Advertisement

Locality and Domination of Semigroups

  • Khalid Akhlil
Article
  • 32 Downloads

Abstract

We characterize all semigroups \((T(t))_{t\ge 0}\) on \(L^2(\Omega )\) sandwiched between Dirichlet and Neumann ones, i.e.:
$$\begin{aligned} e^{t\Delta _D}\le T(t)\le e^{t\Delta _N},\quad \text {for all }t\ge 0 \end{aligned}$$
in the positive operators sense. The proof uses the well-known Beurling–Deny and Lejan formula to drop the locality assumption made usually on the form associated with \((T(t))_{t\ge 0}\). Moreover, we prove that if \(T(t)\le S(t)\) for all \(t\ge 0\), where \((T(t))_{t\ge 0}\) (resp. \((S(t))_{t\ge 0}\)) is a \(C_0\)-semigroup on some \(L^2\)-space associated with a regular Dirichlet form (aD(a)) (resp. with a Dirichlet form (bD(b))), then the locality of b implies the locality of a.

Keywords

Robin boundary conditions locality domination of semigroups 

Mathematics Subject Classification

31C15 31C25 47D07 60H30 60J35 60J60 60J45 

Notes

Acknowledgements

The author would like to thank Wolfgang Arendt Omar El-Mennaoui and Jochen Glück for many stimulating and helpful discussions. This work was achieved during a research stay in Ulm, Germany. Many thanks to the referee who helped a lot to improve the first version of this paper.

References

  1. 1.
    Akhlil, K.: Probabilistic solution of the general Robin boundary value problem on arbitrary domains. Int. J. Stoch. Anal. 2012, 17 (2012)MathSciNetzbMATHGoogle Scholar
  2. 2.
    Allian, G.: Sur la Représentation des Formes de Dirichlet. Ann. Inst. Fourier 25(3–4), 1–10 (1975)MathSciNetCrossRefGoogle Scholar
  3. 3.
    Andersson, L.-E.: On the representation of Dirichlet forms. Ann. Inst. Fourier 25(3–4), 11–25 (1975)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Arendt, W., Warma, M.: The Laplacian with Robin boundary conditions on arbitrary domains. Potential Anal. 19, 341–363 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Arendt, W., Warma, M.: Dirichlet and Neumann boudary conditions: What is in between? J. Evol. Equ 3, 119–135 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Arendt, W.: Private communicationGoogle Scholar
  7. 7.
    Beurling, A., Deny, J.: Espace de Dirichlet. I. Le cas élementaire. Acta Math. 99, 203–224 (1958)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Biroli, M.: Strongly local nonlinear Dirichlet functionals. Ukr. Math. Bul. Tom. 1(4), 485–500 (2004)MathSciNetzbMATHGoogle Scholar
  9. 9.
    Chill, R., Warma, M.: Dirichlet and Neumann boundary conditions for \(p\)-Laplace operator: What is in between? Proc. R. Soc. Edinb. 142A, 975–1002 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Daners, D., Glück, J., Kennedy, J.B.: Eventually and asymptotically positive semigroups on Banach Lattices. J. Differ. Equ. 261, 2607–2649 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Fukushima, M., Oshima, Y., Takeda, M.: Dirichlet Forms and Symmetric Markov Processes. Walter de Gruyter, Berlin (1994)CrossRefzbMATHGoogle Scholar
  12. 12.
    Hu, Z.-C., Ma, Z.-M.: Beurling-Deny formula of semi-Dirichlet forms. C. R. Acad. Sci. Paris Ser. I 338, 521–526 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Hu, Z.-C., Ma, Z.-M., Sun, W.: Extensions of Lévy–Khintchine formula and Beurling–Deny formula in semi-Dirichlet forms setting. J. Funct. Anal. 239, 179–213 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Hu, Z.-C., Ma, Z.-M., Sun, W.: On representation of non-symmetric Dirichlet forms. Potential Anal. 32, 101–131 (2010)MathSciNetCrossRefzbMATHGoogle Scholar
  15. 15.
    Ma, Z.M., Röckner, M.: Introduction to the Theory of (Non-symmetric) Dirichlet Forms. Springer, Berlin (1992)CrossRefzbMATHGoogle Scholar
  16. 16.
    Nagel, R. (ed.): One-parameter semigroups of positive operators. Lecture Notes in Math, vol. 1184. Springer, Berlin (1986)Google Scholar
  17. 17.
    Ouhabaz, E.M.: Invariance of closed convex sets and domination criteria for semigroups. Potential Anal. 5, 611–625 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Rudin, W.: Real and Complex Analysis. McGraw-Hill Inc, New York (1966)zbMATHGoogle Scholar
  19. 19.
    Warma, M.: The Laplacian with general Robin boundary conditions. Ph.D. thesis, University of Ulm (2002)Google Scholar

Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Applied Analysis InstituteUniversity of UlmUlmGermany
  2. 2.Polydisciplinary Faculty of OuarzazateIbn Zohr UniversityOuarzazateMorocco

Personalised recommendations