Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Trace theorems for holomorphic semigroups
and the second order Cauchy problem

Authors: O. El-Mennaoui and V. Keyantuo
Journal: Proc. Amer. Math. Soc. 124 (1996), 1445-1458
MSC (1991): Primary 47D06, 47F05
MathSciNet review: 1301022
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We use the theory of boundary values (also called traces) of holomorphic semigroups as developed by Boyadzhiev-deLaubenfels (1993) and El-Mennaoui (1992) to study the second order Cauchy problem for certain generators of holomorphic semigroups. Our results contain in particular the result of Hieber (Math. Ann. 291 (1991), 1--16) for the Laplace operator on $L^p(\mathbb R^N)$.

References [Enhancements On Off] (What's this?)

  • 1. W. Arendt, Vector-valued Laplace transforms and Cauchy problems, Israel. J. Math. 59 (1987), 327--352. MR 89a:47064
  • 2. W. Arendt and H. Kellermann, Integrated solutions of Volterra integrodifferential equations and Cauchy problems, Integrodifferential Equations (Proc. Conf. Trento, 1987) (G. Da Prato and M. Iannelli, eds.), Pitman Res. Notes Math. Ser., vol. 190, Longman Sci. Tech., Harlow, 1987, pp. 21--51. MR 90d:00047
  • 3. W. Arendt, Sobolev imbeddings and integrated semigroups, 2nd International Conference on Trends in Semigroup Theory and Evolution Equations (Ph. Clément, E. Mitidieri, and B. de Pagter, eds.), Lecture Notes in Pure and Appl. Math., vol. 135, Marcel Dekker, New York, 1991. MR 92m:47001
  • 4. M. Balabane, H. Emamirad, and M. Jazar Spectral distributions and generalization of Stone's theorem, Acta Appl. Math. 31 (1993), 275--295. MR 94f:47038
  • 5. K. Boyadzhiev and R. deLaubenfels Boundary values of holomorphic semigroups, Proc. Amer. Math. Soc. 118 (1993), 113--119. MR 93f:47043
  • 6. P. L. Butzer and H. Berens, Semigroups of operators and approximation, Springer Verlag, Berlin and New York, 1967. MR 37:5588
  • 7. E. B. Davies, Heat kernels and spectral theory, Cambridge Univ. Press, London and New York, 1989. MR 90e:35123
  • 8. R. deLaubenfels, Existence families, functional calculi and evolution equations, Lecture Notes in Math., vol. 1570, Springer Verlag, Berlin and New York, 1994. CMP 94:17
  • 9. O. El-Mennaoui, Traces de semi-groupes holomorphes singuliers à l'origine et comportement asymptotique, Thèse, Besançon, 1992.
  • 10. H. O. Fattorini, Second order linear differential equations in Banach spaces North-Holland, Amsterdam, New York, and London, 1985. MR 87b:34001
  • 11. J. A. Goldstein, Semigroups of linear operators and applications, Oxford Math. Monographs, Oxford Univ. Press, New York, 1985. MR 87c:47056
  • 12. M. Hieber, Integrated semigroups and differential operators on $L^p(\mathbb R^N)$-spaces, Math. Ann. 291 (1991), 1--16. MR 92g:47052
  • 13. E. Hille and R. S. Phillips, Functional analysis and semigroups, Amer. Math. Soc. Colloq. Publ., vol. 31, Amer. Math. Soc., Providence, RI, 1957. MR 19:664d
  • 14. R. W. Hoppe, Interpolation of cosine operator functions, Ann. Mat. Pura Appl. 136 (1984), 183--212. MR 86b:47074
  • 15. L. Hörmander, Estimates for translation invariant operators in $L^p$ spaces, Acta. Math. 104 (1960), pages 93--140. MR 22:12389
  • 16. V. Keyantuo A note on interpolation of semigroups, Proc. Amer. Math. Soc. 123 (1995), 2123--2132. MR 95i:47073
  • 17. ------, The Weierstrass formula and the abstract Cauchy problem, preprint.
  • 18. F. Neubrander Integrated semigroups and their applications to the abstract Cauchy problem, Pacific J. Math. 135 (1988), 233--251. MR 90b:47073
  • 19. E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, NJ, 1971.
  • 20. K. Yosida, Functional analysis, Springer Verlag, Berlin and New York, 1980. MR 82i:46002

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (1991): 47D06, 47F05

Retrieve articles in all journals with MSC (1991): 47D06, 47F05

Additional Information

O. El-Mennaoui
Affiliation: Mathematisches Institut der Universität Tübingen, Auf der Morgenstelle 10, 7400 Tübingen, Germany

V. Keyantuo
Affiliation: Équipe de Mathématiques, Université de Franche-Comté, Route de Gray, 25030 Besançon, France
Address at time of publication: Department of Mathematics, University of Puerto Rico, Box 23355 Rio Piedras, Puerto Rico 00931

Received by editor(s): June 6, 1994
Received by editor(s) in revised form: September 26, 1994
Additional Notes: This work was supported by the DAAD and the European Science Plan “Evolutionary Systems”.
Communicated by: Palle E. T. Jorgensen
Article copyright: © Copyright 1996 American Mathematical Society

American Mathematical Society