Baptiste Devyver

OPTIMAL HARDY WEIGHT FOR SECOND-ORDER ELLIPTIC OPERATOR: AN ANSWER TO A PROBLEM OF AGMON

Abstract 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 − λ W 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 P u = 0 , the existence of which depends on the subcriticality of P in Ω.

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