A phase-field version of the Faber–Krahn theorem

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 $[0,1]$. 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 $\epsilon >0$. Our proof relies on radially symmetric-decreasing rearrangements and corresponding functional inequalities. Moreover, we provide a $\Gamma$-convergence result which allows us to recover a variant of the Faber–Krahn theorem for sets of finite perimeter in the sharp interface limit.