Denseness of domains of differential operators in Sobolev spaces (Q2778490)

Let \(G\) be a bounded domain of \(\mathbb R^r\) of class \(C^l\) (\(l\geq 1)\). Consider the operators NEWLINE\[NEWLINE L_{\nu}u=\sum\limits_{\left|\alpha\right|=m_{\nu}} b_{\nu \alpha}(x')D^{\alpha}u(x') + \sum\limits_{p=0}^{m_{\nu}-1}K_{\nu p}\frac{\partial^pu(x')}{\partial n^p},\;x'\in \partial G,\;\nu=1,\dots ,m. NEWLINE\]NEWLINE Say that \(L_{\nu}\) is normal if \(m_j\neq m_k\) whenever \(j\neq k\) and, for any vector \(\sigma\) normal to \(\partial G\) at \(x' \in \partial G\), NEWLINE\[NEWLINE L_{\nu 0}(x',\sigma)=\sum\limits_{\left|\alpha\right|=m_{\nu}} b_{\nu \alpha}(x')\sigma^{\alpha}\neq 0,\;\nu=1,\dots ,m, NEWLINE\]NEWLINE and the operator \(K_{\nu p}\) is compact from \(W^{m_{\nu}-p,q}(\partial G)\) into \(L^q(\partial G)\), \(1<q<+\infty\). NEWLINENEWLINENEWLINEAssume that \(b_{\nu \alpha}\in C^{l-m_{\nu}}(\overline{G})\), that the operators \(K_{\nu p}\) are compact from \(W^{m_{\nu}-p,q}(\partial G)\) into \(L^q(\partial G)\) and from \(W^{l-p,q}(\partial G)\) into \(W^{l-m_{\nu},q}(\partial G)\), with \(l\geq \max (m_{\nu})+1\). Assume furthermore that \(L_{\nu}\) is normal. Then, for all integer \(0\leq k\leq l\), NEWLINE\[NEWLINE \left. \overline{W^{l,q}\left(G;L_{\nu}u=0,\nu=1,\dots ,m\right)}\right|_{W^{k,q}(G)}=W^{k,q}\left(G;L_{\nu}=0,m_{\nu}\leq k-1\right). NEWLINE\]NEWLINE Similar and related results were presented in [\textit{S.} and \textit{Ya. Yakubov}, ``Differential-operator equations. ordinary and partial differential equations'', Boca Raton (2000; Zbl 0936.35002), p. 568, Theorem 3.4.2/1].