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Golubev series for solutions of elliptic equations

Authors:	Ch Dorschfeldt N N Tarkhanov

Institution:	Max-Planck-Arbeitsgruppe,"Partielle Differentialgleichungen und Komplexe Analysis", Universität Potsdam, Am Neuen Palais 10, D - 14415, Germany ; Max-Planck-Arbeitsgruppe,"Partielle Differentialgleichungen und Komplexe Analysis", Universität Potsdam, Am Neuen Palais 10, D - 14415, Germany

Abstract:	Let 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 Given a locally connected closed set $\sigma \subset X,$ we fix some massive measure on $\sigma$ . Here, a non-negative measure is called massive, if the conditions $s \subset \sigma$ and imply that $\overline{\sigma \setminus s} = \sigma .$ We prove that, if is a solution of the equation in $X \setminus \sigma ,$ then for each relatively compact open subset of and every $1<p<\infty$ there exist a solution $f_{e}$ of the equation in 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

Keywords:	Solutions with singularities real analytic coefficients elliptic systems Golubev series

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