Regularity for an obstacle problem of Hessian equations on Riemannian manifolds

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

“…Once (19) is obtained, the Evans-Krylov theorem [6,18] and the Schauder theory [7] ensure the smooth regularity of admissible solutions of (16), while the existence is guaranteed by the continuity method [7] and the degree theory [22]. We then conclude that there exists a function C 1,1 (M ) satisfying (1) and 2, see [2,30].…”

“…Proof of Theorem 3.1. We are going to prove the following inequality by computation follows essentially [2] that…”

“…Proof of Theorem 4.1. We outline the proof here for completeness, and the reader can refer to [2] for more details and another group of assumptions that guarantees (45). Suppose |∇u|φ −1/2 achieves a maximum at an interior point x 0 ∈ M , where φ is a positive function to be determined.…”

“…which is introduced in [11] was used by Bao, Dong, and Jiao in [2] to consider the problem (1) and (2). This assumption excludes the linear function f = σ 1 .…”

“…Jiao in [16] considered the problem with a more general obstacle, but with A ≡ χ where χ is a smooth tensor on M and ψ ≡ ψ(x). Compared with these, we study the obstacle problem of the general case (1) and (2), and derive a priori estimates without condition (15). Moreover, our problem (1) covers the case that f = (σ k /σ l ) 1/(k−l) , 1 ≤ l < k ≤ n and f = σ 1 .…”