Inequalities and Separation for a Biharmonic Laplace-Beltrami Differential Operator in a Hilbert Space Associated with the Existence and Uniqueness Theorem
Year: 2016
Author: Elsayed M. E. Zayed
Journal of Partial Differential Equations, Vol. 29 (2016), Iss. 1 : pp. 59–70
Abstract
In this paper, we have studied the separation for the biharmonic Laplace-Beltrami differential operator
Au(x)=−ΔΔu(x)+V(x)u(x),
for all x∈Rn, in the Hilbert space H=L2(Rn,H1) with the operator potential V(x)∈C1(Rn,L(H1)), where L(H1) is the space of all bounded linear operators on the Hilbert space H1, while ΔΔu\ is the biharmonic differential operator and
Δu=−n∑i,j=11√detg∂∂xi[√detgg−1(x)∂u∂xj]
is the Laplace-Beltrami differential operator in Rn. Here g(x)=(gij(x)) is the Riemannian matrix, while g−1(x) is the inverse of the matrix g(x). Moreover, we have studied the existence and uniqueness Theorem for the solution of the non-homogeneous biharmonic Laplace-Beltrami differential equation Au=−ΔΔu+V(x)u(x)=f(x) in the Hilbert space H where f(x)∈H as an application of the separation approach.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/jpde.v29.n1.6
Journal of Partial Differential Equations, Vol. 29 (2016), Iss. 1 : pp. 59–70
Published online: 2016-01
AMS Subject Headings:
Copyright: COPYRIGHT: © Global Science Press
Pages: 12
Keywords: Separation biharmonic Laplace-Beltrami operator