Abstract and Applied Analysis
Volume 6 (2001), Issue 8, Pages 441-488
doi:10.1155/S1085337501000586
Factor spaces and implications on Kirchhoff equations with clamped boundary conditions
Department of Mathematics, University of Virginia, Charlottesville 22904, VA, USA
Received 18 June 2001
Copyright © 2001 Irena Lasiecka and Roberto Triggiani. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Abstract
We consider mixed problems for the Kirchhoff elastic and
thermoelastic systems, subject to boundary control in the clamped
boundary conditions BC (clamped control). If w
denotes the
elastic displacement and θ
the temperature, we establish
sharp regularity of {w,wt,wtt}
in the elastic case, and
of {w,wt,wtt,θ}
in the thermoelastic case. Our
results complement those by 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 BC, and only in this case, the duality argument in
question is both delicate and technical. In this respect, the
clamped BC are “exceptional” within the set of canonical
BC (hinged, clamped, free BC). 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.