A phase-field version of the Faber--Krahn theorem

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 $\Gamma$-convergence result which allows us to recover a variant of the Faber--Krahn theorem for sets of finite perimeter in the sharp interface limit.