We prove the interior C2,α regularity of solutions for some nonconvex fully nonlinear elliptic equations F(D2u, x) = f (x), x ∈ B1 ⊂ Rn. 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.  2003 Éditions scientifiques et médicales Elsevier SAS. All rights reserved.

Peer Reviewed