On the boundary value problems for fully nonlinear elliptic equations of second order

Fully nonlinear second-order, elliptic equations F (x, u,Du,D2u) = 0 are considered in a bounded domain Ω ⊂ Rn, n ≥ 2. The class of equations includes the Bellman equations supm(L mu+ fm) = 0, where the functions fm and the coefficients of the linear operators Lm are bounded in the Hölder space Cα(Ω), 0 < α < 1. We prove the interior C2,α-smoothness of solutions in Ω with some small α > 0. Under the Dirichlet boundary condition u = φ on ∂Ω with φ ∈ C2,α(Ω) and ∂Ω ∈ C2,α, the solutions u ∈ C2,α(Ω). Under the oblique derivative condition b0u + b · Du = φ on ∂Ω, where b = (b1, · · · , bn) is not tangent to ∂Ω, the solutions u ∈ C2,α(Ω) if bi, φ ∈ C1,α(Ω), and also ∂Ω ∈ C1,α.