A generalization of Savin's small perturbation theorem for fully nonlinear elliptic equations and applications

In this note, we generalize Savin's small perturbation theorem to nonhomogeneous fully nonlinear equations $F(D^2u, Du, u,x)=f$ provided the coefficients and the right-hand side terms are H\"older small perturbations. As an application, we establish a partial regularity result for the sigma-$k$ Hessian equation $\sigma_{k}(D^2u)=f$.