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).
Publié le : 2010-10-15
Classification:
boundary growth,
nontangential limit,
reverse mean value inequality,
Harnack type inequality,
convergence property,
superharmonic function,
semilinear elliptic equation,
uniform domain,
31B05,
31B25,
31C45,
35J60
@article{1288703096,
author = {HIRATA, Kentaro},
title = {Properties of superharmonic functions satisfying nonlinear inequalities in nonsmooth domains},
journal = {J. Math. Soc. Japan},
volume = {62},
number = {1},
year = {2010},
pages = { 1043-1068},
language = {en},
url = {http://dml.mathdoc.fr/item/1288703096}
}
HIRATA, Kentaro. Properties of superharmonic functions satisfying nonlinear inequalities in nonsmooth domains. J. Math. Soc. Japan, Tome 62 (2010) no. 1, pp. 1043-1068. http://gdmltest.u-ga.fr/item/1288703096/