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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.21629 |
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| _version_ | 1866916715007311872 |
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| author | Cicalese, Marco Kreutz, Leonard Leonardi, Gian Paolo Morselli, Gabriele |
| author_facet | Cicalese, Marco Kreutz, Leonard Leonardi, Gian Paolo Morselli, Gabriele |
| contents | While the classical Faber-Krahn inequality shows that the ball uniquely minimizes the first Dirichlet eigenvalue of the Laplacian in the continuum, this rigidity may fail in the discrete setting. We establish quantitative fluctuation estimates for the first Dirichlet eigenvalue of the combinatorial Laplacian on subsets of $\mathbb{Z}^{d}$ when their cardinality diverges. Our approach is based on a controlled discrete-to-continuum extension of the associated variational problem and the quantitative Faber-Krahn inequality. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_21629 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The Quantitative Faber-Krahn Inequality for the Combinatorial Laplacian in $\mathbb{Z}^{d}$ Cicalese, Marco Kreutz, Leonard Leonardi, Gian Paolo Morselli, Gabriele Functional Analysis While the classical Faber-Krahn inequality shows that the ball uniquely minimizes the first Dirichlet eigenvalue of the Laplacian in the continuum, this rigidity may fail in the discrete setting. We establish quantitative fluctuation estimates for the first Dirichlet eigenvalue of the combinatorial Laplacian on subsets of $\mathbb{Z}^{d}$ when their cardinality diverges. Our approach is based on a controlled discrete-to-continuum extension of the associated variational problem and the quantitative Faber-Krahn inequality. |
| title | The Quantitative Faber-Krahn Inequality for the Combinatorial Laplacian in $\mathbb{Z}^{d}$ |
| topic | Functional Analysis |
| url | https://arxiv.org/abs/2504.21629 |