Unique continuation for over-determined Kirchhoff plate equations and related thermoelastic systems (Q2744624)

The authors continue their long series of important articles on elastic systems, control and stabilization of such systems, and more recent studies of both Kirchhoff and Timoshenko plates. In particular, they follow their own article on boundary control of thermoelastic plates. In this article they discuss a unique continuation of overdetermined system involving specifically a Kirchhoff plate-thermoelastic interaction. The authors assume that zero Cauchy data is given only on a small part of the boundary. The problem they tackle is interesting because exact controllability can be generally established for elastic systems, thus for purely mechanical effects in a plate, but only approximate controllability of thermal variables by means of boundary control. The problem they consider is overdetermined. The governing equations are: \(\psi_{tt}- \gamma\Delta\psi_{tt}+ \Delta^2\psi-\text{div}(\alpha\nabla(\alpha\psi_t))= 0\) in \(Q= (0,T]\times \Omega\), with \(\psi\in H^3(Q)\) and with \(\psi\) and all of its normal derivatives up to order 3 identically vanishing on the boundary \(\Gamma\) of \(\Omega\). The main theorem, whose consequences include unique continuation, asserts that if \(T> 2(\gamma)\text{supdist}(x,\Gamma)_{x\in\Omega}\) then \(\psi(T/2,.)= \psi(T/2,.)= 0\). Minor difficulties are overcome when different boundary conditions are specified, considering alternately hinged support or clamped support. The proof proceeds in steps. Step 1 sets the functional analytic setting. The regularizer, which looks familiar because it was used in Gauss' definition of a normal distribution curve, was also used previously by Tataru. This step leads to the Carleman-type estimates for the Kirchhoff plate. The proof of local unique continuation uses an application of the Carleman estimate, or rather of different forms of Carleman's estimate. The complexity of the arguments can be judged by the fact that at one point the authors have to prove that \(A^{-1}W_{ttttt}\) and \(A^{-1}\Theta_{tttt}\) belong, respectively to some rather complex \(L^2\) spaces.NEWLINENEWLINENEWLINEThis article is a confirmation of the past excellent work of the authors. The 45 pages contain condensed proofs of several important results, aside from the main theorem on unique continuation. For example the use of the Gaussian regularizer: \(\exp(-D^2_t/2\tau)\) was previously introduced by Tataru, but was also used by Weierstrass in his study of transforms, instead of the simpler but most likely less effective exponential approximation to the Dirac delta function. This idea may require a separate study, rather than being mentioned just as a detail on a much broader canvas.NEWLINENEWLINENEWLINEKnowing the authors' talent in applied mechanics and in control theory, the reviewer expects that they shall apply this result, hopefully in the near future, to the control or stabilization of Kirchhoff plates or related structures whose dynamics are affected by thermal effects.