Separation of the two dimensional Grushin operator by the disconjugacy property

摘要

In this work we have introduced a new proof of the separation of the Grushin differential operator of the form in the Hilbert space Ha??=a??L 2(??), with potential q(x,a??y)C 1(??), by the disconjugacy property.We show that certain properties of positive solutions of the disconjugate second order differential expression G[u] imply the separation of minimal and maximal operators determined by G i.e. the property that G[u]L 2(??)a??a??quL 2(??), ??R 2, see [W.N. Everitt and M. Giertz, Some properties of the domains of certain differential operators, Proc. London Math. Soc. 23 (1971), pp. 301-324]; a property leading to a new proof and generalizing of a 1971 separation criterion due to Everitt and Giertz. A final result of this article shows that the disconjugacy of Ga??a??a????q 2 for some ??a??>a??0 implies the separation of G. This work is a generalizing of our work in [H.A. Atia and R.A. Mahmoud, Separation of two dimensional Laplace operator by disconjugacy property, Panamerican Math. J. 20(2) (2009), pp. 93-103].View full textDownload full textKeywordsseparation, Grushin differential operator, Disconjugacy, Hilbert spaceAMS Subject Classifications47F05, 58J99Related var addthis_config = { ui_cobrand: "Taylor & Francis Online", services_compact: "citeulike,netvibes,twitter,technorati,delicious,linkedin,facebook,stumbleupon,digg,google,more", pubid: "ra-4dff56cd6bb1830b" }; var addthis_config = {"data_track_addressbar":true,"ui_click":true}; Add to shortlist Link Permalink http://dx.doi.org/10.1080/00036811.2011.593173