Advances in Differential Equations

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

Shigeaki Koike and Toshimi Takahashi

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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

Permanent link to this document
https://projecteuclid.org/euclid.ade/1356651805

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.


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