Atia, H. (2019). Separation Problem for Bi-Harmonic Differential Operators in Lp− spaces on Manifolds. test, 27(1), 1-10. doi: 10.1186/s42787-019-0029-6
H. A. Atia. "Separation Problem for Bi-Harmonic Differential Operators in Lp− spaces on Manifolds". test, 27, 1, 2019, 1-10. doi: 10.1186/s42787-019-0029-6
Atia, H. (2019). 'Separation Problem for Bi-Harmonic Differential Operators in Lp− spaces on Manifolds', test, 27(1), pp. 1-10. doi: 10.1186/s42787-019-0029-6
Atia, H. Separation Problem for Bi-Harmonic Differential Operators in Lp− spaces on Manifolds. test, 2019; 27(1): 1-10. doi: 10.1186/s42787-019-0029-6

Separation Problem for Bi-Harmonic Differential Operators in Lp− spaces on Manifolds

Journal of the Egyptian Mathematical Society
Volume 27, Issue 1, 2019, Pages 1-10    PDF (433.93 K)
DOI: 10.1186/s42787-019-0029-6
Author
H. A. Atia
Faculty of Science, Department of Mathematics, Zagazig University, Zagazig, Egypt
Abstract
Consider the bi-harmonic differential expression of the form
A = 2M2 + q
on a manifold of bounded geometry (M, g) with metric g, where M is the scalar
Laplacian on M and q ≥ 0 is a locally integrable function on M.
In the terminology of Everitt and Giertz, the differential expression A is said to be
separated in Lp (M), if for all u ∈ Lp (M) such that Au ∈ Lp (M), we have qu ∈ Lp (M). In
this paper, we give sufficient conditions for A to be separated in Lp (M),where
1 < p < ∞
Keywords
Separation problem; Bi-harmonic differential operator; Manifold
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