- Research
- Open Access
- Published:

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

*Journal of Inequalities and Applications*
**volume 2023**, Article number: 17 (2023)

## 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

with the boundary condition

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

\(\nabla ^{2} u\) 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

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:

and

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

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

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

and

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

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

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

and the growth condition

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

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

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

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

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

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

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*

*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* \(M_{\underline{u}}\),

*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

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

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

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

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*

### 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 \(x_{0}\),

and

It follows that, at \(x_{0}\),

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

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 \(e_{1}, \ldots, e_{n}\),

Let *F* be the function defined by

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

Following the literature we denote throughout this paper

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

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*

### Lemma 2.3

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

*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 \(x_{0}\),

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

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

Since \(u - \underline{u} = 0\) on *∂M* we have

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

Next, we establish the estimate

Define the linear operator *L* by

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

### Lemma 3.1

*Let*

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

*and*

*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

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

for *δ* sufficiently small. It follows that

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

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

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

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

for some index \(1 \leq r \leq n\). Let

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

and

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

Since

we obtain (3.4).

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

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

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

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

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

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

Note that *F̃* is concave by (1.5). Let

We find

By (3.2) we have on *∂M* near \(x_{0}\),

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

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

where

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

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

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 \(x_{0}\),

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 \(x_{0}\),

Combining (3.18) and (3.19) we obtain

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

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

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

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

It follows that

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

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

Therefore,

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).

## Availability of data and materials

Not applicable.

## References

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)Caffarelli, L.A., McCann, R.: Free boundaries in optimal transport and Monge–Ampère obstacle problems. Ann. Math.

**171**, 673–730 (2010)Caffarelli, L.A., Nirenberg, L., Spruck, J.: The Dirichlet problem for nonlinear second-order elliptic equations III: functions of eigenvalues of the Hessians. Acta Math.

**155**, 261–301 (1985)Gerhardt, C.: Hypersurfaces of prescribed mean curvature over obstacles. Math. Z.

**133**, 169–185 (1973)Chu, J.C., Jiao, H.M.: Curvature estimates for a class of Hessian type equations. Calc. Var. Partial Differ. Equ.

**60**, 90 (2021)Guan, B.: The Dirichlet problem for Hessian equations on Riemannian manifolds. Calc. Var. Partial Differ. Equ.

**8**, 45–69 (1999)Guan, B.: Second order estimates and regularity for fully nonlinear elliptic equations on Riemannian manifolds. Duke Math. J.

**163**, 1491–1524 (2014)Guan, B.: The Dirichlet problem for fully nonlinear elliptic equations on Riemannian manifolds. arXiv:1403.2133

Guan, B., Jiao, H.M.: Second order estimates for Hessian type fully nonlinear elliptic equations on Riemannian manifolds. Calc. Var. Partial Differ. Equ.

**54**(3), 2693–2712 (2015)Guan, B., Jiao, H.M.: The Dirichlet problem for Hessian type elliptic equations on Riemannian manifolds. Discrete Contin. Dyn. Syst., Ser. A

**36**, 701–714 (2016)Guan, P.F., Ren, C.Y., Wang, Z.Z.: Global \(C^{2}\) estimates for convex solutions of curvature equations. Commun. Pure Appl. Math.

**68**, 1287–1325 (2015)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)Jiao, H.M., Liu, J.X.: On a class of Hessian type equations on Riemannian manifolds. Proc. Am. Math. Soc. arXiv:2202.05067. https://doi.org/10.1090/proc/15508

Jiao, H.M., Wang, Y.: The obstacle problem for Hessian equations on Riemannian manifolds. Nonlinear Anal.

**95**, 543–552 (2014)Kinderlehrer, D.: How a minimal surface leaves an obstacle. Acta Math.

**130**, 221–242 (1973)Lee, K.: The obstacle problem for Monge–Ampère equation. Commun. Partial Differ. Equ.

**26**, 33–42 (2001)Lee, K., Lee, T., Park, J.: The obstacle problem for the Monge–Ampère equation with the lower obstacle. Nonlinear Anal.

**210**, 112374 (2021)Li, Y.Y.: Some existence results of fully nonlinear elliptic equations of Monge–Ampere type. Commun. Pure Appl. Math.

**43**, 233–271 (1990)Liu, J.K., Zhou, B.: An obstacle problem for a class of Monge–Ampère type functionals. J. Differ. Equ.

**254**, 1306–1325 (2013)Ren, C., Wang, Z.: On the curvature estimates for Hessian equation. Am. J. Math.

**141**(5), 1281–1315 (2019)Ren, C., Wang, Z.: The global curvature estimate for the \(n-2\) Hessian equation. arXiv:2002.08702. Preprint

Savin, O.: The obstacle problem for Monge Ampere equation. Calc. Var. Partial Differ. Equ.

**22**, 303–320 (2005)Trudinger, N.S.: On the Dirichlet problem for Hessian equations. Acta Math.

**175**, 151–164 (1995)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)

Xiong, J.G., Bao, J.G.: The obstacle problem for Monge–Ampère type equations in non-convex domains. Commun. Pure Appl. Anal.

**10**(1), 59–68 (2011)

## Acknowledgements

We thank Associate Professor Heming Jiao for the idea and helpful comments and suggestions.

## Funding

No funding.

## Author information

### Authors and Affiliations

### Contributions

JL conceptualized the idea and wrote the first draft. YW reviewed and edited the manuscript. All authors read and approved the final manuscript.

### Corresponding author

## Ethics declarations

### Competing interests

The authors declare that they have no competing interests.

## Rights and permissions

**Open Access** This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.

## About this article

### Cite this article

Liu, J., Wang, Y. On a class of obstacle problem for Hessian equations on Riemannian manifolds.
*J Inequal Appl* **2023**, 17 (2023). https://doi.org/10.1186/s13660-023-02926-0

Received:

Accepted:

Published:

DOI: https://doi.org/10.1186/s13660-023-02926-0

### Keywords

- Obstacle problem
- Hessian equations
- Second-order estimates