Optimal Regularity of Elastic and Thermoelastic Kirchhoff Plates with Clamped Boundary Control

Abstract

We consider mixed problems for the 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 in [L-L.1], 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., the duality argument in question is both delicate and technical. 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 Kirchhoff problems with clamped controls.