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Paper details
Number 4 - October 2001
Volume 11 - 2001
Sharp regularity of the second time derivative wtt of solutions to Kirchhoff equations with clamped boundary conditions
Irena Lasiecka, Roberto Triggiani
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, wt, wtt} in the elastic case, and of {w, wt, wtt, 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.
Keywords
Kirchhoff elastic and thermoelastic plate equations, clamped boundary conditions