Gevrey class semigroups arising from elastic systems with gentle dissipation: the case
Authors:
Shu Ping Chen and Roberto Triggiani
Journal:
Proc. Amer. Math. Soc. 110 (1990), 401415
MSC:
Primary 47D05; Secondary 34G10, 73D30
MathSciNet review:
1021208
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Abstract: Let (the elastic operator) be a positive, selfadjoint operator with domain in the Hilbert space , and let (the dissipation operator) be another positive, selfadjoint operator satisfying for some constants and . Consider the operator (corresponding to the elastic model written as a first order system), which (once closed) is plainly the generator of a strongly continuous semigroup of contractions on the space . In [CT.l] [CT.3] we showed that, for , such a semigroup is analytic (holomorphic) on on a triangular sector of containing the positive real axis and, moreover, that the property of analyticity is false for , say for . We now complete the description of in the range by showing that such semigroup is in fact of Gevrey class , hence differentiable on for all .
 [B.1 ]
A. V. Balakrishnan, Damping operators in continuum models of flexible structures, preprint.
 [CR.1]
G.
Chen and D.
L. Russell, A mathematical model for linear elastic systems with
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no. 4, 433–454. MR 644099
(83f:70034)
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Shu
Ping Chen and Roberto
Triggiani, Proof of two conjectures by G. Chen and D. L. Russell on
structural damping for elastic systems, Approximation and optimization
(Havana, 1987) Lecture Notes in Math., vol. 1354, Springer, Berlin,
1988, pp. 234–256. MR 996678
(90j:34088), http://dx.doi.org/10.1007/BFb0089601
 [CT.2]
Shu
Ping Chen and Roberto
Triggiani, Proof of extensions of two conjectures on structural
damping for elastic systems, Pacific J. Math. 136
(1989), no. 1, 15–55. MR 971932
(90g:47071)
 [CT.3]
Shu
Ping Chen and Roberto
Triggiani, Proof of extensions of two conjectures on structural
damping for elastic systems, Pacific J. Math. 136
(1989), no. 1, 15–55. MR 971932
(90g:47071)
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(91a:47052), http://dx.doi.org/10.1007/BFb0043280
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, Improving stability properties of hyperbolic damped equations by boundary feedback, SpringerVerlag Lecture Notes (LNCIS), 1985, 400409.
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Triggiani, Finite rank, relatively bounded perturbations of
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stabilization, Differential Integral Equations 3
(1990), no. 3, 503–522. MR 1047750
(91f:93091)
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S. Taylor, Ph.D. thesis, Chapter 'Gevrey semigroups', School of Mathematics, University of Minnesota, 1989.
 [B.1 ]
 A. V. Balakrishnan, Damping operators in continuum models of flexible structures, preprint.
 [CR.1]
 G. Chen and D. L. Russell, A mathematical model for linear elastic systems with structural damping, Quart. Appl. Math. 39 (1982), 433454. MR 644099 (83f:70034)
 [CT.1]
 S. Chen and R. Triggiani, Proof of two conjectures of G. Chen and D. L. Russell on structural damping for elastic systems: the case , in SpringerVerlag Lecture Notes in Math., vol. 1354, pp. 234256 (Proceedings of the Seminar in Approximation and Optimization, University of Habana, Cuba, January 1012, 1987). MR 996678 (90j:34088)
 [CT.2]
 , Proof of extensions of two conjectures on structural damping for elastic systems: the case , Pacific J. Math. 39 (1989), 1555. MR 971932 (90g:47071)
 [CT.3]
 S. Chen and R. Triggiani, Analyticity of elastic systems with non selfadjoint dissipation: the general case, preprint, 1989. MR 971932 (90g:47071)
 [K.1]
 S. G. Krein, Linear differential equations in Banach space, Trans. Amer. Math. Soc. 29 (1971). MR 0342804 (49:7548)
 [K.2]
 T. Kato, Perturbation theory for linear operators, SpringerVerlag, 1966. MR 0203473 (34:3324)
 [L.1]
 W. Littman, A generalization of a theorem of Datko and Pazy, University of Minnesota, preprint, 1988; presented at International Conference, Baton Rouge, Louisiana, October 1921, 1988. MR 1029070 (91a:47052)
 [L.2]
 J. L. Lions, Controlabilite exacte des systemes distribues, Masson, Paris, 1989.
 [LM.1]
 J. L. Lions and E. Magenes, Non homogeneous boundary value problems and applications I, SpringerVerlag, New York, 1972.
 [P.1]
 A. Pazy, Semigroups of linear operators and applications to partial differential equations, SpringerVerlag, New York, 1983. MR 710486 (85g:47061)
 [T.l]
 R. Triggiani, On the stabilizability problem in Banach space, J. Math. Anal. Appl. 52 (1975), 383403. Addendum J. Math. Anal. Appl. 56 (1976), 492493. MR 0445388 (56:3730)
 [T.2]
 , Improving stability properties of hyperbolic damped equations by boundary feedback, SpringerVerlag Lecture Notes (LNCIS), 1985, 400409.
 [T.3]
 , Finite rank, relatively bounded perturbations of semigroup generators, Part III: a sharp result on the lack of uniform stabilization, Differential and Integral Equations 3 (1990), 503522. MR 1047750 (91f:93091)
 [T.4]
 S. Taylor, Ph.D. thesis, Chapter 'Gevrey semigroups', School of Mathematics, University of Minnesota, 1989.
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Additional Information
DOI:
http://dx.doi.org/10.1090/S00029939199010212084
PII:
S 00029939(1990)10212084
Article copyright:
© Copyright 1990
American Mathematical Society
