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Boundary behavior of superharmonic functions satisfying nonlinear inequalities in uniform domains


Author: Kentaro Hirata
Journal: Trans. Amer. Math. Soc. 363 (2011), 4007-4025
MSC (2010): Primary 31B25; Secondary 31B05, 31A05, 31A20, 31C45, 35J61
DOI: https://doi.org/10.1090/S0002-9947-2011-05071-3
Published electronically: March 10, 2011
MathSciNet review: 2792977
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Abstract: In a uniform domain $ \Omega$, we investigate the boundary behavior of positive superharmonic functions $ u$ satisfying the nonlinear inequality

$\displaystyle -\Delta u(x) \le c\delta_\Omega(x)^{-\alpha}u(x)^p$   for a.e. $\displaystyle x\in\Omega $

with some constants $ c>0$, $ \alpha\in\mathbb{R}$ and $ p>0$, where $ \Delta$ is the Laplacian and $ \delta_\Omega(x)$ is the distance from a point $ x$ to the boundary of $ \Omega$. In particular, we present a Fatou type theorem concerning the existence of nontangential limits and a Littlewood type theorem concerning the nonexistence of tangential limits.


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Additional Information

Kentaro Hirata
Affiliation: Faculty of Education and Human Studies, Akita University, Akita 010-8502, Japan
Email: hirata@math.akita-u.ac.jp

DOI: https://doi.org/10.1090/S0002-9947-2011-05071-3
Keywords: Nontangential limit, minimal fine limit, superharmonic function, nonlinear elliptic equation, uniform domain
Received by editor(s): March 20, 2008
Received by editor(s) in revised form: April 1, 2009
Published electronically: March 10, 2011
Additional Notes: This work was partially supported by Grant-in-Aid for Young Scientists (B) (No. 19740062), Japan Society for the Promotion of Science.
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

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