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