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

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• 作者：Baptiste Devyver^a ; ^{baptiste.devyver@univ-nantes.fr" class="auth_mail} ; Martin Fraas^b ; ^{martin.fraas@gmail.com" class="auth_mail} ; Yehuda Pinchover^a ; ^{pincho@techunix.technion.ac.il" class="auth_mail}
• 关键词：Agmon metric ; Ground state ; Hardy inequality ; Logarithmic Caccioppoli inequality ; Minimal growth ; Positive solutions ; Rellich inequality ; Weighted Poincaré ; inequality
• 刊名：Journal of Functional Analysis
• 出版年：1 April, 2014
• 年：2014
• 卷：266
• 期：7
• 页码：4422-4489
• 全文大小：693 K

文摘

For a general subcritical second-order elliptic operator P in a domain (or noncompact manifold), we construct Hardy-weight W which is optimal in the following sense. The operator is subcritical in 惟 for all , null-critical in 惟 for , and supercritical near any neighborhood of infinity in 惟 for any . Moreover, if P is symmetric and , then the spectrum and the essential spectrum of are equal to , 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 , the existence of which depends on the subcriticality of P in 惟.