Approximation durch Lösungen elliptischer Randwertprobleme auf offenen Mengen. I. (Approximation by solutions of elliptic boundary value problems on open sets. I.) (Q2276664)

Summary: Let \(\Omega \subset {\mathbb{R}}^ n\) be a bounded, smooth domain, \(\Omega_ 1,{\bar \Omega}_ 1\subset \Omega\), an arbitrary open set and L an elliptic differential operator of order 2m on \(\Omega\). It is proved that every function v from the Sobolev space \(W^ k_ p(\Omega_ 1)\) \((- \infty <k\leq 0,1<p<\infty)\) with \(Lv=0\) in \(\Omega_ 1\) can be approximated in the \(W^ k_ p(\Omega_ 1)\)-norm by solutions of elliptic boundary value problems with respect to \(\Omega\), if \(C_ 0^{\infty}(\Omega_ 1)\) is dense in \(\{f\in W_{p'}^{2m- k}({\mathbb{R}}^ n):\) supp \(f\subseteq {\bar \Omega}_ 1\}\) \((p'=p/(p-1))\). For \(k\geq 1\) this assertion is true for those \(v\in W^ k_ p(\Omega_ 1)\), which have an extension to a function from \(W^ k_ p({\mathbb{R}}^ n)\).