The Regularity Theory for the Double Obstacle Problem for Fully Nonlinear Operator

In this paper, we prove the existence and uniqueness of $W^{2,p}$ ($n<p<\infty$) solutions of a double obstacle problem with $C^{1,1}$ obstacle functions. Moreover, we show the optimal regularity of the solution and the local $C^1$ regularity of the free boundary. In the study of the regularity of the free boundary, we deal with a general problem, the no-sign reduced double obstacle problem with an upper obstacle $\psi$, $F(D^2 u,x) =f\chi_{\Omega(u) \cap\{ u< \psi\} } + F(D^2\psi,x) \chi_{\Omega(u)\cap \{u=\psi\}}, u\le \psi \text { in } B_1$, where $\Omega(u)=B_1 \setminus \left( \{u=0\} \cap \{ \nabla u =0\}\right)$.