A phase-field version of the Faber--Krahn theorem
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| Format: | Preprint |
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2022
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| _version_ | 1866913337135071232 |
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| author | Hüttl, Paul Knopf, Patrik Laux, Tim |
| author_facet | Hüttl, Paul Knopf, Patrik Laux, Tim |
| contents | We investigate a phase-field version of the Faber--Krahn theorem based on a phase-field optimization problem introduced in Garcke et al. [ESAIM Control Optim. Calc. Var. 29 (2023), Paper No. 10] formulated for the principal eigenvalue of the Dirichlet--Laplacian. The shape, that is to be optimized, is represented by a phase-field function mapping into the interval $[0,1]$. We show that any minimizer of our problem is a radially symmetric-decreasing phase-field attaining values close to $0$ and $1$ except for a thin transition layer whose thickness is of order $\varepsilon>0$. Our proof relies on radially symmetric-decreasing rearrangements and corresponding functional inequalities. Moreover, we provide a $Γ$-convergence result which allows us to recover a variant of the Faber--Krahn theorem for sets of finite perimeter in the sharp interface limit. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2207_10946 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A phase-field version of the Faber--Krahn theorem Hüttl, Paul Knopf, Patrik Laux, Tim Analysis of PDEs Optimization and Control Spectral Theory 35P05, 35P15, 49Q10, 49R05 We investigate a phase-field version of the Faber--Krahn theorem based on a phase-field optimization problem introduced in Garcke et al. [ESAIM Control Optim. Calc. Var. 29 (2023), Paper No. 10] formulated for the principal eigenvalue of the Dirichlet--Laplacian. The shape, that is to be optimized, is represented by a phase-field function mapping into the interval $[0,1]$. We show that any minimizer of our problem is a radially symmetric-decreasing phase-field attaining values close to $0$ and $1$ except for a thin transition layer whose thickness is of order $\varepsilon>0$. Our proof relies on radially symmetric-decreasing rearrangements and corresponding functional inequalities. Moreover, we provide a $Γ$-convergence result which allows us to recover a variant of the Faber--Krahn theorem for sets of finite perimeter in the sharp interface limit. |
| title | A phase-field version of the Faber--Krahn theorem |
| topic | Analysis of PDEs Optimization and Control Spectral Theory 35P05, 35P15, 49Q10, 49R05 |
| url | https://arxiv.org/abs/2207.10946 |