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

Abstract

In this paper, we prove the existence and uniqueness of W2,p (n<p<∞) 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 ψ, F(D2u,x)=fχΩ(u)∩{u<ψ}+F(D2ψ,x)χΩ(u)∩{u=ψ},u≤ψinB1, where Ω(u)=B1∖{u=0}∩{∇u=0}.