Spectral convergence for the Reissner-Mindlin system in arbitrary dimension

Abstract

We establish the convergence of the resolvent of the Reissner-Mindlin system in any dimension $N \geq 2$, 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 ${\mathbb R}^N$ with $1 \leq d < N$ 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 $Ω\subset {\mathbb{R}}^{N-d}$ as $δ\to 0^+$. 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 $Ω\times B_d(0, δ)$.