A priori estimates for the obstacle problem of Hessian type equations on Riemannian manifolds

We are concerned with a priori estimates for the obstacle problem of a wide class of fully nonlinear equations on Riemannian manifolds. We use new techniques introduced by Bo Guan and derive new results for a priori second order estimates of its singular perturbation problem under fairly general conditions. By approximation, the existence of a C^{1,1} viscosity solution is proved.


Introduction
This is one of a series of papers in which we study the obstacle problem for Hessian type equations on Riemannian manifolds. Let (M n , g) be a compact Riemannian manifold of dimension n ≥ 2 with smooth boundary ∂M,M := M ∪ ∂M, and ∇ denote its Levi-Civita connection. In this paper we study the obstacle problem where h ∈ C 3 (M ) is called an obstacle, ϕ ∈ C 4 (∂M), h > ϕ on ∂M, ψ[u] = ψ(x, u, ∇u) is a positive function of C 3 and A[u] = A(x, u, ∇u) is a smooth (0, 2) tensor which may depend on u and ∇u, f is a symmetric function of λ ∈ R n , and for a (0, 2) tensor X on M, λ(X) denotes the eigenvalues of X with respect to the metric g.
Following [4], the function f ∈ C 2 (Γ) ∩ C 0 (Γ) is assumed to be defined in an open, convex, symmetric cone Γ R n , with vertex at the origin, containing the positive cone: {λ ∈ R n : each component λ i > 0}, and to satisfy the fundamental structure A function u ∈ C 2 (M) is called admissible at x ∈ M if λ(∇ 2 u + A[u])(x) ∈ Γ and we call it admissible in M if it is admissible at each x ∈ M. It is shown in [4] that (1.3) implies that (1.1) is elliptic for admissible solutions, and (1.4) ensures that F defined by F (r) = f (λ(r)) for r = {r ij } ∈ S n×n with λ(r) ∈ Γ is concave, where S n×n is the set of n × n symmetric matrices.
In this paper, we prove the existence of a viscosity solution of (1.1) and (1.2) in C 1,1 (M ) (see [5,27] for the definition of viscosity solution). Our motivation to study equation (1.1) comes partly from its geometric applications. In [8] Gerhardt considered hypersurfaces having prescribed mean curvature H that are bounded from below by an obstacle. The case H = 0 (minimal surfaces) had been studied by for example Kinderlehrer [18,19] and Giusti [15]. Xiong and Bao [29] studied the problem of finding the greatest hypersurface below a given obstacle, whose Gauss-Kronecker curvature (accordingly, f = σ 1/n n ) is bounded from below by a positive function, and established C 1,1 regularity in nonconvex domains in R n . Lee [20] considered obstacle problem for Monge-Ampère equation of the case when A ≡ 0, ψ ≡ 1, ϕ ≡ 0, and proved the C 1,1 regularity of the viscosity solution and C 1,α regularity of free boundary in a strictly convex domain in R n . The interest to (1.1) is also arising from its connection to optimal transportation problem, see e.g. Savin [25,26], Caffarelli and McCann [3]. Moreover, Liu and Zhou [22] treated an obstacle problem for Monge-Ampère type functionals whose Euler-Lagrange equations including the affine maximal surface equation and Abreu's equation. Oberman [23,24] showed that the convex envelope is a viscosity solution of a partial differential equation in the form of a nonlinear obstacle problem.
The obstacle problem for Hessian equations on Riemannian manifolds has been studied by Jiao and Wang [16], where they considered the case when A ≡ κug under conditions on f which however exclude the case that f = (σ k /σ l ) 1/(k−l) , 1 ≤ l < k ≤ n. Bao, Dong and Jiao [2] considered (1.1) and (1.2) under a condition (see the condition (2.4) in [2], see also [11]) which was essential for a priori second order estimates. Compared with these, we study the obstacle problem of the general case (1.1) and (1.2), and derive a priori estimates without such a condition, using the new technique introduced by Guan [12], see also [13,14]. Moreover, our problem (1.1) covers the case that f = (σ k /σ l ) 1/(k−l) , 1 ≤ l < k ≤ n.

