Spectral convergence for the Reissner-Mindlin system in arbitrary dimension
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909443740925952 |
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| author | Buoso, Davide Ferraresso, Francesco |
| author_facet | Buoso, Davide Ferraresso, Francesco |
| contents | 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 $Ω\times B_d(0, δ)$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2412_20094 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spectral convergence for the Reissner-Mindlin system in arbitrary dimension Buoso, Davide Ferraresso, Francesco Analysis of PDEs Spectral Theory 35J30, 35P15, 49R05, 74K20 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 $Ω\times B_d(0, δ)$. |
| title | Spectral convergence for the Reissner-Mindlin system in arbitrary dimension |
| topic | Analysis of PDEs Spectral Theory 35J30, 35P15, 49R05, 74K20 |
| url | https://arxiv.org/abs/2412.20094 |