Journal of the Mathematical Society of Japan
- J. Math. Soc. Japan
- Volume 62, Number 4 (2010), 1043-1068.
Properties of superharmonic functions satisfying nonlinear inequalities in nonsmooth domains
In a uniform domain Ω, we present a certain reverse mean value inequality and a Harnack type inequality for positive superharmonic functions satisfying a nonlinear inequality -Δu(x) ≤ cδΩ(x)-αu(x)p for x ∈ Ω, where c > 0, α ≥ 0 and p > 1 and δΩ(x) is the distance from a point x to the boundary of Ω. These are established by refining a boundary growth estimate obtained in our previous paper (2008). Also, we apply them to show the existence of nontangential limits of quotients of such functions and to give an extension of a certain minimum principle studied by Dahlberg (1976).
J. Math. Soc. Japan, Volume 62, Number 4 (2010), 1043-1068.
First available in Project Euclid: 2 November 2010
Permanent link to this document
Digital Object Identifier
Mathematical Reviews number (MathSciNet)
Zentralblatt MATH identifier
Primary: 31B05: Harmonic, subharmonic, superharmonic functions
Secondary: 31B25: Boundary behavior 31C45: Other generalizations (nonlinear potential theory, etc.) 35J60: Nonlinear elliptic equations
HIRATA, Kentaro. Properties of superharmonic functions satisfying nonlinear inequalities in nonsmooth domains. J. Math. Soc. Japan 62 (2010), no. 4, 1043--1068. doi:10.2969/jmsj/06241043. https://projecteuclid.org/euclid.jmsj/1288703096