Separation of bi-harmonic differential operators on Riemannian manifolds

Abstract

Abstract. Consider the bi-harmonic differential expression of the form A = ▵ ▵ + q ${A=\triangle \triangle +q}$ on a complete Riemannian manifold (M,g) with metric g , ${g,}$ where ▵ ${\triangle }$ is the Laplacian on M and q ≥ 0 ${q\ge 0}$ 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 L 2 ( M ) ${L^{2}( M) }$ if for all u ∈ L 2 ( M ) ${u\in L^{2}( M) }$ such that A u ∈ L 2 ( M ) ${Au\in L^{2}( M) }$ , we have q u ∈ L 2 ( M ) ${ qu\in L^{2}( M)}$ . In this paper we give sufficient conditions for A to be separated in L 2 ( M ) ${L^{2}( M)}$ .