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

Abstract

We study the comparison principle and interior Hölder continuity of viscosity solutions of $$ F(x,u(x),Du(x),D^2u(x))+H(x,Du(x))-f(x)=0\quad\mbox{in }\Omega , $$ where $F$ satisfies the standard "structure condition'' and $H$ has superlinear growth with respect to $Du$. Following Caffarelli, Crandall, Kocan and Święch [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 $2$.