Ahlfors-regularity for minimizers of a multiphase optimal design problem
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909653785378816 |
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| author | Esposito, Luca Lamberti, Lorenzo Pisante, Giovanni |
| author_facet | Esposito, Luca Lamberti, Lorenzo Pisante, Giovanni |
| contents | We establish an Alhfors-regularity result for minimizers of a multiphase optimal design problem. It is a variant of the classical variational problem which involves a finite number of chambers $\mathcal{E}(i)$ of prescribed volume that partition a given domain $Ω\subset\mathbb{R}^n$. The cost functional associated with a configuration $\left(\{\mathcal{E}(i)\}_i,u\right)$ is made up of the perimeter of the partition interfaces and a Dirichlet energy term, which is discontinuous across the interfaces. We prove that the union of the optimal interfaces is $(n-1)$-Alhfors-regular via a penalization method and decay estimates of the energy. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2506_15861 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Ahlfors-regularity for minimizers of a multiphase optimal design problem Esposito, Luca Lamberti, Lorenzo Pisante, Giovanni Optimization and Control Analysis of PDEs 49Q10, 49N60, 49Q20 We establish an Alhfors-regularity result for minimizers of a multiphase optimal design problem. It is a variant of the classical variational problem which involves a finite number of chambers $\mathcal{E}(i)$ of prescribed volume that partition a given domain $Ω\subset\mathbb{R}^n$. The cost functional associated with a configuration $\left(\{\mathcal{E}(i)\}_i,u\right)$ is made up of the perimeter of the partition interfaces and a Dirichlet energy term, which is discontinuous across the interfaces. We prove that the union of the optimal interfaces is $(n-1)$-Alhfors-regular via a penalization method and decay estimates of the energy. |
| title | Ahlfors-regularity for minimizers of a multiphase optimal design problem |
| topic | Optimization and Control Analysis of PDEs 49Q10, 49N60, 49Q20 |
| url | https://arxiv.org/abs/2506.15861 |