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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Golubev series for solutions of elliptic equations


Authors: Ch. Dorschfeldt and N. N. Tarkhanov
Journal: Trans. Amer. Math. Soc. 351 (1999), 581-594
MSC (1991): Primary 35A20, 35C10
MathSciNet review: 1433116
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Abstract: Let $P$ be an elliptic system with real analytic coefficients on an open set $X\subset {\Bbb R}^{n},$ and let $\Phi$ be a fundamental solution of $P.$ Given a locally connected closed set $\sigma \subset X,$ we fix some massive measure $m$ on $\sigma$. Here, a non-negative measure $m$ is called massive, if the conditions $s \subset \sigma $ and $m(s)=0$ imply that $\overline{\sigma \setminus s} = \sigma .$ We prove that, if $f$ is a solution of the equation $Pf =0$ in $X \setminus \sigma ,$ then for each relatively compact open subset $U$ of $X$ and every $1<p<\infty$ there exist a solution $f_{e} $ of the equation in $U$ and a sequence $f_{\alpha }$ ($\alpha \in {\Bbb N}^{n}_{0} $) in $L^{p} (\sigma \cap U, m) $ satisfying $\| \alpha ! f_{\alpha } \|^{1/|\alpha|}_{L^{p} (\sigma \cap U,m)} \rightarrow 0$ such that $f(x) = f_{e} (x) +\sum _{\alpha}\int _{\sigma \cap U} D^{\alpha }_{y} \Phi (x,y) f_{\alpha } (y) dm(y)$ for $x \in U \setminus \sigma .$ This complements an earlier result of the second author on representation of solutions outside a compact subset of $X.$


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Additional Information

Ch. Dorschfeldt
Email: christoph@mpg-ana.uni-potsdam.de

N. N. Tarkhanov
Email: tarkhan@mpg-ana.uni-potsdam.de

DOI: http://dx.doi.org/10.1090/S0002-9947-99-01988-1
PII: S 0002-9947(99)01988-1
Keywords: Solutions with singularities, real analytic coefficients, elliptic systems, Golubev series
Received by editor(s): February 15, 1995
Received by editor(s) in revised form: November 20, 1996
Additional Notes: This research was supported in part by the Alexander von Humboldt Foundation
Article copyright: © Copyright 1999 American Mathematical Society