Huettl, Paul and Knopf, Patrik and Laux, Tim (2024) A phase-field version of the Faber-Krahn theorem. INTERFACES AND FREE BOUNDARIES, 26 (4). pp. 587-623. ISSN 1463-9963, 1463-9971
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We investigate a phase-field version of the Faber-Krahn theorem based on a phase-field optimization problem introduced by Garcke et al. in their 2023 paper 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 e > 0. Our proof relies on radially symmetric-decreasing rearrangements and corresponding functional inequalities. Moreover, we provide a P-convergence result which allows us to recover a variant of the Faber-Krahn theorem for sets of finite perimeter in the sharp interface limit.
| Item Type: | Article |
|---|---|
| Uncontrolled Keywords: | INEQUALITIES; MINIMIZERS; REGULARITY; SHAPE; Faber-Krahn inequality; radially symmetric-decreasing rearrangements; phase-field models; shape optimization; sharp interface limit; Gamma-limit |
| Subjects: | 500 Science > 510 Mathematics |
| Divisions: | Mathematics > Prof. Dr. Harald Garcke |
| Depositing User: | Dr. Gernot Deinzer |
| Date Deposited: | 29 Jul 2025 08:41 |
| Last Modified: | 29 Jul 2025 08:41 |
| URI: | https://pred.uni-regensburg.de/id/eprint/64089 |
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