On Liouville-type theorems and the uniqueness of the positive Cauchy problem for a class of hypoelliptic operators

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• 作者：Alessia E. Kogoj ; Yehuda Pinchover ; Sergio Polidoro
• 关键词：Mathematics Subject Classification35K70 ; 35B09 ; 35B53 ; 35K15 ; 35K65
• 刊名：Journal of Evolution Equations
• 出版年：2016
• 出版时间：December 2016
• 年：2016
• 卷：16
• 期：4
• 页码：905-943
• 全文大小：
• 刊物类别：Mathematics and Statistics
• 刊物主题：Analysis;
• 出版者：Springer International Publishing
• ISSN：1424-3202
• 卷排序：16

文摘

The paper contains a representation formula for positive solutions of linear degenerate second-order equations of the form$$\partial_t u (x,t) = \sum_{j=1}^m X_j^2 u(x,t) + X_0 u(x,t) \quad (x,t)\in \mathbb{R}^N \times\, ]- \infty ,T[,$$where the X_j are smooth vector fields satisfying the Hörmander condition. It is assumed that X_j are invariant under left translations of a Lie group and the corresponding paths satisfy a local admissibility criterion. The representation formula is established by an analytic approach based on Choquet theory. As a consequence we obtain Liouville-type theorems and uniqueness results for the positive Cauchy problem.Mathematics Subject Classification35K7035B0935B5335K1535K65KeywordsHarnack inequalityHypoelliptic operatorsPositive Cauchy problemLiouville-type theoremsUltraparabolic operatorsReferences1.A. A. Agrachev and Y. L. 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Soc. 55, 85–95 (1944)MathSciNetCrossRefMATHGoogle ScholarCopyright information© Springer International Publishing 2016Authors and AffiliationsAlessia E. Kogoj1Email authorYehuda Pinchover2Sergio Polidoro31.Dipartimento di Ingegneria dell’Informazione, Ingegneria Elettrica e Matematica ApplicataUniversità degli Studi di SalernoFiscianoItaly2.Department of MathematicsTechnion - Israel Institute of TechnologyHaifaIsrael3.Dipartimento di Scienze Fisiche, Informatiche e MatematicheUniversità di Modena e Reggio EmiliaModenaItaly About this article CrossMark Publisher Name Springer International Publishing Print ISSN 1424-3199 Online ISSN 1424-3202 About this journal Reprints and Permissions Article actions .buybox { margin: 16px 0 0; position: relative; } .buybox { font-family: Source Sans Pro, Helvetica, Arial, sans-serif; font-size: 14px; font-size: .875rem; } .buybox { zoom: 1; } .buybox:after, .buybox:before { content: ''; display: table; } .buybox:after { clear: both; } /*---------------------------------*/ .buybox .buybox__header { border: 1px solid #b3b3b3; 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