Stabilization of the damped plate equation under general boundary conditions

Abstract

We consider a damped plate equation on an open bounded subset of R, or a smooth manifold, with boundary, along with general boundary operators fulfilling the Lopatinskĭı-Šapiro condition. The damping term acts on a region without imposing a geometrical condition. We derive a resolvent estimate for the generator of the damped plate semigroup that yields a logarithmic decay of the energy of the solution to the plate equation. The resolvent estimate is a consequence of a Carleman inequality obtained for the bi-Laplace operator involving a spectral parameter under the considered boundary conditions. The derivation goes first though microlocal estimates, then local estimates, and finally a global estimate.