A beam that can only bend on the Cantor set
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866917631574933504 |
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| author | Paroni, Roberto Seguin, Brian |
| author_facet | Paroni, Roberto Seguin, Brian |
| contents | In this work we address the following question: is it possible for a one-dimensional, linearly elastic beam to only bend on the Cantor set and, if so, what would the bending energy of such a beam look like? We answer this question by considering a sequence of beams, indexed by $n$, each one only able to bend on the set associated with the $n$-th step in the construction of the Cantor set and compute the $Γ$-limit of the bending energies. The resulting energy in the limit has a structure similar to the traditional bending energy, a key difference being that the measure used for the integration is the Hausdorff measure of dimension $\ln 2/\ln 3$, which is the dimension of the Cantor set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_04463 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A beam that can only bend on the Cantor set Paroni, Roberto Seguin, Brian Analysis of PDEs 49S05, 49J05 In this work we address the following question: is it possible for a one-dimensional, linearly elastic beam to only bend on the Cantor set and, if so, what would the bending energy of such a beam look like? We answer this question by considering a sequence of beams, indexed by $n$, each one only able to bend on the set associated with the $n$-th step in the construction of the Cantor set and compute the $Γ$-limit of the bending energies. The resulting energy in the limit has a structure similar to the traditional bending energy, a key difference being that the measure used for the integration is the Hausdorff measure of dimension $\ln 2/\ln 3$, which is the dimension of the Cantor set. |
| title | A beam that can only bend on the Cantor set |
| topic | Analysis of PDEs 49S05, 49J05 |
| url | https://arxiv.org/abs/2404.04463 |