The regularity theory for the double obstacle problem for fully nonlinear operator

In this paper, we prove the existence and uniqueness of W2,p (n < p < & INFIN;) solutions of a double obstacle problem. Moreover, we show the optimal regularity of the solution and the local C1 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 & psi;, F(D2u, x) = f & chi;& OHM;(u)& AND;{u<& psi;} + F(D2 & psi;, x)& chi;& OHM;(u)& AND;{u=& psi;}, u & LE; & psi;inB1,where & OHM;(u) = B1 \ ({u = 0} & AND; { backward difference u = 0}).& COPY; 2023 Elsevier Ltd. All rights reserved.