We consider a function which is a viscosity solution of a uniformly elliptic equation only at those points where the gradient is large. We prove that the Hölder estimates and the Harnack inequality, as in the theory of Krylov and Safonov, apply to these functions.
R, we considerThe main theorem of this paper is the following Hölder estimate.Theorem 1.1 (Hölder estimate). For any continuous function u :where C depends on λ, Λ, dimension and γ/C 0 and α depends on λ, Λ and dimension.Remark 1.2. The constant C in Theorem 1.1 grows like (γ/C 0 ) α as γ/C 0 tends to +∞. That isNote that when γ = 0, then the constant C becomes independent of C 0 and we recover the classical estimate for uniformly elliptic equations.Our second main result is the following Harnack inequality.