Spectral convergence for the Reissner-Mindlin system in arbitrary dimension

TL;DR

This paper proves spectral convergence of the Reissner-Mindlin system to the biharmonic operator in arbitrary dimensions, including thin domain limits, with implications for eigenvalues and spectral projections.

Contribution

It establishes the resolvent convergence of the Reissner-Mindlin system to the biharmonic operator in any dimension and for various boundary conditions, extending previous results to higher dimensions and thin domains.

Findings

Resolvent convergence in operator norm for arbitrary dimensions

Spectral convergence including eigenvalues and projections

Conjecture on convergence rate verified in cylindrical domains

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 $\Omega \times B_d(0, \delta)$.