标题： 频谱收敛性对于任意维数的Reissner-Mindlin系统 显示英文标题

标题： Spectral convergence for the Reissner-Mindlin system in arbitrary dimension

摘要： 我们建立了在任意维度$N \geq 2$中 Reissner-Mindlin 系统的预解算子，对于任何物理上相关的边界条件，在厚度趋于零的极限下，收敛到具有适当定义边界条件的双调和算子的预解算子。 此外，给定一个在${\mathbb R}^N$中的薄域$\Omega_\delta$，具有$1 \leq d < N$个薄方向，我们证明了带有自由边界条件的 Reissner-Mindlin 系统的预解算子在厚度趋于零时收敛到极限域$\Omega \subset {\mathbb{R}}^{N-d}$中一个适当定义的 Reissner-Mindlin 系统的预解算子，当$\delta \to 0^+$时。 在两种情况下，收敛都是在算子范数下进行的，因此也意味着所有特征值和谱投影的收敛。 在薄域情况下，我们提出了一个关于收敛速度的猜想，该猜想用$\delta$表示，在圆柱体$\Omega \times B_d(0, \delta)$的情况下得到了验证。 显示英文摘要

摘要： 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 $\Omega \times B_d(0, \delta)$.