Separation of the general second order elliptic differential operator with an operator potential in the weighted Hilbert spaces

doi:10.1016/j.amc.2003.12.091

Applied Mathematics and Computation

Volume 162, Issue 1, 4 March 2005, Pages 155-163

https://doi.org/10.1016/j.amc.2003.12.091 Get rights and content

Abstract

The purpose of this paper is to study the separation for the general second order elliptic differential operator $G=G_{0} +V(x),x∈R^{n},$ in the weighted Hilbert space H=L_2,k(Rⁿ,H₁), where G₀=−∑_i,j=1ⁿa_ij(x)D_i^j is the differential operator with the real positive coefficients a_ij(x)∈C²(Rⁿ) and $D_{i}^{j} = ∂^{2} ∂ x_{i} ∂ x_{j},i,j=1,…,n$ . The operator potential V(x)∈C¹(Rⁿ,L(H₁)), where L(H₁) is the space of all bounded linear operators on the arbitrary Hilbert space H₁. Moreover, we study the existence and uniqueness of the solution of the second order differential equation −∑_i,j=1ⁿa_ij(x)D_i^ju(x)+V(x)u(x)=f(x), where f(x)∈H, in the weighted Hilbert space H=L_2,k(Rⁿ,H₁).

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: $P[y]=y^{″} (x)+q(x)y(x)$ 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 operator $S[u]=− Δ u(x)+q(x)u(x), x∈R^{n},$ is separated in the space $L_{p} (Ω)^{ℓ}$ , 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=L_2,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=L_2,k(Rⁿ,H₁) of all vector functions u(x), x∈Rⁿ, equipped with the norm $∥u∥_{k} = ∫_{R^{n}} k^{2} (x)∥u(x)∥_{1}^{2} d x^{12} .$ The symbol 〈u,v〉_k where u, v∈H denotes the scalar product in H which is defined by 〈u,v〉_k=〈ku,kv〉 where $〈u,v〉=∫_{R^{n}} 〈u(x),v(x)〉_{1} d x, x∈R^{n},$ is the scalar

Discussion of the results

Definition 3.1

The differential operator $Gu(x)=G_{0} u(x)+V(x)u(x), x∈R^{n}$ is said to be separated in the weighted Hilbert space H if the following statement holds: If u(x)∈H∩W_2,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: $a_{ij}^{12} (x)V_{0}^{−12} (x)k^{−1} (x)D_{i} k(x) ⩽ 12 σ_{1},$ $a_{ij}^{12} (x)V_{0}^{−12} (x)V^{−1} (x)D_{i} V(x) ⩽σ_{2}$ and $a_{ij}^{−12} (x)V_{0}^{−12} (x)D_{i} a_{ij} ⩽σ_{3}, i,j=1,…,n,$

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.

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Separation of the general second order elliptic differential operator with an operator potential in the weighted Hilbert spaces

Abstract

Introduction

Section snippets

Notations

Discussion of the results

Acknowledgements

Two separation criteria for second order ordinary or partial differential operators

Math. Bohemica

Some separation criteria and inequalities associated with linear second order differential operators

Some properties of the domains of certain differential operators

Proc. London Math. Soc

Some inequalities associated with certain differential operators

Math. Z

On some properties of the powers of a familly self-adjoint differential expressions

Proc. London Math. Soc