𝐶^{1,1} regularity for an obstacle problem of Hessian equations on Riemannian manifolds

Abstract

In this paper, we study an obstacle problem for a class of fully nonlinear equations on Riemannian manifolds. Using some new ideas, the C 1 , 1 C^{1,1} regularity for the greatest viscosity solution is established under essentially optimal structure conditions.

The obstacle problem for conformal metrics on compact Riemannian manifolds

“…Because of the term ( ), here we only need a minimal amount of assumptions. For other works, see [ 4 , 14 , 15 , 18 – 21 ].…”

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

“…Recently, Jiao [16] studied an obstacle problem for Hessian equations on Riemannian manifolds using the ideas from the theory of the a priori estimates for fully nonlinear elliptic equations introduced by Guan [12] (see [14] for a general form). 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].…”

“…Acknowledgments: The authors would like to thank Heming Jiao for drawing the authors' attention to the work about the obstacle problem on Riemannian manifolds and many useful suggestions and comments. We also thank him for sending us his preprint [16].…”

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

“…Here, in this paper the assumption is taken off due to the new Lemma 2.1 which together with (25) is essential for a priori second order estimates. 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).…”