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

Abstract

Abstract Consider the bi-harmonic differential expression of the form A=△M2+q $ A=\triangle _{M}^{2}+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 L p (M), if for all u∈L p (M) such that Au∈L p (M), we have qu∈L p (M). In this paper, we give sufficient conditions for A to be separated in L p (M),where 1<p<∞.

Keywords

• Separation problem
• Bi-harmonic differential operator
• Manifold