### 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 $2$.

#### Article information

Source
Adv. Differential Equations Volume 7, Number 4 (2002), 493-512.

Dates
First available in Project Euclid: 27 December 2012

Mathematical Reviews number (MathSciNet)
MR1869522

Zentralblatt MATH identifier
1223.35156

Subjects
Primary: 35J60: Nonlinear elliptic equations
Secondary: 35D10 49L25: Viscosity solutions

#### Citation

Koike, Shigeaki; Takahashi, Toshimi. Remarks on regularity of viscosity solutions for fully nonlinear uniformly elliptic PDEs with measurable ingredients. Adv. Differential Equations 7 (2002), no. 4, 493--512. https://projecteuclid.org/euclid.ade/1356651805