We first investigate spectral properties of the Neumann–Poincaré (NP) operator for the Lamé system of elasto-statics. We show that the elasto-static NP operator can be symmetrized in the same way as that for the Laplace operator. We then show that even if elasto-static NP operator is not compact even on smooth domains, it is polynomially compact and its spectrum on two-dimensional smooth domains consists of eigenvalues that accumulate to two different points determined by the Lamé constants. We then derive explicitly eigenvalues and eigenfunctions on discs and ellipses. Using these resonances occurring at eigenvalues is considered. We also show on ellipses that cloaking by anomalous localized resonance takes place at accumulation points of eigenvalues.

We show results on Lp-spectral multipliers for Maxwell operators with bounded measurable coefficients. We also present similar results for the Stokes operator with Hodge boundary conditions and the Lamé system. Here, we rely on resolvent estimates recently established by Mitrea and Monniaux.

Exact controllabilityresults for a multilayer plate system are obtained from the method of Carleman estimates.The multilayer plate system is a natural multilayer generalization of a classical three-layer “sandwichplate” system due to Rao and Nakra. The multilayer version involves a number ofLamé systems for plane elasticity coupled with a scalar Kirchhoff plate equation. The plate is assumed to be either clamped or hinged and controlsare assumed to be locally distributed in a neighborhood of a portion of the boundary. The Carleman estimates developed for thecoupled system are based on some new Carleman estimates for the Kirchhoff plate as well as some known Carleman estimates due to Imanuvilov and Yamamoto for the Lamé system.

This paper studies the strong unique continuation property for theLamé system of elasticity with variable Lamé coefficientsλ, µ in three dimensions, ${\rm{div}}(\mu(\nabla u+\nablau^t))+ \nabla(\lambda{\rm{div}} u)+Vu=0$where λ and μ are Lipschitz continuous and V ∈ L∞. The method is based on the Carleman estimate with polynomial weights for the Lamé operator.

In this paper, we establish Carleman estimates for the twodimensional isotropic non-stationary Lamé system with the zero Dirichlet boundaryconditions. Using this estimate, we prove the uniqueness and thestability in determining spatially varying density and two Lamécoefficients by a single measurement of solution over (0,T) x ω, where T > 0 is a sufficiently large time interval and a subdomainω satisfies a non-trapping condition.