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oai:recercat.cat:2117/12242025-07-17T02:12:50Zcom_2072_1033col_2072_452950

On L^p-solutions to the Laplace equation and zeros of holomorphic functions Bruna, Joaquim Ortega Cerdà, Joaquim Universitat Politècnica de Catalunya. Departament de Matemàtica Aplicada I Potential theory (Mathematics) Partial differential equations Laplace equation holomorphic functions zeros Potencial, Teoria del (Matemàtica) Equacions en derivades parcials Classificació AMS::31 Potential theory::31A Two-dimensional theory Classificació AMS::31 Potential theory::31B Higher-dimensional theory Classificació AMS::35 Partial differential equations::35J Partial differential equations of elliptic type 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. 1996 Article https://hdl.handle.net/2117/1224 eng http://creativecommons.org/licenses/by-nc-nd/2.5/es/ Open Access Attribution-NonCommercial-NoDerivs 2.5 Spain 20 pages application/pdf