On L^p-solutions to the Laplace equation and zeros of holomorphic functions
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Cita com:
hdl:2117/1224
Tipus de documentArticle
Data publicació1996
Condicions d'accésAccés obert
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Reconeixement-NoComercial-SenseObraDerivada 2.5 Espanya
Abstract
The problem we solve in this paper is to characterize, in a smooth domain
$\Omega$ in $\Bbb R^n$ and for $1\le p\le\infty$, those positive Borel
measures on $\Omega$ for which there exists a subharmonic function $u\in
L^p(\Omega)$ such that $\Delta u=\mu$.
The motivation for this question is mainly for $n=2$, in which case it is
related with problems about distributions of zeros of holomorphic
functions: If $\{a_n\}^{\infty}_{n=1}$ is a sequence in $\Omega\subset\Bbb
C$ with no accumulation points in a simply connected domain $\Omega$, and
$\mu=2\pi\sum_n\delta_{a_n}$, then all solutions $u$ of $\Delta u=\mu$ are
of the form $u=\log |f|$, with $f$ holomorphic vanishing exactly on the
poits $a_n$. Thus our results give the characterization of the zero
sequences of holomorphic functions with $\log |f|\in L^p(\Omega)$. A
related class had been considered by Beller.
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