Stabilization of the damped plate equation under general boundary conditions

TL;DR

This paper establishes energy decay for a damped plate equation with general boundary conditions by deriving resolvent estimates through Carleman inequalities, microlocal, local, and global analysis.

Contribution

It introduces a novel resolvent estimate for the damped plate operator under general boundary conditions without geometric restrictions.

Findings

Logarithmic decay of the solution's energy.

Derivation of a Carleman inequality for the bi-Laplace operator.

Effective resolvent estimate under broad boundary conditions.

Abstract

We consider a damped plate equation on an open bounded subset of R^d, or a smooth manifold, with boundary, along with general boundary operators fulfilling the Lopatinskii-Sapiro 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.