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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/2503.20439 |
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| _version_ | 1866912295122108416 |
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| author | Del Nin, Giacomo De Luca, Lucia |
| author_facet | Del Nin, Giacomo De Luca, Lucia |
| contents | We consider a variant of the sticky disk energy where distances between particles are evaluated through the sup norm $\lVert\cdot\rVert_\infty$ in the plane. We first prove crystallization of minimizers in the square lattice, for any fixed number $N$ of particles. Then we consider the limit as $N\to\infty$: in contrast to the standard sticky disk, there is only one orientation in the limit, and we are able to compute explicitly the $Γ$-limit to be an anisotropic perimeter with octagonal Wulff shape. The results are based on an energy decomposition for graphs that generalizes the one proved by De Luca-Friesecke [J. Nonlinear Sci. 28 (2018), 69-90] in the triangular case. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2503_20439 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The square sticky disk: crystallization and Gamma-convergence to the octagonal anisotropic perimeter Del Nin, Giacomo De Luca, Lucia Analysis of PDEs Mathematical Physics 49J45 (primary), 05C10, 70C20, 82D25 (secondary) We consider a variant of the sticky disk energy where distances between particles are evaluated through the sup norm $\lVert\cdot\rVert_\infty$ in the plane. We first prove crystallization of minimizers in the square lattice, for any fixed number $N$ of particles. Then we consider the limit as $N\to\infty$: in contrast to the standard sticky disk, there is only one orientation in the limit, and we are able to compute explicitly the $Γ$-limit to be an anisotropic perimeter with octagonal Wulff shape. The results are based on an energy decomposition for graphs that generalizes the one proved by De Luca-Friesecke [J. Nonlinear Sci. 28 (2018), 69-90] in the triangular case. |
| title | The square sticky disk: crystallization and Gamma-convergence to the octagonal anisotropic perimeter |
| topic | Analysis of PDEs Mathematical Physics 49J45 (primary), 05C10, 70C20, 82D25 (secondary) |
| url | https://arxiv.org/abs/2503.20439 |