In this paper we consider minimizers of the functional$\min \{{\lambda }_1(\mathrm{\Omega })+\cdots +{\lambda }_k(\mathrm{\Omega })+\mathrm{\Lambda }|\mathrm{\Omega }|,:\mathrm{\Omega }\subset D\text{ open}\}$ where $D\subset {\mathbb{R}}^d$ is a bounded open set and where $0<{\lambda }_1(\mathrm{\Omega })\le \cdots \le {\lambda }_k(\mathrm{\Omega })$ 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 ${\mathrm{\Omega }}^{⁎}$ have finite perimeter and that their free boundary $\partial {\mathrm{\Omega }}^{⁎}\cap D$ is composed of a regular part, which is locally the graph of a $C^{1,\alpha }$-regular function, and a singular part, which is empty if $d<d^{⁎}$, discrete if $d=d^{⁎}$ and of Hausdorff dimension at most $d-d^{⁎}$ if $d>d^{⁎}$, for some $d^{⁎}\in \{5,6,7\}$.

在本文中，我们考虑了泛函的极小值$\mathrm{分钟}\{{\lambda }_1(\mathrm{\Omega })+\cdots +{\lambda }_{克}(\mathrm{\Omega })+\mathrm{\Lambda }|\mathrm{\Omega }|,：\mathrm{\Omega }\subset D\text{ 打开}\}$ 在哪里 $D\subset {\mathbb{电阻}}^d$ 是一个有界开集，其中 $0<{\lambda }_1(\mathrm{\Omega })\le \cdots \le {\lambda }_{克}(\mathrm{\Omega })$是具有 Dirichlet 边界条件和 Hölder 连续系数的散度形式的算子 Ω 上的前k 个特征值。我们证明最优集合${\mathrm{\Omega }}^{⁎}$ 具有有限周长并且它们的自由边界 $\partial {\mathrm{\Omega }}^{⁎}\cap D$由一个规则部分组成，它是一个局部图$C^{1,\alpha }$-regular 函数，以及一个单数部分，如果是空的$d<d^{⁎}$, 离散如果 $d=d^{⁎}$ 和至多豪斯多夫维数 $d-d^{⁎}$ 如果 $d>d^{⁎}$， 对于一些 $d^{⁎}\in \{5,6,7\}$.