Partial regularity of solutions of fully nonlinear, Uniformly elliptic equations

We prove that a viscosity solution of a uniformly elliptic, fully nonlinear equation is C ^2,α on the complement of a closed set of Hausdorff dimension at most ε{lunate} less than the dimension. The equation is assumed to be C ¹, and the constant ε{lunate} > 0 depends only on the dimension and the ellipticity constants. The argument combines the W ^2,ε{lunate} estimates of Lin with a result of Savin on the C ^2,α regularity of viscosity solutions that are close to quadratic polynomials.