq\delta\}$, where δ is sufficiently small. Thus,

$$ \textstyle\begin{cases} f (\lambda( g^{-1} T[u] )) = \psi[u] & \mbox{in } M_{\delta},\\ u = \varphi& \mbox{on } \partial M. \end{cases} $$

(2.23)

By the same arguments of Sect. 4 in [8], we obtain that

$$ \sup_{\partial M} \bigl\vert \nabla^{2} u \bigr\vert \leq C, $$

(2.24)

where C depends on $|u|_{C^{1}(\bar{M})}$ and other known data.

Combining (2.1) and (2.24), we therefore get the full estimates for second order derivatives.

3 Gradient estimates, maximum principle, and existence

For the gradient estimates, we have the following theorem.

Theorem 3.1

Assume that (1.5)–(1.8) hold. Let $u \in C^{3} (\bar{M})$ be an admissible solution to (1.16). Then

$$ \sup_{M} \vert \nabla u \vert \leq C \Bigl(1 + \sup_{\partial M} \vert \nabla u \vert \Bigr), $$

(3.1)

where C depends on $|u|_{C^{0} (\bar{M})}$ and other known data.

Proof

Suppose that $we^{\phi}$, where $w = \frac{|\nabla u|^{2}}{2}$ and $\phi = \phi(u)$ to be determined satisfying that $\phi' (u) > 0$, achieves a maximum at an interior point $x_{0} \in M$. As before, we choose a smooth orthonormal local frame $e_{1}, \ldots, e_{n}$ about $x_{0}$ such that $\nabla_{e_{i}} e_{j} = 0$ at $x_{0}$ and $\{T_{ij} (x_{0})\}$ is diagonal. Differentiating $we^{\phi}$ at $x_{0}$ twice, we have

$$ \nabla_{i} w + w \nabla_{i} \phi= 0 $$

(3.2)

and

$$ \nabla_{ii} w - w (\nabla_{i} \phi)^{2} + w \nabla_{ii} \phi\leq0. $$

(3.3)

Differentiating w, we see

$$\nabla_{i} w = \sum_{k} \nabla_{k} u \nabla_{ik} u, \qquad \nabla_{ii} w = \sum_{k} (\nabla_{ik} u)^{2} + \sum_{k} \nabla_{k} u \nabla_{iik} u . $$

Using (3.2) it follows from (3.3) that

$$ F^{ii} \biggl(\delta_{kl} - \frac{\nabla_{k} u \nabla_{l} u}{2 w} \biggr) \nabla_{ik} u \nabla_{il} u + F^{ii} \nabla_{k} u \nabla_{iik} u - w F^{ii} \biggl( \frac{(\nabla_{i} \phi)^{2}}{2} - \nabla_{ii} \phi \biggr) \leq0 $$

(3.4)

and

$$ \sum_{i,k,l} \biggl( \delta_{kl} - \frac{\nabla_{k} u \nabla_{l} u}{2 w} \biggr) \nabla_{ik} u \nabla_{il} u + \nabla_{k} u \Delta(\nabla_{k} u) - \frac{w}{2} \vert \nabla\phi \vert ^{2} + w \Delta\phi \leq0. $$

(3.5)

Note that the first term in (3.4) and (3.5) is nonnegative. Multiply $\gamma\sum F^{ii}$ to (3.5) and add what we got to (3.4). Thus, by (2.9) we obtain

$$ \begin{aligned}[b] & F^{ii} \nabla_{k} u (\nabla_{kii} u + \gamma\nabla_{k} \Delta u ) - \frac{w}{2} F^{ii} \bigl( \vert \nabla_{i} \phi \vert ^{2} + \gamma \vert \nabla\phi \vert ^{2} \bigr) \\ &\quad{}+ w F^{ii} ( \nabla_{ii} \phi+ \gamma\Delta\phi ) \leq C \vert \nabla u \vert ^{2} \sum F^{ii} . \end{aligned} $$

(3.6)

Now we compute the first term in (3.6). Firstly, we have

$$\nabla_{i} \phi= \phi' \nabla_{i} u, \qquad \nabla_{ii} \phi= \phi' \nabla_{ii} u + \phi'' (\nabla_{i} u)^{2}. $$

Using (3.2), we easily get that

$$ \begin{aligned}[b] & F^{ii} \nabla_{k} u (\nabla_{kii} u + \gamma\nabla_{k} \Delta u ) \\ & \quad= F^{ii} \nabla_{k} u \nabla_{k} \biggl( T_{ii} - s \vert \nabla_{i} u \vert ^{2} + \frac{t}{2} \vert \nabla u \vert ^{2} - \chi_{ii} \biggr) \\ & \quad= \nabla_{k} u \nabla_{k} (\psi+ \beta_{\varepsilon}) + w \phi ' F^{ii} \bigl( 2 s \vert \nabla_{i} u \vert ^{2} - t \vert \nabla u \vert ^{2} \bigr) - F^{ii} \nabla_{k} u \nabla_{k} \chi_{ii}. \end{aligned} $$

