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Hodge Decompositions on Weakly Lipschitz Domains

  • Andreas Axelsson
  • Alan McIntosh
Chapter
Part of the Trends in Mathematics book series (TM)

Abstract

We survey the L 2 theory of boundary value problems for exterior and interior derivative operators \( {d_{k1}} = d + {k_1}eo \wedge \) and \( {\delta _{k2}} = \delta + {k_2}eo \) on a bounded, weakly Lipschitz domain \(\Omega \subset {{R}^{n}} \), for k 1, k 2C. The boundary conditions are that the field be either normal or tangential at the boundary. The well-posedness of these problems is related to a Hodge decomposition of the space L 2(Ω) corresponding to the operators d and δ In developing this relationship, we derive a theory of nilpotent operators in Hilbert space.

Mathematics Subject Classification (2000). 35J55, 35Q60, 47B99.

Keywords

Maxwell’s equations boundary value problem Hodge decomposition Lipschitz domain first order system nilpotent operator exterior derivative 

Mathematics Subject Classification (2000)

35J55 35Q60 47B99 

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Copyright information

© Springer Basel AG 2004

Authors and Affiliations

  • Andreas Axelsson
    • 1
  • Alan McIntosh
    • 1
  1. 1.Centre for Mathematics and its Applications, Mathematical Sciences InstituteAustralian National University CanberraAustralia

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