Beginning of Proof
We use ideas from [12], see also [13,14]. Suppose, in addition to (1.3)-(1.5), that there exists an admissible subsolution u ∈ C 2 (M ) satisfying and u ≤ h in M. We remark here that the existence of u in some special cases can be found in [16].
To prove the existence of viscosity solutions to (1.1) and (1.2), we use a penalization technique and consider the following singular perturbation problem where the penalty function β ε is defined by for ε ∈ (0, 1). Obviously, see [29], β ε ∈ C 2 (R) satisfies Observe that u is also a subsolution to (2.2). Let We aim to derive the uniform bound where C is independent of ε. Once (2.5) is obtained, we conclude that there exists a function C 1,1 (M ) satisfying (1.1) and (1.2), see [2,29].
Remark 2.1. For simplicity, we may drop the subscript ε in the following when there is no possible confusion.
In the proof of the second order estimates, we adapt new methods introduced by Guan [12]. We use notations in [12].
We use the notation . Under a local frame e 1 , . . . , e n , U ij := U(e i , e j ) = ∇ ij u + A ij [u] and Let L be the linear operator locally defined by . In the process of deriving a priori second order estimates, see Section 3 below, we apply Lemma 2.2 with ζ = ζ 0 in (2.6) (we will explain this in Remark 2.4), and an immediate result shows that: Proof. For any x ∈ M, choose a smooth orthonormal local frames e 1 , . . . , e n about x It follows from (2.8) and (2.9) that Thus (2.10) is obtained.
Remark 2.4. In another case |ν µ − ν λ | < ζ 0 , we have by (2.6) that ν λ − ζ 0 1 ∈ Γ n , and therefore We also have in this case that, by the concavity of F , Then combining with (2.11) we obtain Remark 2.5. Note that (2.10) and (2.13) are the highlight of the paper.

Estimates for second order derivatives
In this section, we prove a priori estimates of second order derivatives for an admissible solution u ∈ U . We see that tr(A[u]) ≤ C onM , where C is independent of ε and C depends on |u| and hence by the arguments of Section 3 in [14], we obtain the boundary estimates for second order derivatives for some K 0 ≥ 0, where the constant C in (3.2) is independent of ε and depends on |u| C 1 (M ) . Note that the condition (3.3) is used to overcome the difficulty caused by the presence of curvature in the boundary estimates (3.2) (see [11,12,14]). Therefore, it remains to estimate the interior second order derivatives |∇ 2 u| C 0 (M ) for the global estimates of second derivatives |∇ 2 u| C 0 (M ) . The following lemma will be needed which is key in both the second derivative estimates and the gradient estimates.
Lemma 3.1 ( [2,29]). There exists a positive constant c 0 , which is independent of ε and depends on |u| C 0 (M ) , such that Now we are ready to prove the following theorem.  Proof. Set where φ is a function to be determined. Assume that W is achieved at an interior point x 0 ∈ M in a unit direction ξ ∈ T x 0 M. Choose a smooth orthonormal local frame e 1 , . . . , e n about x 0 such that ξ = e 1 , ∇ i e j (x 0 ) = 0 and that U ij (x 0 ) is diagonal. We assume U 11 (x 0 ) > 0 and At the point x 0 where the function log U 11 +φ (defined near x 0 ) attains its maximum, we have Differentiating equation (2.2) twice and using (3.7), we obtain at x 0 , provided U 11 is sufficiently large. Recall the formula for interchanging order of covariant derivatives It follows Differentiating equation (2.2) once, we obtain Moreover, we use the formula Thus, by substituting (3.10) into (3.8) and using (3.9) and (3.13), we obtain where b, δ are undetermined constants satisfying 0 < δ < 1 ≤ b. Direct computation yields and . We then have by (3.12) that Now we estimate E in (3.14) following [11] (see also [28]) by using an inequality shown by Andrews [1] and Gerhardt [9]. For fixed 0 < s ≤ 1/3, let Similar to [11], we have Then, Combining (3.14), (3.15) and (3.16), we obtain Taking δ < 1 small enough such that Then we may assume otherwise, we have U 11 ≤ C/c 1 and we are done. Therefore, So far, the proof above follows essentially [2]. From now on we use the new method introduced by Guan [12]. Letμ = µ(x 0 ) andλ = λ(U(x 0 )). If |νμ − νλ| ≥ ζ 0 , we apply (2.10) to (3.17) and obtain that . Fix b > 1 sufficiently large such that bθ − C > 0, and it follows from Lemma 3.1 that when ε is small. Note that |U ii | ≥ sU 11 for i ∈ J. It follows that This implies a bound U 11 (x 0 ) ≤ Cb 2 /(c 1 s 2 ). Next suppose |νμ − νλ| < ζ 0 . We then obtain by applying (2.13) to (3.17) that Again we can choose ε small enough such that bβ ε (u − h) − c 1 β ′ ε (u − h) ≤ 0. Thus we have by (2.12), By the concavity of f , we have where c 0 comes from Lemma 3.1. Therefore, when |λ| is large enough satisfying f (|λ|1) ≥ 2+c 0 +max x∈M ψ[u] by (3.5). Combining (3.19) and (3.20) we have which gives |λ| ≤ Cb 2 .
The gradient estimates (4.5) can be derived as in [2] using condition (3.3) in place of (2.6) in [2]. We outline the proof here for completeness, and the reader can refer to [2] for more details and another group of assumptions that guarantees (4.5).
Finally, by applying Theorem 4.1 in [2] which gives uniform bounds for |u| C 0 (M ) and |∇u| C 0 (∂M ) , provided (i) A(x, z, p) ≡ A(x, p) and A ξξ (x, p) is concave in p for each ξ ∈ T x M or (ii) trA(x, z, 0) ≤ 0 when z is sufficiently large and