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

Abstract

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 $\Omega (u)=B_1\setminus \left(\{u=0\}\cap \{\nabla u=0\}\right)$.