论文标题
适用于某些功能的最佳集合，涉及运算符的特征值以发散形式
Regularity of optimal sets for some functional involving eigenvalues of an operator in divergence form
论文作者
论文摘要
在本文中，我们考虑了功能\ begin {equation*} \ min \ big \ big \ \ {λ_1（ω）+\ cdots+cdots+λ_k（ω）+λ|ω|，\ \ c. $D\subset\mathbb{R}^d$ is a bounded open set and where $0<λ_1(Ω)\leq\cdots\leqλ_k(Ω)$ are the first $k$ eigenvalues on $Ω$ of an operator in divergence form with Dirichlet boundary condition and with Hölder continuous coefficients.我们证明，最佳集合$ω^\ ast $具有有限的周围，并且它们的自由边界$ \partialΩ^\ ast \ cap d $由常规零件组成，在本地是$ c^{1，α} $的图形，n是常规函数，以及$ d <d^$ divt^aSt $ divit d = divet d^af ah af af，如果$ d> d^\ ast $，则最多是$ d-d-d-d^\ ast $，对于某些$ d^\ ast \ in \ in \ {5,6,7 \} $。
In this paper we consider minimizers of the functional \begin{equation*} \min \big\{ λ_1(Ω)+\cdots+λ_k(Ω) + Λ|Ω|, \ : \ Ω\subset D \text{ open} \big\} \end{equation*} where $D\subset\mathbb{R}^d$ is a bounded open set and where $0<λ_1(Ω)\leq\cdots\leqλ_k(Ω)$ are the first $k$ eigenvalues on $Ω$ of an operator in divergence form with Dirichlet boundary condition and with Hölder continuous coefficients. We prove that the optimal sets $Ω^\ast$ have finite perimeter and that their free boundary $\partialΩ^\ast\cap D$ is composed of a regular part, which is locally the graph of a $C^{1,α}$-regular function, and a singular part, which is empty if $d<d^\ast$, discrete if $d=d^\ast$ and of Hausdorff dimension at most $d-d^\ast$ if $d>d^\ast$, for some $d^\ast\in\{5,6,7\}$.
