Approximation by linear combinations of fundamental solutions of elliptic systems of partial differential operators
The author considers the elliptic system \(L(D) = (L_{ij}(D))_{i,j = 1,\dots,N}\) of partial differential operators of order \(2m\). Let \(E\) be a fundamental solution of \(L\) and \(E_ j\) the column vectors of \(E\). Let \(\Gamma\) be the smooth boundary of a bounded domain \(\Omega \subset \mathbb{R}^ n\) and \(b(x,D) = (b_{hj}(x,D): h = 1,\dots,m; j=1,\dots,N)\) a normal system of boundary operators on \(\Gamma\). Let \(b_ h u =\sum_{1 \leq j \leq N}\) \(b_{hj}u_ j\) \((u = (u_ 1,\dots,u_ N)^ T)\). Furthermore let \((y_ k)^ \infty_ 1 \subset \mathbb{R}^ n\setminus \overline{\Omega}\) be a sequence of points and \(D_ h(\Gamma)\) \((h = 1,\dots,m)\) suitable function spaces over \(\Gamma\) (e.g. \(C^ s\)-spaces or Sobolev spaces). It is investigated, under which conditions on the sequence \((y_ k)^ \infty_ 1\) the set \(\text{span}\{(b_ 1 D^ \alpha E_ j(x-y_ k)|_{x\in \Gamma}, \dots, b_ m D^ \alpha E_ j(x-y_ k)|_{x\in \Gamma}): k\in \mathbb{N}, | \alpha | \geq 0; j = 1,\dots,N\}\) is dense in \(\prod_{1 \leq h \leq m} D_ h(\Gamma)\). Previously similar results for Lamé systems were obtained by the author [Math. Nachr. 164, 271-281 (1993)] and for elliptic operators with variable coefficients by I. Ya. Roitberg and the reviewer [Dokl. Akad. Nauk Ukr. 12, 15-20 (1992)].
scientific article; zbMATH DE number 549186