(3.7)

By the homogeneity of F, we also get

$$ \begin{aligned}[b] & F^{ii} ( \nabla_{ii} \phi+ \gamma\Delta\phi ) \\ & \quad= \phi'' F^{ii} \bigl( \vert \nabla_{i} u \vert ^{2} + \gamma \vert \nabla u \vert ^{2} \bigr) + \phi' F^{ii} \biggl( T_{ii} - s \vert \nabla_{i} u \vert ^{2} + \frac {t}{2} \vert \nabla u \vert ^{2} - \chi_{ii} \biggr) \\ & \quad= \phi'' F^{ii} \bigl( \vert \nabla_{i} u \vert ^{2} + \gamma \vert \nabla u \vert ^{2} \bigr) + \phi' \biggl(F - s F^{ii} \vert \nabla_{i} u \vert ^{2} + \frac{t}{2} F^{ii} \vert \nabla u \vert ^{2} - F^{ii} \chi_{ii} \biggr). \end{aligned} $$

(3.8)

According to (3.7) and (3.8), it follows from (3.6)

$$ \begin{aligned}[b] & \gamma \vert \nabla u \vert ^{2} \biggl( \phi'' - \frac{1}{2} \bigl(\phi'\bigr)^{2} - \frac {t}{2 \gamma} \phi' \biggr) \sum F^{ii} + \biggl( \phi'' - \frac{1}{2} \bigl(\phi' \bigr)^{2} + s \phi' \biggr) F^{ii} ( \nabla_{i} u)^{2} \\ & \quad\leq- \phi' \bigl(\psi+ \beta_{\varepsilon} - F^{ii} \chi_{ii} \bigr) + C \sum F^{ii} - \frac{\nabla_{k} u \nabla_{k} (\psi+ \beta_{\varepsilon})}{w} \\ & \quad\leq- \biggl( \phi' \beta_{\varepsilon} + \frac{\nabla_{k} u \nabla_{k} (u - h) \beta'_{\varepsilon}}{w} \biggr) + C \sum F^{ii} + C . \end{aligned} $$

(3.9)

Let

$$\phi(u) = v^{- a}, \qquad v = 1 - u + \sup_{M} u. $$

We have

$$\phi' (u) = a v^{- a - 1}, \qquad\phi'' (u) = \frac{(a + 1 )\phi '}{v} , $$

and

$$\phi'' - \frac{1}{2} \bigl( \phi'\bigr)^{2} = \phi' \biggl( \frac{a + 1}{v} - \frac{a v^{- a}}{2v} \biggr) \geq\frac{\phi' a}{2v} > 0 $$

since $v^{- a} \leq1$. When $|\nabla u (x_{0})|$ is sufficiently large, we see $\nabla_{k} u \nabla_{k} (u - h) > 0$. Hence we have that the first term on the right-hand side of (3.9) is negative as $\beta_{\varepsilon}, \beta'_{\varepsilon} > 0$. From (3.9) and (1.9) when a is sufficiently large, we then obtain that

$$ \frac{\phi' a \gamma \vert \nabla u \vert ^{2} }{4 v} \leq C, $$

(3.10)

from which we conclude that (3.1) holds. □

In order to prove (1.19), it remains to bound $\sup_{M} |u| + \sup_{\partial M} |\nabla u|$. We quote two lemmas in [8], the ingredients of whose proofs are the maximum principle.

Lemma 3.2

If either (1.10) or (1.11) holds, then any admissible solution u of (1.16) admits the a priori bound

$$ \sup_{M} u \leq c_{0}. $$

(3.11)

Lemma 3.3

If u is admissible such that $\operatorname{tr} T[u] \geq0$ and $|u|_{C^{0}(M)} \leq\mu$, then

$$ \sup_{\partial M} \nabla_{\nu}u \leq c_{1} (\mu), $$

(3.12)

where ν is the interior unit normal to ∂M.

Now with the above two lemmas and the fact $\nabla_{\nu}u \geq\nabla _{\nu}\underline{u}$ on ∂M when $u \in\mathscr{U}$, we then have the following.

Theorem 3.4

Suppose that (1.5)–(1.8), and either (1.10) or (1.11) hold. Then, for $u \in\mathscr{U}$, (1.19) holds.

Therefore, the uniform estimates (1.19) ensure that there exist a subsequence $\{u_{\varepsilon_{k}}\}$ of $\{u_{\varepsilon}\}$ and a function $u \in C^{1,1} (\bar{M})$ such that $u_{\varepsilon_{k}} \rightarrow u$ in M as $\varepsilon_{k} \rightarrow0$. It is easy to verify that u satisfies (1.3) and (1.4) and $u \in C^{3, \alpha} (E)$ for any $\alpha\in(0, 1)$. Consequently, Theorem 1.1 is established.