Optimal hardy weight for second-order elliptic operator: An answer to a problem of Agmon

摘要

For a general subcritical second-order elliptic operator P in a domain Ω ? R~n (or noncompact manifold), we construct Hardy-weight W which is optimal in the following sense. The operator P ? λW is subcritical in Ω for all λ < 1, null-critical in Ω for λ = 1, and supercritical near any neighborhood of infinity in Ω for any λ > 1. Moreover, if P is symmetric and W > 0, then the spectrum and the essential spectrum of W~(-1) P are equal to [1, ∞), and the corresponding Agmon metric is complete. Our method is based on the theory of positive solutions and applies to both symmetric and nonsymmetric operators. The constructed Hardy-weight is given by an explicit simple formula involving two distinct positive solutions of the equation Pu = 0, the existence of which depends on the subcriticality of P in Ω.