A phase-field version of the Faber–Krahn theorem
Paul Hüttl
Universität Regensburg, Regensburg, GermanyPatrik Knopf
Universität Regensburg, Regensburg, GermanyTim Laux
University of Bonn, Bonn, Germany
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Abstract
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 . We show that any minimizer of our problem is a radially symmetric-decreasing phase-field attaining values close to and except for a thin transition layer whose thickness is of order . Our proof relies on radially symmetric-decreasing rearrangements and corresponding functional inequalities. Moreover, we provide a -convergence result which allows us to recover a variant of the Faber–Krahn theorem for sets of finite perimeter in the sharp interface limit.
Cite this article
Paul Hüttl, Patrik Knopf, Tim Laux, A phase-field version of the Faber–Krahn theorem. Interfaces Free Bound. (2024), published online first
DOI 10.4171/IFB/519