Boundary behavior of superharmonic functions satisfying nonlinear inequalities in uniform domains

Hirata, Kentaro

doi:10.1090/S0002-9947-2011-05071-3

Boundary behavior of superharmonic functions satisfying nonlinear inequalities in uniform domains
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by Kentaro Hirata

Trans. Amer. Math. Soc. 363 (2011), 4007-4025

DOI: https://doi.org/10.1090/S0002-9947-2011-05071-3

Published electronically: March 10, 2011

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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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