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 $\Omega_\delta$ 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 $\Omega \subset {\mathbb{R}}^{N-d}$ as $\delta \to 0^+$. 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 $\delta$, which is verified in the case of the cylinder $\Omega \times B_d(0, \delta)$.