论文标题
第二个Dirichlet特征值的最佳集合的规律性
Regularity of the optimal sets for the second Dirichlet eigenvalue
论文作者
论文摘要
本文专门针对Dirichlet Laplacian的第二特征值的最佳集合的规律性。 Precisely, we prove that if the set $Ω$ minimizes the functional \[ \mathcal F_Λ(Ω)=λ_2(Ω)+Λ|Ω|, \] among all subsets of a smooth bounded open set $D\subset \mathbb{R}^d$, where $λ_2(Ω)$ is the second eigenvalue of the Dirichlet Laplacian on $ω$和$λ> 0 $是一个固定常数，然后$ω$等于两个不相交的打开集合$ω_+$和$ω_- $，它是$ c^{1，α} $ - 正常的（可能是空的）hausdorff dimensies $ d-d-d-5 $ with的hausdorff d-5 $，含有一个$ d-5 $，含有一个 - \partialΩ_+\ setMinus \partialΩ_- $和$ d \ cap \partialΩ_- \ setMinus \partialΩ_+$。
This paper is dedicated to the regularity of the optimal sets for the second eigenvalue of the Dirichlet Laplacian. Precisely, we prove that if the set $Ω$ minimizes the functional \[ \mathcal F_Λ(Ω)=λ_2(Ω)+Λ|Ω|, \] among all subsets of a smooth bounded open set $D\subset \mathbb{R}^d$, where $λ_2(Ω)$ is the second eigenvalue of the Dirichlet Laplacian on $Ω$ and $Λ>0$ is a fixed constant, then $Ω$ is equivalent to the union of two disjoint open sets $Ω_+$ and $Ω_-$, which are $C^{1,α}$-regular up to a (possibly empty) closed set of Hausdorff dimension at most $d-5$, contained in the one-phase free boundaries $D\cap \partialΩ_+\setminus\partialΩ_-$ and $D\cap\partialΩ_-\setminus\partialΩ_+$.
