Golubev series for solutions of elliptic equations

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 $X.$