On a selfadjoint realization of curl in exterior domains
Many applications require to have a domain for the operator curl where it becomes selfadjoint. Its initial, natural domain is the linear space of smooth vector fields compactly supported in an open subset \(\Omega\) of \(\mathbb{R}^m\), and thus the question is how to construct its adequate extension. General aspects of the problem are analyzed in the Introduction. In Section 1, various Hilbert spaces are defined which are needed to precisely formulate the results, and certain results which are valid for any open subset, are collected there. How to ensure the selfadjointness of curl for the case of exterior domains, is decribed in Section 2: the author proves that it holds under a certain ``General assumption'' (as he calls it) which says that some sets generated by the domains of curl and div are compactly imbedded into \(L_2\). In Section 3 some consequential spectral results are presented including the so-called limiting absorption principle.
scientific article; zbMATH DE number 1268069