Partial regularity of solutions of fully nonlinear uniformly elliptic equations

Abstract

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.

13 pages