Regularity Theory for L\u3csup\u3eN\u3c/sup\u3e-Viscosity Solutions to Fully Nonlinear Elliptic Equations With Asymptotical Approximate Convexity

We develop interior W2,p,μ and W2,BMO regularity theories for Ln-viscosity solutions to fully nonlinear elliptic equations T(D2u,x)=f(x), where T is approximately convex at infinity. Particularly, W2,BMO regularity theory holds if operator T is locally semiconvex near infinity and all eigenvalues of D2T(M) are at least −C‖M‖−(1+σ0) as M→∞. W2,BMO regularity for some Isaacs equations is given. We also show that the set of fully nonlinear operators of W2,BMO regularity theory is dense in the space of fully nonlinear uniformly elliptic operators