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

In this paper, we establish the a prioriC2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C^{2}$\end{document} 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.


Introduction
Let (M, g) be a compact manifold with smooth boundary ∂M. In this paper, we are concerned with the obstacle problem with the boundary condition u = ϕ on ∂M, (1.2) where f is a smooth, symmetric function defined in an open convex cone ⊂ R n with a vertex at the origin and n = λ = (λ 1 , . . . , λ n ) ∈ R n : each λ i > 0 ⊆ = R n , ∇ 2 u denotes the Hessian of u, χ is a (0, 2)-tensor field, λ(h) denotes the eigenvalues of a (0, 2)-tensor field h with respect to the metric g and ϕ ∈ C 4 (∂M). In this work, we assume the obstacle function φ ∈ C 3 (M) satisfies φ = ϕ 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 φ > ϕ 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 ∈ C 2 ( ) ∩ C 0 ( ) satisfies the structure conditions: (1.4) f is concave in , (1.5) and ⎧ ⎨ ⎩ f > 0 in , (1. 6) In addition, f is also assumed to satisfy that for any positive constants μ 1 , μ 2 with 0 < μ 1 < μ 2 < sup f there exists a positive constant c 0 depending on μ 1 and μ 2 such that for any λ ∈ μ 1 ,μ 2 := {λ ∈ : Furthermore, f is supposed to satisfy that for any A > 0 and any compact set K ⊂ , there exists R = R(A, K) > 0 such that (1.10) Following [3], we assume that there exists a large number R > 0 such that at each x ∈ ∂M, where (κ 1 (x), . . . , κ 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 u ∈ C 2 (M) satisfying (1.12) As in [6], the function ψ(x, z, p) ∈ C 2 (T * M × R) > 0 satisfies and the growth condition when |p| is sufficiently large, where γ 1 < 2, γ 2 < 4 are positive constants andψ is a positivecontinuous function of (x, z) ∈ × R.
Our main results are stated as follows. Note that in Theorem 1.2, the function φ is assumed to be admissible. Under the homogeneous boundary condition, i.e., ϕ ≡ 0, and that χ ≡ 0, we can remove this assumption. Typical examples are given by f = σ 1/k k , 1 ≤ k ≤ n, defined on the cone k = {λ ∈ R n : σ j (λ) > 0, j = 1, . . . , k}, where σ k (λ) are the elementary symmetric functions (1.16) Other interesting examples satisfying (1.4)-(1.11) (see [13]) are defined on the cone = {λ ∈ R n : (μ 1 , . . . , μ n ) ∈ k }, where μ 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).
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 be a uniformly convex bounded domain in R n . Given a function K(x, z) ∈ C 2 ( × R) > 0 satisfying that there exists a positive constant A such that and a piece of uniformly convex graphic hypersurface M φ , suppose there exists a uniformly convex graphic hypersurface M u under M φ satisfying the Gauss curvature of M u , 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 β in (2.1). It is also why the conditions (1.9)-(1.11) are needed.
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.

Preliminaries
As in [14] and [25], we consider the singular perturbation problem where the penalty function β is defined by for ∈ (0, 1). Obviously, β ∈ C 2 (R) satisfies Since u ≤ φ, u is also a subsolution to (2.1). Let u ∈ C 3 (M) ∩ C 4 (M) be an admissible solution of (2.1) with u ≥ u. We shall show that there exists a constant C independent of such that for small . The C 0 estimates can be easily derived from the fact that ⊂ 1 and u ≥ 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 such that
(

2.4)
Proof We consider the maximal value of uφ on M. We may assume it is achieved at an interior point and It follows that, at x 0 , for some positive constant c 2 depending only on φ C 2 (M) and (2.4) holds.
Then, we see u is an admissible solution of (1.1) and (1.2) as in [25]. The fact that u ∈ C 3,α (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 ∈ C 4 (M) be an admissible function. For simplicity we shall use the notation U = χ + ∇ 2 u and, under an orthonormal local frame e 1 , . . . , e n , (2.7) 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 , . . . , e n . The matrix {F ij } has eigenvalues f 1 , . . . , 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].
In the following section, we will drop the subscript for convenience.

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 ∈ ∂M. We choose smooth orthonormal local frames e 1 , . . . , e n around x 0 such that when restricted to ∂M, e n is normal to ∂M.
Since ∂M is smooth we may assume the distance function to ∂M is smooth in M δ 0 for fixed δ 0 > 0 sufficiently small (depending only on the curvature of M and the principal curvatures of ∂M). Since ∇ ij ρ 2 (x 0 ) = 2δ ij , we may assume ρ is smooth in M δ 0 and 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].
Then, there exist positive constants t, δ sufficiently small and N sufficiently large such that and v ≥ 0 (3.6) in M δ for some uniform constat 0 > 0.
Proof First, there exists a positive constant θ 0 such that uθ 0 ρ 2 is also admissible. By (2.4) and the concavity of F, we have where the constant C depends on u C 1 (M) and the constant c 2 in (2.4). Recall that f i = ∂f ∂λ i , where λ = λ(∇ 2 u + χ) for i = 1, . . . , n. Without loss of generality, we may assume f n = min i {f i }. Next, since ∇d ≡ 1 on the boundary, we have for δ sufficiently small. It follows that Thus, we can choose N sufficiently large and t, δ sufficiently small such that We may further make δ sufficiently small such that v ≥ 0 in M δ . Since β ≥ 0 we obtain (3.5).
Since u + tr(χ) > 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 We wish to showm R → +∞ as R → +∞. Without loss of generality we assumem R <c R /2 (otherwise we are done by (3.12) (3.14) where σ αβ = ∇ α e β , e n ; note that σ αβ = Π(e α , e β ) on ∂M. Define where η =F αβ 0 σ αβ . From (3.13) and (3.14) we see that Q(x 0 ) = 0 and Q ≥ 0 on ∂M near x 0 . Furthermore, we have in M δ 0 .