Journal of Evolution Equations

, Volume 11, Issue 4, pp 907–924 | Cite as

The regular part of sectorial forms

Article

Abstract

We study the regular part of a densely defined sectorial form, first in the abstract setting and then under mild conditions for a differential sectorial form. The regular part of the latter turns out to be again a differential sectorial form. Moreover, we characterize when taking the real part of a differential sectorial form commutes with taking the regular part. An example shows that these two operations do not commute in general.

Mathematics Subject Classification (2000)

Primary 47A07 Secondary 35J70 

Keywords

Sectorial forms non-closable forms regular part degenerate elliptic operators 

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References

  1. 1.
    Arendt, W. and ter Elst, A. F. M., Sectorial forms and degenerate differential operators. J. Operator Theory (2011). To appear.Google Scholar
  2. 2.
    Brezis, H., Analyse fonctionnelle, Théorie et applications. Collection Mathématiques appliquées pour la maîtrise. Masson, Paris etc., 1983.Google Scholar
  3. 3.
    Fukushima, M., Oshima, Y. and Takeda, M., Dirichlet forms and symmetric Markov processes, vol. 19 of de Gruyter Studies in Mathematics. Walter de Gruyter & Co., Berlin, 1994.Google Scholar
  4. 4.
    Gantmacher F.R.: The theory of matrices. Vol. 1. Chelsea Publishing Co., New York (1959)Google Scholar
  5. 5.
    Gilbarg, D. and Trudinger, N. S., Elliptic partial differential equations of second order. Second edition, Grundlehren der mathematischen Wissenschaften 224. Springer, Berlin etc., 1983.Google Scholar
  6. 6.
    Kato, T., Perturbation theory for linear operators. Second edition, Grundlehren der mathematischen Wissenschaften 132. Springer, Berlin etc., 1980.Google Scholar
  7. 7.
    M. Röckner and N. Wielens, Dirichlet forms—closability and change of speed measure, Infinite-dimensional analysis and stochastic processes (Bielefeld, 1983), Res. Notes in Math., vol. 124, Pitman, Boston, MA, 1985, pp. 119–144.Google Scholar
  8. 8.
    Simon B.: A canonical decomposition for quadratic forms with applications to monotone convergence theorems. J. Funct. Anal. 28, 377–385 (1978)MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    Vogt H.: The regular part of symmetric forms associated with second order elliptic differential expressions. Bull. Lond. Math. Soc. 41, 441–444 (2009)MathSciNetMATHCrossRefGoogle Scholar

Copyright information

© Springer Basel AG 2011

Authors and Affiliations

  1. 1.Department of MathematicsThe University of AucklandAucklandNew Zealand

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