Inequalities and separation for the Laplace–Beltrami differential operator in Hilbert spaces

• Published in: Journal of mathematical analysis and applications Vol. 336; no. 1; pp. 81 - 92
• Main Authors: Zayed, E.M.E., Mohamed, A.S., Atia, H.A.
• Format: Journal Article
• Language: English
• Published: San Diego, CA Elsevier Inc 01.12.2007; Elsevier
• Subjects: Coercive estimate; Exact sciences and technology; Functional analysis; Hilbert space; Laplace–Beltrami differential operator; Mathematical analysis; Mathematics; Operator potential; Operator theory; Ordinary differential equations; Partial differential equations; Sciences and techniques of general use; Separation; Laplace–Beltrami differential operator; Coercive estimate; Operator potential; Hilbert space; Separation; Laplace operator; Differential equation; Arbitrary space; Bounded operator; Laplace equation; Linear operator; Laplace-Beltrami differential operator; Mathematical analysis; Differential operator; Application
• ISSN: 0022-247X; 1096-0813
• DOI: 10.1016/j.jmaa.2006.07.031

In this paper we have studied the separation for the Laplace–Beltrami differential operator of the form A u = − 1 det g ( x ) ∂ ∂ x i [ det g ( x ) g −1 ( x ) ∂ u ∂ x j ] + V ( x ) u ( x ) , ∀ x = ( x 1 , x 2 , … , x n ) ∈ Ω ⊂ R n , in the Hilbert space H = L 2 ( Ω , H 1 ) , with the operator potential V ( x ) ∈ C 1 ( Ω , L ( H 1 ) ) , where L ( H 1 ) is the space of all bounded linear operators on the arbitrary Hilbert space H 1 and g ( x ) = ( g i j ( x ) ) is the Riemannian matrix, while g −1 ( x ) is the inverse of the matrix g ( x ) . Also we have studied the existence and uniqueness of the solution for the Laplace–Beltrami differential equation of the form − 1 det g ( x ) ∂ ∂ x i [ det g ( x ) g −1 ( x ) ∂ u ∂ x j ] + V ( x ) u ( x ) = f ( x ) , f ( x ) ∈ H , in the Hilbert space H = L 2 ( Ω , H 1 ) .