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 < ∞) solutions of a double obstacle problem. Moreover, we show the optimal regularity of the solution and the local C 1 regularity of the free boundary under a thickness assumption at the free boundary point on the intersection of two free boundaries. 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 ψ , F (D 2 u , x) = f χ Ω (u) ∩ { u < ψ } + F (D 2 ψ , x) χ Ω (u) ∩ { u = ψ } , u ≤ ψ in B 1 , where Ω (u) = B 1 ∖ { u = 0 } ∩ { ∇ u = 0 } . [ABSTRACT FROM AUTHOR]