Regularity for an obstacle problem of hessian equations on Riemannian manifolds (Q473061)

Let \((M^n, g)\) be a compact Riemannian manifold of dimension \(n\geq2\) with smooth boundary \(\partial M\) and \(\overline{M}=M\cup \partial M.\) The authors study the obstacle problem \[ \max \big\{u-h,(f(\lambda (\nabla^2u+A[u]))-\psi (x,u,\nabla u))\big\}=0\quad \text{in}\;M \] with boundary condition \[ u=\varphi\quad \text{on}\;\partial M. \] Here \(h\in C^2(\overline{M}),\) \(\varphi\in C^4(\partial M),\) \(h>\varphi\) on \(\partial M,\) \(f\) is a symmetric function of \(\lambda\in \mathbb{R}^n,\) \(\nabla^2u\) stands for the Hessian of a function \(u\) on \(M,\) \(A[u]=A(x,u,\nabla u)\) is a smooth \((0,2)\) tensor and, for any \((0,2)\) tensor \(X\) on \(M\) \(\lambda(X)\) stands for the eigenvalues of \(X\) with respect to the metric \(g\). The authors study the regularity for solutions to the obstacle problem and, as application, prove existence of a \(C^{1,1}\) solution.