## Journal of the Mathematical Society of Japan

### Weak Harnack inequality for fully nonlinear uniformly elliptic PDE with unbounded ingredients

#### Abstract

The weak Harnack inequality for $L^{p}$-viscosity solutions is shown for fully nonlinear, second order uniformly elliptic partial differential equations with unbounded coefficients and inhomogeneous terms. This result extends those of Trudinger for strong solutions [21] and Fok for $L^{p}$-viscosity solutions [13]. The proof is a modification of that of Caffarelli [5], [6]. We apply the weak Harnack inequality to obtain the strong maximum principle, boundary weak Harnack inequality, global $C^{\alpha}$ estimates for solutions of fully nonlinear equations, strong solvability of extremal equations with unbounded coefficients, and Aleksandrov-Bakelman-Pucci maximum principle in unbounded domains.

#### Article information

Source
J. Math. Soc. Japan, Volume 61, Number 3 (2009), 723-755.

Dates
First available in Project Euclid: 30 July 2009

https://projecteuclid.org/euclid.jmsj/1248961477

Digital Object Identifier
doi:10.2969/jmsj/06130723

Mathematical Reviews number (MathSciNet)
MR2552914

Zentralblatt MATH identifier
1228.35104

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

#### Citation

KOIKE, Shigeaki; ŚWIĘCH, Andrzej. Weak Harnack inequality for fully nonlinear uniformly elliptic PDE with unbounded ingredients. J. Math. Soc. Japan 61 (2009), no. 3, 723--755. doi:10.2969/jmsj/06130723. https://projecteuclid.org/euclid.jmsj/1248961477

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