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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Boundary behavior of superharmonic functions satisfying nonlinear inequalities in uniform domains
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by Kentaro Hirata PDF
Trans. Amer. Math. Soc. 363 (2011), 4007-4025 Request permission

Abstract:

In a uniform domain $\Omega$, we investigate the boundary behavior of positive superharmonic functions $u$ satisfying the nonlinear inequality \[ -\Delta u(x) \le c\delta _\Omega (x)^{-\alpha }u(x)^p \quad \text {for a.e. } x\in \Omega \] 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
  • 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.
  • © Copyright 2011 American Mathematical Society
    The copyright for this article reverts to public domain 28 years after publication.
  • 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
  • MathSciNet review: 2792977