# 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

## Access Full Article

top## Abstract

top## How to cite

topLasiecka, 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

top- Aubin J.P. (1972): Approximation of Elliptic Boundary-Value Problems. — New York: Wiley- Interscience. Zbl0248.65063
- Giles J.R. (2000): Introduction to the Analysis of Normed Linear Spaces. — Cambridge: Cambridge University Press. Zbl0941.46001
- De Simon L. (1964): Un’ applicazione della teoria degli integrali singulari allo studio delle equazioni differenziali astratte del primo ordine. — Rendic. Semin. Mat. Univ. Padova, Vol.34, pp.205–223. Zbl0196.44803
- Grisvard P. (1967): Caracterization de quelques espaces d’interpolation. — Arch. Rational Mech. Anal., Vol.25, pp.40–63. Zbl0187.05901
- Eller M., Lasiecka I. and Triggiani R. (2001a): Simultaneous exact/approximate boundary controllability of thermo-elastic plates with variable transmission coefficients, In: Lecture Notes in Pure and Applied Mathematics (J. Cagnol, M. Polis, J. P.Zolesio, Eds.). — New York: Marcel Dekker, pp.109–230. Zbl0980.74043
- Eller M., Lasiecka I. and Triggiani R. (2001b): Simultaneous exact/approximate controllability of thermoelastic plates with variable thermal coefficient and clamped/Dirichlet boundary controls. — Cont. Discr. Dynam. Syst., Vol.7, No.2, pp.283–302. Zbl1031.93101
- Lagnese J. (1989): Boundary Stabilization of Thin Plates. — Philadelphia: SIAM. Zbl0696.73034
- Lagnese J. and Lions J.L. (1988): Modelling, Analysis and Control of Thin Plates. — Paris: Masson. Zbl0662.73039
- Lasiecka I. (1989): Controllability of a viscoelastic Kirchhoff plate. — Int. Series Num. Math., Vol.91, Basel: Birkhäuser, pp.237–247. Zbl0685.93007
- 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
- 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
- Lasiecka I. and Triggiani R. (2000c): Control Theory for Partial Differential Equations: Continuous and Approximation Theories, Vol.I: Abstract Parabolic Systems; Vol.II: Abstract Hyperbolic-Like Systems over a Finite Time Horizon. — Cambridge: Cambridge University Press. Zbl0961.93003
- 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
- Lions J.L. and Magenes E. (1972): Nonhomogeneous Boundary Value Problems and Applications, Vol.1. — New York: Springer. Zbl0223.35039
- Taylor A.E. and Lay D.C. (1980): Introduction to Functional Analysis, 2nd Ed. — New York: Wiley.
- Triggiani R. (1993): Regularity with interior point control, Part II: Kirchhoff Equations. — J. Diff. Eqns., Vol.103, No.2, pp.394–420. Zbl0800.93596
- Triggiani R. (2000): Sharp regularity theory of thermoelastic mixed problems. — Applicable Analysis (to appear).

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.