The obstacle problem for conformal metrics on compact Riemannian manifolds

We prove a priori estimates up to their second order derivatives for solutions to the obstacle problem of curvature equations on Riemannian manifolds \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$(M^{n}, g)$\end{document}(Mn,g) arising from conformal deformation. With the a priori estimates the existence of a \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C^{1,1} $\end{document}C1,1 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 ≥ 3 with smooth boundary ∂M,M := M ∪ ∂M. In conformal geometry, it is interesting to find a complete metricg ∈ [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 metricg of negative Ricci curvature on M satisfying f -λ g -1 Ricg = ψ in M, (1.1) where Ricg is the Ricci tensor ofg, and λ(g -1 Ricg) = (λ 1 , . . . , λ n ) are the eigenvalues of g -1 Ricg . The transformation formula for the Ricci tensor under conformal deformatioñ g = e 2u g is given by where ∇u, ∇ 2 u, and u 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 λ g -1 ∇ 2 u + u n -2 g + |∇u| 2 gdu ⊗ du -Ric g n -2 = ψ(x) n -2 e 2u . (1.2) In this paper, we study the obstacle problem of equation (1.2). More generally, let where χ is a smooth (0, 2) tensor, γ > 0 is a constant, and s, t ∈ R. We consider the following equation: with the Dirichlet boundary condition 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 ∈ C 2 ( ) ∩ C 0 (¯ ) is a symmetric function of λ ∈ R n , defined in an open, convex, and symmetric cone R n , with vertex at the origin, which contains the positive cone: + n := {λ ∈ R n : each component λ i > 0} and satisfies the following fundamental structure conditions: f is a concave function, (1.6) and Here, for convenience, we also assume that f is homogeneous of degree one. (1.8) We observe that by the concavity and homogeneity of f , Important classes of f are the elementary symmetric functions and their quotients, i.e., and Let F be defined by F(r) = f (λ(r)) for r = {r ij } ∈ S n×n with λ(r) ∈ , where S n×n is the set of n × 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.
)(x) ∈ , 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 (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 R n . Lee [17] considered the obstacle problem for the Monge-Ampère equation (i.e., f = (σ n ) 1 n ) for the case that T[u] = D 2 u, ψ ≡ 1, and ϕ ≡ 0, and proved the C 1,1 regularity of the viscosity solution in a strictly convex domain in R n . Xiong and Bao [27] extended the work of Lee to a nonconvex domain in 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] = ∇ 2 u + A(x, u, ∇u), under a certain technical assumption. Because of the term γ u (γ > 0), here we only need a minimal amount of assumptions. For other works, see [4,14,15,[18][19][20][21].
Our main result is the following theorem.

Theorem 1.1 Assume that (1.5)-(1.8) and either the following condition
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 ∇ ν 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 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 ∈ ∂M, where κ 1 , . . . , κ 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 there is a number 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: where the penalty function β ε ∈ C 2 (R) satisfies (1.17) An example given in [27] is for ε ∈ (0, 1). Observe that u is also a subsolution to (1.16). Let is an admissible solution of (1.16) with u ε ≥ u onM .
We aim to derive the uniform bound  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.
where C depends on |u| C 1 (M) and other known data.
Proof Set where φ is a function to be determined. Assume that W is achieved at an interior point x 0 ∈ M and 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 T ij (x 0 ) is diagonal. We write G = ∇ 11 u + s|∇ 1 u| 2 . Assume G(x 0 ) > 0 (otherwise we are done). At the point x 0 , where the function log G + φ (defined near x 0 ) attains its maximum, we have and

By (2.3) we have
and Since γ > 0, we obtain By calculation, we get and Recall the formula for interchanging order of covariant derivatives and It follows from (2.10) and Differentiating equation (1.16) once at x 0 , we obtain for 1 ≤ k ≤ n, It is easy to see that (2.14) and that With (2.9) we see (2.16) and similarly With (2.12), (2.14)-(2.17), and the concavity of F, we derive (2.18) By (1.9) and β ε > 0 it follows from (2.6) and (2.18) that where a > sup M w is a constant to be determined. We have Next, by (2.14) We could assume that G ≥ 2C. When a > 2C, the coefficient of β ε (uh) is negative. Then we can derive G ≤ 4aC γ .
To derive the boundary estimates for ∇ 2 u, we note that tr(sdu ⊗ du -t 2 |∇u| 2 g + χ) ≤ C onM, where C is independent of ε, though it may depend on |u| C 1 (M) . As in [1,4], let H be the solution to Then we have u ≤ H in M by the maximum principle and (2.23) By the same arguments of Sect. 4 in [8], we obtain that where C depends on |u| C 1 (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. where C depends on |u| C 0 (M) and other known data.
Proof Suppose that we φ , where w = |∇u| 2 2 and φ = φ(u) to be determined satisfying that φ (u) > 0, achieves a maximum at an interior point x 0 ∈ M. As before, we choose a smooth orthonormal local frame e 1 , . . . , e n about x 0 such that ∇ e i e j = 0 at x 0 and {T ij (x 0 )} is diagonal. Differentiating we φ at x 0 twice, we have Differentiating w, we see Using (3.2) it follows from (3.3) that Note that the first term in (3.4) and (3.5) is nonnegative. Multiply γ F ii to (3.5) and add what we got to (3.4). Thus, by (2.9) we obtain Now we compute the first term in (3.6). Firstly, we have Using (3.2), we easily get that By the homogeneity of F, we also get According to (3.7) and (3.8), it follows from (3.6) We have since v -a ≤ 1. When |∇u(x 0 )| is sufficiently large, we see ∇ k u∇ k (uh) > 0. Hence we have that the first term on the right-hand side of (3.9) is negative as β ε , β ε > 0. From (3.9) and (1.9) when a is sufficiently large, we then obtain that φ aγ |∇u| 2 4v ≤ 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 ∂M |∇u|. We quote two lemmas in [8], the ingredients of whose proofs are the maximum principle. where ν is the interior unit normal to ∂M. Now with the above two lemmas and the fact ∇ ν u ≥ ∇ ν u on ∂M when u ∈ U, we then have the following. Therefore, the uniform estimates (1.19) ensure that there exist a subsequence {u ε k } of {u ε } and a function u ∈ C 1,1 (M) such that u ε k → u in M as ε k → 0. It is easy to verify that u satisfies (1.3) and (1.4) and u ∈ C 3,α (E) for any α ∈ (0, 1). Consequently, Theorem 1.1 is established.

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