Geodesic
Sharp Regularity of the Second Time Derivative wtt of Solutions to Kirchhoff Equations With Clamped Boundary Conditions
International Journal of Applied Mathematics and Computer Science, Tome 11 (2001) no. 4, pp. 753-772.

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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 theta temperature, we establish optimal regularity of w,w_t,w_tt in the elastic case, and of w,w_t,w_tt,theta 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.
Keywords: Kirchhoff elastic and thermoelastic plate equations, clamped boundary conditions
Mots-clés : warunek brzegowy, matematyka 
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     title = {Sharp {Regularity} of the {Second} {Time} {Derivative} wtt of {Solutions} to {Kirchhoff} {Equations} {With} {Clamped} {Boundary} {Conditions}},
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Lasiecka, I.; Triggiani, R. Sharp Regularity of the Second Time Derivative wtt of Solutions to Kirchhoff Equations With Clamped Boundary Conditions. International Journal of Applied Mathematics and Computer Science, Tome 11 (2001) no. 4, pp. 753-772. http://geodesic.mathdoc.fr/item/IJAMCS_2001_11_4_a0/