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On explicit representation and approximations of Dirichlet-to-Neumann semigroup

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Abstract

In his book (Functional Analysis, Wiley, New York, 2002), P. Lax constructs an explicit representation of the Dirichlet-to-Neumann semigroup, when the matrix of electrical conductivity is the identity matrix and the domain of the problem in question is the unit ball in ℝn. We investigate some representations of Dirichlet-to-Neumann semigroup for a bounded domain. We show that such a nice explicit representation as in Lax book, is not possible for any domain except Euclidean balls. It is interesting that the treatment in dimension 2 is completely different than other dimensions. Finally, we present a natural and probably the simplest numerical scheme to calculate this semigroup in full generality by using Chernoff’s theorem.

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References

  1. Arendt, W., Ter Elst, A.F.M.: The Dirichlet-to-Neumann operator on rough domains. arXiv:1010.1703 (2010)

  2. Blair, D.E.: Inversion Theory and Conformal Mapping. American Mathematical Society, Providence (2000)

    MATH  Google Scholar 

  3. Chernoff, P.R.: Note on product formulas for operator semigroups. J. Funct. Anal. 2, 238–242 (1968)

    Article  MathSciNet  MATH  Google Scholar 

  4. deLaubenfels, R.: Well-behaved derivations on C[0 1]. Pac. J. Math. 115, 73–80 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  5. Emamirad, H., Laadnani, I.: An approximating family for the Dirichlet-to-Neumann semigroup. Adv. Differ. Equ. 11, 241–257 (2006)

    MathSciNet  MATH  Google Scholar 

  6. Engel, K.-J.: The Laplacian on \(C(\overline{\Omega})\) with generalized Wenzell boundary conditions. Arch. Math. 81, 548–558 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Escher, J.: The Dirichlet-Neumann operator on continuous functions. Ann. Sc. Norm. Super. Pisa 21, 235–266 (1994)

    MathSciNet  MATH  Google Scholar 

  8. Lax, P.D.: Functional Analysis. Wiley, New York (2002)

    MATH  Google Scholar 

  9. Vrabie, I.I.: C 0-Semigroups and Applications. North-Holland, Amsterdam (2003)

    MATH  Google Scholar 

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Acknowledgements

We wish to thank Professor Ralph deLaubenfels who was the instigator of this method, for his collaboration with the first author which ends up with this paper.

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Authors and Affiliations

  1. School of Mathematics, Institute for Research in Fundamental Sciences (IPM), P.O. Box 19395-5746, Tehran, Iran

    H. Emamirad & M. Sharifitabar

  2. Laboratoire de Mathématiques, Université de Poitiers, Teleport 2, BP 179, 86960, Chassneuil du Poitou Cedex, France

    H. Emamirad

Authors
  1. H. Emamirad
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  2. M. Sharifitabar
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Corresponding author

Correspondence to H. Emamirad.

Additional information

Communicated by Jerome A. Goldstein.

This research was in part supported by a grant from IPM.

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Emamirad, H., Sharifitabar, M. On explicit representation and approximations of Dirichlet-to-Neumann semigroup. Semigroup Forum 86, 192–201 (2013). https://doi.org/10.1007/s00233-012-9380-8

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  • DOI: https://doi.org/10.1007/s00233-012-9380-8

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