Optimal hardy weight for second-order elliptic operator: An answer to a problem of Agmon
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Date
2014-04-01
Publication Type
Journal Article
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Abstract
For a general subcritical second-order elliptic operator P in a domain Ω⊂Rn (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⁻¹P 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. © 2014 Elsevier Inc.
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published
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Journal / series
Journal of Functional Analysis
Volume
266 (7)
Pages / Article No.
4422 - 4489
Publisher
Elsevier
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Methods
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Subject
Agmon metric; Ground state; Hardy inequality; Logarithmic Caccioppoli inequality; Minimal growth; Positive solutions; Rellich inequality; Weighted Poincaré inequality
Organisational unit
03355 - Graf, Gian Michele / Graf, Gian Michele