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Ricerche di Matematica

, Volume 67, Issue 2, pp 285–320 | Cite as

Dirichlet and Neumann eigenvalue problems on CR manifolds

  • Amine Aribi
  • Sorin DragomirEmail author
Article
  • 68 Downloads

Abstract

We study the properties of Carnot–Carathéodory spaces attached to a strictly pseudoconvex CR manifold M, in a neighborhood of each point \(x \in M\), versus the pseudohermitian geometry of M arising from a fixed positively oriented contact form \(\theta \) on M. The weak Dirichlet problem for the sublaplacian \(\Delta _b\) on \((M, \theta )\) is solved on domains \(\Omega \subset M\) supporting the Poincaré inequality. The solution to Neumann problem for the sublaplacian \(\Delta _b\) on a \(C^{1,1}\) connected \((\epsilon , \delta )\)-domain \(\Omega \subset {{\mathbb {G}}}\) in a Carnot group (due to Danielli et al. in: Memoirs of American Mathematical Society 2006) is revisited for domains in a CR manifold. As an application we prove discreetness of the Dirichlet and Neumann spectra of \(\Delta _b\) on \(\Omega \subset M\) in a Carnot–Carthéodory complete pseudohermitian manifold \((M, \theta )\).

Keywords

CR manifold Hörmander system Carnot–Carathéodory metric Sublaplacian 

Mathematics Subject Classification

32V20 35H20 53C99 

Notes

Acknowledgements

The second named author acknowledges the influence on the present work of the thought line of the school of PDEs in University of Bologna and is especially grateful to Ermanno Lanconelli, Nicola Garofalo and Giovanna Citti for patiently explaining to him their results.

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Copyright information

© Università degli Studi di Napoli "Federico II" 2018

Authors and Affiliations

  1. 1.Laboratoire de Mathématiques et Physique ThéoriqueUniversité François RabelaisToursFrance
  2. 2.Dipartimento di Matematica e InformaticaUniversità degli Studi della BasilicataPotenzaItaly

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