Separation of the general second order elliptic differential operator with an operator potential in the weighted Hilbert spaces
Introduction
The separation of differential expressions and its fundamental results have been obtained by Everitt and Giertz [5], [6], [7], [8]. They have obtained in [5] the separation results for the Sturm–Liouville differential expression:in L2(−∞,∞). They studied the following question: What are the conditions on q(x) such that if y(x)∈L2(−∞,∞), −y″(x)+q(x)y(x)∈L2(−∞,∞) imply both of y″(x) and q(x)y(x)∈L2(−∞,∞)?
A number of results concerning a property referred to as separation of differential expressions have been discussed by Biomatov [2], Otelbaev [13], Zettle [14] and Mohamed [9], [10], [11], [12].
Separation for the differential expressions with the matrix coefficients was first examined by Bergbaev [1]. He has obtained the conditions on q(x) in order that the Schrodinger operatoris separated in the space , where Δ is the Laplace operator in Rn and q(x) is an λ×λ positive hermitian matrix.
Some separation criteria and inequalities associated with linear second order differential operators have been studied by Brown et al. [3], [4].
In this paper, we obtain new results, namely Theorem 3.2, Theorem 3.3 on the separation of the general second order elliptic differential operator G=G0+V(x),x∈Rn in the weighted Hilbert space H=L2,k(Rn,H1) when the potential V(x) is a bounded linear operator on the arbitrary Hilbert space H1, which have not been discussed elsewhere.
Section snippets
Notations
In the following we introduce the terminology that will be used in the subsequent sections:
For some positive weight function k∈C1(Rn) and arbitrary separable Hilbert space H1, with norm ∥·∥1 and scalar product 〈,〉1 we introduce the weighted Hilbert space H=L2,k(Rn,H1) of all vector functions u(x), x∈Rn, equipped with the normThe symbol 〈u,v〉k where u, v∈H denotes the scalar product in H which is defined by 〈u,v〉k=〈ku,kv〉 whereis the scalar
Discussion of the results
Definition 3.1 The differential operatoris said to be separated in the weighted Hilbert space H if the following statement holds: If u(x)∈H∩W2,loc2(Rn,H1) and Gu(x)∈H, implies G0u(x) and V(x)u(x)∈H. For some implications of this definition see [5], [6].
The main results of our paper can be formulated in the following theorems: Theorem 3.2 If the following conditions are satisfied for all x∈Rn:and
Acknowledgements
The authors would like to thank Professor E.M.E. Zayed, head of Mathematics Department, Zagazig University, Egypt for his revision of this paper and his interesting suggestions. They also, would like to thank the referees for their suggestions and comments on this paper to be in a good final form.
References (14)
- A. Bergbaev, Smooth solution of non-linear differential equation with matrix potential, the VIII scientific conference...
- K.Kh. Biomatov, Coercive estimates and separation for second order elliptic differential equations, Sov. Math. Dokl. 38...
- et al.
Two separation criteria for second order ordinary or partial differential operators
Math. Bohemica
(1999) - et al.
Some separation criteria and inequalities associated with linear second order differential operators
- et al.
Some properties of the domains of certain differential operators
Proc. London Math. Soc
(1971) - et al.
Some inequalities associated with certain differential operators
Math. Z
(1972) - et al.
On some properties of the powers of a familly self-adjoint differential expressions
Proc. London Math. Soc
(1972)
Cited by (11)
Separation for the biharmonic differential operator in the Hilbert space associated with the existence and uniqueness theorem
2008, Journal of Mathematical Analysis and ApplicationsInequalities and separation for the Laplace-Beltrami differential operator in Hilbert spaces
2007, Journal of Mathematical Analysis and ApplicationsSolution estimates for one class of elliptic and parabolic nonlinear equations
2023, Complex Variables and Elliptic EquationsCoercive estimates and separation for partial differential operators in hilbert spaces associated with the existence and uniqueness theorem
2017, Dynamics of Continuous, Discrete and Impulsive Systems Series A: Mathematical AnalysisInequalities and separation for a biharmonic elliptic partial differential equation in the weighted hilbert space with an application
2015, Panamerican Mathematical Journal