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

Abstract

Consider the bi-harmonic differential expression of the form A = 2 M + 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 < ∞.

Essential self-adjointness for covariant tri-harmonic operators on manifolds and the separation problem

“…. Atia was studied the separation problem of bi-harmonic differential operators on Riemannian manifolds in [3,4].…”

“…Gaffney studied essential self-adjointness for differential operators on Riemannian manifolds in [15]. This problem has lead to many works, such as [4,5,10,17,18,21,25]. The study of the separation property for Schrodinger operators on R n was studied through Everitt and Giertz, see [15].…”

Essential Self-adjointness of Differential Operators on Riemannian Manifolds

“…For a study of separation in the context of a perturbation of the magnetic Bi-Laplacian on L 2 (M ), see the paper [1]. Atia studied the separation problem on Riemannian manifolds in [2] and [3]. In this paper, we consider the operator, 8 + V , acting on sections of a Hermitian vector bundle E over a complete Riemannian manifold M , where = r + r denotes a Bochner Laplacian associated to a Hermitian connection r, V is a potential satis…ed that V 2 L 1 loc (End E), and satis…es a bound from below by a non-positive function depending on the distance from a point.…”