Sharp regularity of the second time derivative w_tt of solutions to Kirchhoff equations with clamped Boundary Conditions

Irena Lasiecka; Roberto Triggiani

International Journal of Applied Mathematics and Computer Science (2001)

  • Volume: 11, Issue: 4, page 753-772
  • ISSN: 1641-876X

Abstract

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We consider mixed problems for Kirchhoff elastic and thermoelastic systems, subject to boundary control in the clamped Boundary Conditions B.C. (“clamped control”). If w denotes elastic displacement and θ temperature, we establish optimal regularity of {w, w_t, w_tt} in the elastic case, and of {w, w_t, w_tt, θ} in the thermoelastic case. Our results complement those presented in (Lagnese and Lions, 1988), where sharp (optimal) trace regularity results are obtained for the corresponding boundary homogeneous cases. The passage from the boundary homogeneous cases to the corresponding mixed problems involves a duality argument. However, in the present case of clamped B.C., and only in this case, the duality argument in question is both delicate and technical. In this respect, the clamped B.C. are ‘exceptional’ within the set of canonical B.C. (hinged, clamped, free B.C.). Indeed, it produces new phenomena which are accounted for by introducing new, untraditional factor (quotient) spaces. These are critical in describing both interior regularity and exact controllability of mixed elastic and thermoelastic Kirchhoff problems with clamped controls.

How to cite

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Lasiecka, Irena, and Triggiani, Roberto. "Sharp regularity of the second time derivative w_tt of solutions to Kirchhoff equations with clamped Boundary Conditions." International Journal of Applied Mathematics and Computer Science 11.4 (2001): 753-772. <http://eudml.org/doc/207530>.

@article{Lasiecka2001,
abstract = {We consider mixed problems for Kirchhoff elastic and thermoelastic systems, subject to boundary control in the clamped Boundary Conditions B.C. (“clamped control”). If w denotes elastic displacement and θ temperature, we establish optimal regularity of \{w, w\_t, w\_tt\} in the elastic case, and of \{w, w\_t, w\_tt, θ\} in the thermoelastic case. Our results complement those presented in (Lagnese and Lions, 1988), where sharp (optimal) trace regularity results are obtained for the corresponding boundary homogeneous cases. The passage from the boundary homogeneous cases to the corresponding mixed problems involves a duality argument. However, in the present case of clamped B.C., and only in this case, the duality argument in question is both delicate and technical. In this respect, the clamped B.C. are ‘exceptional’ within the set of canonical B.C. (hinged, clamped, free B.C.). Indeed, it produces new phenomena which are accounted for by introducing new, untraditional factor (quotient) spaces. These are critical in describing both interior regularity and exact controllability of mixed elastic and thermoelastic Kirchhoff problems with clamped controls.},
author = {Lasiecka, Irena, Triggiani, Roberto},
journal = {International Journal of Applied Mathematics and Computer Science},
keywords = {Kirchhoff elastic and thermoelastic plate equations; clamped boundary conditions; Timoshenko-type beams; untraditional function spaces; boundary control; Kirchhoff elastic and thermoelastic system; clamped boundary; optimal regularity; Kirchhoff plate; exact controllability; thermal effects},
language = {eng},
number = {4},
pages = {753-772},
title = {Sharp regularity of the second time derivative w\_tt of solutions to Kirchhoff equations with clamped Boundary Conditions},
url = {http://eudml.org/doc/207530},
volume = {11},
year = {2001},
}

TY - JOUR
AU - Lasiecka, Irena
AU - Triggiani, Roberto
TI - Sharp regularity of the second time derivative w_tt of solutions to Kirchhoff equations with clamped Boundary Conditions
JO - International Journal of Applied Mathematics and Computer Science
PY - 2001
VL - 11
IS - 4
SP - 753
EP - 772
AB - We consider mixed problems for Kirchhoff elastic and thermoelastic systems, subject to boundary control in the clamped Boundary Conditions B.C. (“clamped control”). If w denotes elastic displacement and θ temperature, we establish optimal regularity of {w, w_t, w_tt} in the elastic case, and of {w, w_t, w_tt, θ} in the thermoelastic case. Our results complement those presented in (Lagnese and Lions, 1988), where sharp (optimal) trace regularity results are obtained for the corresponding boundary homogeneous cases. The passage from the boundary homogeneous cases to the corresponding mixed problems involves a duality argument. However, in the present case of clamped B.C., and only in this case, the duality argument in question is both delicate and technical. In this respect, the clamped B.C. are ‘exceptional’ within the set of canonical B.C. (hinged, clamped, free B.C.). Indeed, it produces new phenomena which are accounted for by introducing new, untraditional factor (quotient) spaces. These are critical in describing both interior regularity and exact controllability of mixed elastic and thermoelastic Kirchhoff problems with clamped controls.
LA - eng
KW - Kirchhoff elastic and thermoelastic plate equations; clamped boundary conditions; Timoshenko-type beams; untraditional function spaces; boundary control; Kirchhoff elastic and thermoelastic system; clamped boundary; optimal regularity; Kirchhoff plate; exact controllability; thermal effects
UR - http://eudml.org/doc/207530
ER -

References

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  10. Lasiecka I. and Triggiani R. (2000a): Optimal regularity of elastic and thermoelastic Kirchhoff plates with clamped boundary control. — Proc. Oberwohlfach Conf. Control of Complex Systems, Birkhauser (to be published). Zbl1214.74011
  11. Lasiecka I. and Triggiani R. (2000b): Factor spaces and implications on Kirchhoff equations with clamped boundary conditions. — Abstract and Applied Analysis (to appear). Zbl1006.35018
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  13. Lasiecka I. and Triggiani R. (2000d): Structural decomposition of thermoelastic semigroups with rotational forces. — Semigroup Forum, Vol.60, No.1, pp.1–61. Zbl0990.74034
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  16. Triggiani R. (1993): Regularity with interior point control, Part II: Kirchhoff Equations. — J. Diff. Eqns., Vol.103, No.2, pp.394–420. Zbl0800.93596
  17. Triggiani R. (2000): Sharp regularity theory of thermoelastic mixed problems. — Applicable Analysis (to appear). 

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