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 ε>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.
1
Universität Regensburg, Regensburg, Germany
2
University of Bonn, Bonn, Germany
Paul Hüttl; Patrik Knopf; Tim Laux. A phase-field version of the Faber–Krahn theorem. Interfaces and free boundaries, Tome 26 (2024) no. 4, pp. 587-623. doi: 10.4171/ifb/519
@article{10_4171_ifb_519,
author = {Paul H\"uttl and Patrik Knopf and Tim Laux},
title = {A phase-field version of the {Faber{\textendash}Krahn} theorem},
journal = {Interfaces and free boundaries},
pages = {587--623},
year = {2024},
volume = {26},
number = {4},
doi = {10.4171/ifb/519},
url = {http://geodesic.mathdoc.fr/articles/10.4171/ifb/519/}
}
TY - JOUR
AU - Paul Hüttl
AU - Patrik Knopf
AU - Tim Laux
TI - A phase-field version of the Faber–Krahn theorem
JO - Interfaces and free boundaries
PY - 2024
SP - 587
EP - 623
VL - 26
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.4171/ifb/519/
DO - 10.4171/ifb/519
ID - 10_4171_ifb_519
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%P 587-623
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%U http://geodesic.mathdoc.fr/articles/10.4171/ifb/519/
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