Remarks on regularity of viscosity solutions for fully nonlinear uniformly elliptic PDEs with measurable ingredients

We study the comparison principle and interior Hölder continuity of viscosity solutions of F(x, u(x),Du(x),D²u(x)) + H(x,Du(x))-f(x) = 0 in O, where F satisfies the standard "structure condition" and H has superlinear growth with respect to Du. Following Caffarelli, Crandall, Kocan and Świȩch [3], we first present the comparison principle between L^p-viscosity subsolution and L^p-strong supersolutions. We next show the interior Hölder continuity for L^p-viscosity solutions of the above equation. For this purpose, modifying some arguments in [1] by Caffarelli, we obtain the Harnack inequality for them when the growth order of H with respect to Du is less than 2.