# 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.

## 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 , Guan studied the existence of a complete conformal metric 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 , 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, 912, 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  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 , Gerhardt considered a hypersurface bounded from below by an obstacle with prescribed mean curvature in $$\mathbb{R}^{n}$$. Lee  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  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  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, 1821].

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  following ideas from  and  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  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 ,  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  or , 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.

## 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 , 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 , 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.

## 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 , 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.

## References

1. Bao, G.-J., Dong, W.-S., Jiao, H.-M.: Regularity for an obstacle problem of Hessian equations on Riemannian manifolds. J. Differ. Equ. 258, 696–716 (2015)

2. Caffarelli, L.A., Nirenberg, L., Spruck, J.: The Dirichlet problem for nonlinear second order elliptic equations III, functions of the eigenvalues of the Hessian. Acta Math. 155, 261–301 (1985)

3. Crandall, M., Ishii, H., Lions, P.: User’s guide to viscosity solutions of second order partial differential equations. Bull. Am. Math. Soc. 27, 1–67 (1992)

4. Dong, W.-S., Wang, T.-T., Bao, G.-J.: A priori estimates for the obstacle problem of Hessian type equations on Riemannian manifolds. Commun. Pure Appl. Anal. 15, 1769–1780 (2016)

5. Evans, L.C.: Classical solutions of fully nonlinear, convex, second order elliptic equations. Commun. Pure Appl. Math. 35, 333–363 (1982)

6. Gerhardt, C.: Hypersurfaces of prescribed mean curvature over obstacles. Math. Z. 133, 169–185 (1973)

7. Guan, B.: The Dirichlet problem for Hessian equations on Riemannian manifolds. Calc. Var. Partial Differ. Equ. 8, 45–69 (1999)

8. Guan, B.: Complete conformal metrics of negative Ricci curvature on compact manifolds with boundary. Int. Math. Res. Not. 2008, rnn105 (2008) Addendum, IMRN 2009 (2009), 4354–4355, rnp166.

9. Guan, B.: Second order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds. Duke Math. J. 163, 1491–1524 (2014)

10. Guan, B.: The Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds. arXiv:1403.2133

11. Guan, B., Jiao, H.-M.: Second order estimates for Hessian type fully nonlinear elliptic equations on Riemannian manifolds. Calc. Var. Partial Differ. Equ. 54, 2693–2712 (2015)

12. Guan, B., Jiao, H.-M.: The Dirichlet problem for Hessian type fully nonlinear elliptic equations on Riemannian manifolds. Discrete Contin. Dyn. Syst. 36, 701–714 (2016)

13. Guan, P.-F., Wang, G.-F.: Local estimates for a class of fully nonlinear equations arising from conformal geometry. Int. Math. Res. Not. 26, 1413–1532 (2003)

14. Jiao, H.-M.: $$C^{1,1}$$ regularity for an obstacle problem of Hessian equations on Riemannian manifolds. Proc. Am. Math. Soc. 144, 3441–3453 (2016)

15. Jiao, H.-M., Wang, Y.: The obstacle problem for Hessian equations on Riemannian manifolds. Nonlinear Anal. TMA 95, 543–552 (2014)

16. Krylov, N.V.: Boundedly inhomogeneous elliptic and parabolic equations in a domain. Izvestia Math. Ser. 47, 75–108 (1983)

17. Lee, K.: The obstacle problem for Monge–Ampère equation. Commun. Partial Differ. Equ. 26, 33–42 (2001)

18. Liu, J.-K., Zhou, B.: An obstacle problem for a class of Monge–Ampère type functionals. J. Differ. Equ. 254, 1306–1325 (2013)

19. Oberman, A.: The convex envelope is the solution of a nonlinear obstacle problem. Proc. Am. Math. Soc. 135, 1689–1694 (2007)

20. Oberman, A., Silvestre, L.: The Dirichlet problem for the convex envelope. Trans. Am. Math. Soc. 363, 5871–5886 (2011)

21. Savin, O.: The obstacle problem for Monge–Ampere equation. Calc. Var. Partial Differ. Equ. 22, 303–320 (2005)

22. Schoen, R.: Conformal deformation of a Riemannian metric to constant scalar curvature. J. Differ. Geom. 20, 479–495 (1984)

23. Trudinger, N.S.: On the Dirichlet problem for Hessian equations. Acta Math. 175, 151–164 (1995)

24. Urbas, J.: Hessian equations on compact Riemannian manifolds. In: Nonlinear Problems in Mathematical Physics and Related Topics, II, pp. 367–377. Kluwer/Plenum, New York (2002)

25. Viaclovsky, J.A.: Conformal geometry, contact geometry, and the calculus of variations. Duke Math. J. 101, 283–316 (2000)

26. Viaclovsky, J.A.: Conformal Geometry and Fully Nonlinear Equations. Nankai Tracts in Mathematics, vol. 11, pp. 435–460. World Scientific, Hackensack (2006)

27. Xiong, J.-G., Bao, J.-G.: The obstacle problem for Monge–Ampère type equations in non-convex domains. Commun. Pure Appl. Anal. 10, 59–68 (2011)

## Funding

The research was supported by the National Natural Science Foundation of China (No. 11771107).

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Correspondence to Sijia Bao.

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