Separation of bi-harmonic differential operators on Riemannian manifolds

摘要

Consider the bi-harmonic differential expression of the form A = ΔΔ + q on a complete Riemannian manifold (M, g) with metric g, where Δ is the Laplacian on M and q ≥ 0 is a locally square integrable function on M. In the terminology of Everitt and Giertz, the differential expression A is said to be separated in L2(M) if for all u ∈ L2(M) such that Au e L2(M),we have qu ∈ L2(M). In this paper we give sufficient conditions for A to be separated in L2(M).