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October, 2010 Properties of superharmonic functions satisfying nonlinear inequalities in nonsmooth domains
Kentaro HIRATA
J. Math. Soc. Japan 62(4): 1043-1068 (October, 2010). DOI: 10.2969/jmsj/06241043


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).


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Kentaro HIRATA. "Properties of superharmonic functions satisfying nonlinear inequalities in nonsmooth domains." J. Math. Soc. Japan 62 (4) 1043 - 1068, October, 2010.


Published: October, 2010
First available in Project Euclid: 2 November 2010

zbMATH: 1210.31002
MathSciNet: MR2761913
Digital Object Identifier: 10.2969/jmsj/06241043

Primary: 31B05
Secondary: 31B25 , 31C45 , 35J60

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

Rights: Copyright © 2010 Mathematical Society of Japan


Vol.62 • No. 4 • October, 2010
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