We study the problem of minimizing the first eigenvalue of the p-Laplacian operator for a two-phase material in a bounded open domain $\Omega \subset {\mathbb{R}}^N$, $N⩾2$ assuming that the amount of the best material is limited. We provide a relaxed formulation of the problem and prove some smoothness results for these solutions. As a consequence we show that if $\Omega$ is of class $C^{1,1}$, simply connected with connected boundary, then the unrelaxed problem has a solution if and only if $\Omega$ is a ball. We also provide an algorithm to approximate the solutions of the relaxed problem and perform some numerical simulations.

中文翻译：

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两相材料的 p-Laplacian 第一特征值的最小化

我们研究了在有界开放域中最小化两相材料的 p-Laplacian 算子的第一特征值的问题 $\Omega \subset {\mathbb{电阻}}^N$, $N⩾2$假设最佳材料的数量是有限的。我们提供了一个轻松的问题公式，并证明了这些解决方案的一些平滑结果。因此我们证明如果$\Omega$ 是一流的 $C^{1,1}$，仅与连通边界相连，则非松弛问题有解当且仅当 $\Omega$是一个球。我们还提供了一种算法来逼近松弛问题的解并进行一些数值模拟。