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Lions’ Lemma, Korn’s Inequalities and the Lamé Operator on Hypersurfaces

  • Roland Duduchava
Chapter
Part of the Operator Theory: Advances and Applications book series (OT, volume 210)

Abstract

We investigate partial differential equations on hypersurfaces written in the Cartesian coordinates of the ambient space. In particular, we generalize essentially Lions’ Lemma, prove Korn’s inequality and establish the unique continuation property from the boundary for Killing’s vector fields, which are analogues of rigid motions in the Euclidean space. The obtained results, the Lax-Milgram lemma and some other results are applied to the investigation of the basic Dirichlet and Neumann boundary value problems for the Lamé equation on a hypersurface.

Mathematics Subject Classification (2000)

35J5 74J35 58J32 

Keywords

Lions’s Lemma Korn’s inequality Killing’s fields Lax-Milgram lemma Lamé equation Boundary value problems 

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Copyright information

© Springer Basel AG 2010

Authors and Affiliations

  • Roland Duduchava
    • 1
    • 2
    • 3
  1. 1.A. Raznmadze Mathematical InstituteTbilisi
  2. 2.I. Javakhishvili State UniversityTbilisi
  3. 3.Georgian Mathematical UnionTbilisi

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