Trace theorems for holomorphic semigroups and the second order Cauchy problem
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- by O. El-Mennaoui and V. Keyantuo
- Proc. Amer. Math. Soc. 124 (1996), 1445-1458
- DOI: https://doi.org/10.1090/S0002-9939-96-03133-4
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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
- Wolfgang Arendt, Vector-valued Laplace transforms and Cauchy problems, Israel J. Math. 59 (1987), no. 3, 327–352. MR 920499, DOI 10.1007/BF02774144
- G. Da Prato and M. Iannelli (eds.), Volterra integrodifferential equations in Banach spaces and applications, Pitman Research Notes in Mathematics Series, vol. 190, Longman Scientific & Technical, Harlow; copublished in the United States with John Wiley & Sons, Inc., New York, 1989. Papers from the conference held in Trento, February 2–7, 1987. MR 1018869
- Philippe Clément, Ben de Pagter, and Enzo Mitidieri (eds.), Semigroup theory and evolution equations, Lecture Notes in Pure and Applied Mathematics, vol. 135, Marcel Dekker, Inc., New York, 1991. MR 1164637
- M. Balabane, H. Emamirad, and M. Jazar, Spectral distributions and generalization of Stone’s theorem, Acta Appl. Math. 31 (1993), no. 3, 275–295. MR 1232940, DOI 10.1007/BF00997121
- Khristo Boyadzhiev and Ralph deLaubenfels, Boundary values of holomorphic semigroups, Proc. Amer. Math. Soc. 118 (1993), no. 1, 113–118. MR 1128725, DOI 10.1090/S0002-9939-1993-1128725-X
- Paul L. Butzer and Hubert Berens, Semi-groups of operators and approximation, Die Grundlehren der mathematischen Wissenschaften, Band 145, Springer-Verlag New York, Inc., New York, 1967. MR 0230022, DOI 10.1007/978-3-642-46066-1
- E. B. Davies, Heat kernels and spectral theory, Cambridge Tracts in Mathematics, vol. 92, Cambridge University Press, Cambridge, 1989. MR 990239, DOI 10.1017/CBO9780511566158
- R. deLaubenfels, Existence families, functional calculi and evolution equations, Lecture Notes in Math., vol. 1570, Springer Verlag, Berlin and New York, 1994.
- O. El-Mennaoui, Traces de semi-groupes holomorphes singuliers à l’origine et comportement asymptotique, Thèse, Besançon, 1992.
- H. O. Fattorini, Second order linear differential equations in Banach spaces, North-Holland Mathematics Studies, vol. 108, North-Holland Publishing Co., Amsterdam, 1985. Notas de Matemática [Mathematical Notes], 99. MR 797071
- Jerome A. Goldstein, Semigroups of linear operators and applications, Oxford Mathematical Monographs, The Clarendon Press, Oxford University Press, New York, 1985. MR 790497
- Matthias Hieber, Integrated semigroups and differential operators on $L^p$ spaces, Math. Ann. 291 (1991), no. 1, 1–16. MR 1125004, DOI 10.1007/BF01445187
- Saunders MacLane and O. F. G. Schilling, Infinite number fields with Noether ideal theories, Amer. J. Math. 61 (1939), 771–782. MR 19, DOI 10.2307/2371335
- Ronald H. W. Hoppe, Interpolation of cosine operator functions, Ann. Mat. Pura Appl. (4) 136 (1984), 183–212. MR 765922, DOI 10.1007/BF01773383
- Lars Hörmander, Estimates for translation invariant operators in $L^{p}$ spaces, Acta Math. 104 (1960), 93–140. MR 121655, DOI 10.1007/BF02547187
- V. Kéyantuo, A note on interpolation of semigroups, Proc. Amer. Math. Soc. 123 (1995), no. 7, 2123–2132. MR 1243170, DOI 10.1090/S0002-9939-1995-1243170-6
- —, The Weierstrass formula and the abstract Cauchy problem, preprint.
- Frank Neubrander, Integrated semigroups and their applications to the abstract Cauchy problem, Pacific J. Math. 135 (1988), no. 1, 111–155. MR 965688, DOI 10.2140/pjm.1988.135.111
- E. M. Stein, Singular integrals and differentiability properties of functions, Princeton Univ. Press, Princeton, NJ, 1971.
- Kôsaku Yosida, Functional analysis, 6th ed., Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 123, Springer-Verlag, Berlin-New York, 1980. MR 617913
Bibliographic 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
- © Copyright 1996 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 124 (1996), 1445-1458
- MSC (1991): Primary 47D06, 47F05
- DOI: https://doi.org/10.1090/S0002-9939-96-03133-4
- MathSciNet review: 1301022