Spectral convergence analysis for the Reissner-Mindlin system in any dimension

We establish the convergence of the resolvent of the Reissner-Mindlin system in any dimension N≥2, with any of the physically relevant boundary conditions, to the resolvent of the biharmonic operator with suitably defined boundary conditions in the vanishing thickness limit. Moreover, given a thin domain Ω_δ in R^N with 1≤d<N thin directions, we prove that the resolvent of the Reissner-Mindlin system with free boundary conditions converges to the resolvent of a suitably defined Reissner-Mindlin system in the limiting domain Ω⊂R^N−d as δ→0⁺. In both cases, the convergence is in operator norm, implying therefore the convergence of all the eigenvalues and spectral projections. In the thin domain case, we formulate a conjecture on the rate of convergence in terms of δ, which is verified in the case of the cylinder Ω×B_d(0,δ).