Journal of the Mathematical Society of Japan

Properties of superharmonic functions satisfying nonlinear inequalities in nonsmooth domains

Kentaro HIRATA

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Abstract

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)u(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).

Article information

Source
J. Math. Soc. Japan, Volume 62, Number 4 (2010), 1043-1068.

Dates
First available in Project Euclid: 2 November 2010

Permanent link to this document
https://projecteuclid.org/euclid.jmsj/1288703096

Digital Object Identifier
doi:10.2969/jmsj/06241043

Mathematical Reviews number (MathSciNet)
MR2761913

Zentralblatt MATH identifier
1210.31002

Subjects
Primary: 31B05: Harmonic, subharmonic, superharmonic functions
Secondary: 31B25: Boundary behavior 31C45: Other generalizations (nonlinear potential theory, etc.) 35J60: Nonlinear elliptic equations

Keywords
boundary growth nontangential limit reverse mean value inequality Harnack type inequality convergence property superharmonic function semilinear elliptic equation uniform domain

Citation

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


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