Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
4590132 | Journal of Functional Analysis | 2014 | 68 Pages |
For a general subcritical second-order elliptic operator P in a domain Ω⊂RnΩ⊂Rn (or noncompact manifold), we construct Hardy-weight W which is optimal in the following sense. The operator P−λWP−λW is subcritical in Ω for all λ<1λ<1, null-critical in Ω for λ=1λ=1, and supercritical near any neighborhood of infinity in Ω for any λ>1λ>1. Moreover, if P is symmetric and W>0W>0, then the spectrum and the essential spectrum of W−1PW−1P are equal to [1,∞)[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=0Pu=0, the existence of which depends on the subcriticality of P in Ω.