Separation of the two dimensional Grushin operator by the disconjugacy property

Abstract

In this work we have introduced a new proof of the separation of the Grushin differential operator of the form

in the Hilbert space H = L ²(Ω), with potential q(x, y)∈C ¹(Ω), 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(Ω) ⇒ qu∈L ²(Ω), Ω∈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 G − λq ² for some λ > 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].

Keywords:

• separation
• Grushin differential operator
• Disconjugacy
• Hilbert space

AMS Subject Classifications:

• 47F05
• 58J99