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

By Baptiste Devyver, M. Fraas, Yehuda Pinchover. 2012

144 citations recorded by the source corpus.

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.

Subfields: Applied Mathematics

View Paper: https://www.semanticscholar.org/paper/fad110dcd82e05e91f05caa384b3cb0310d67b18

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In Applied Mathematics

1. D. Ganguly
2. Elvise Berchio
3. G. Grillo
4. Juan Luis V'azquez
5. Qiaohua Yang
6. Guozhen Lu
7. M. Porzio

Citations in the corpus

• Sharp Poincar\'e-Hardy and Poincar\'e-Rellich inequalities on the hyperbolic space (2015, methodology): “[36] Y. Pinchover, K. Tintarev, A ground state alternative for singular Schrdinger operators, J. Funct.”