Journal of Evolution Equations

, Volume 13, Issue 4, pp 737–749 | Cite as

The regular part of second-order differential sectorial forms with lower-order terms

  • A. F. M. ter Elst
  • Manfred Sauter


We present a formula for the regular part of a sectorial form that represents a general linear second-order differential expression that may include lower-order terms. The formula is given in terms of the original coefficients. It shows that the regular part is again a differential sectorial form and allows to characterise when also the singular part is sectorial. While this generalises earlier results on pure second-order differential expressions, it also shows that lower-order terms truly introduce new behaviour.

Mathematics Subject Classification (2000)

Primary: 47A07 Secondary: 35J70 


Differential sectorial forms non-closable forms regular part degenerate elliptic operators 


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Arendt W., ter Elst A.F.M.: Sectorial forms and degenerate differential operators. J. Operator Theory 67, 33–72 (2012)MathSciNetGoogle Scholar
  2. 2.
    A. Braides, Γ -convergence for beginners, Oxford Lecture Series in Mathematics and its Applications, no.22, Oxford University Press, Oxford, 2002.Google Scholar
  3. 3.
    G. Dal Maso, An introduction to Γ -convergence, Progress in Nonlinear Differential Equations and their Applications, no.8, Birkhäuser, Boston, MA, 1993.Google Scholar
  4. 4.
    ter Elst A.F.M., Sauter M.: The regular part of sectorial forms. J. Evol. Equ. 11, 907–924 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Hassi S., Sebestyén Z., de Snoo H.: Lebesgue type decompositions for nonnegative forms. J. Funct. Anal. 257, 3858–3894 (2009)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Mosco U.: Composite media and asymptotic Dirichlet forms. J. Funct. Anal. 123, 368–421 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Simon B.: A canonical decomposition for quadratic forms with applications to monotone convergence theorems. J. Funct. Anal. 28, 377–385 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Vogt H.:The regular part of symmetric forms associated with second-order elliptic differential expressions. Bull. Lond. Math. Soc. 41, 441–444 (2009)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer Basel 2013

Authors and Affiliations

  1. 1.Department of MathematicsThe University of AucklandAucklandNew Zealand

Personalised recommendations