Interior C2,α regularity theory for a class of nonconvex fully nonlinear elliptic equations

Abstract We prove the interior C2,α regularity of solutions for some nonconvex fully nonlinear elliptic equations F(D2u,x)=f(x), x∈B 1 ⊂ R n . Our hypothesis is that, for every x∈B1, F(· ,x) is the minimum of a concave operator and a convex operator of D2u. This extends the Evans–Krylov theory for convex equations to some nonconvex operators of Isaacs type. For instance, our results apply to the 3-operator equation F3(D2u)=min{L1u,max{L2u,L3u}} =0 (here Li are linear operators), which motivated the present work.