论文标题
Faber的相位版本-Krahn定理
A phase-field version of the Faber--Krahn theorem
论文作者
论文摘要
我们根据Garcke等人在Garcke等人中引入的相位优化问题研究了Faber-Krahn定理的相位版本。 [ESAIM控制最佳。计算。 var。 29（2023），第10号论文]为Dirichlet-Laplacian的主要特征值制定。要优化的形状由映射到Interval $ [0,1] $的相位函数表示。我们表明，我们问题的任何最小化器都是一个径向对称的相位相位场，达到接近$ 0 $和$ 1 $的相位，除了一个薄的过渡层，其厚度为$ \ varepsilon> 0 $。我们的证明依赖于径向对称的重排和相应的功能不平等。此外，我们提供了$γ$ - 融合的结果，使我们能够在尖锐的接口限制下恢复Faber的变体 - Krahn定理的有限外围。
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 $Γ$-convergence result which allows us to recover a variant of the Faber--Krahn theorem for sets of finite perimeter in the sharp interface limit.
