We are interested in the optimal distribution of two isolating materials in the wall of a cavity in order to get the best isolation when the amount of the best isolating material is limited. We assume that the wall is of width $\epsilon >0$ small. The problem can be formulated as the minimization of the first eigenvalue of a certain diffusion operator where the diffusion constant is of order one inside the cavity and of order $\epsilon$ in the wall. As it is usual in optimal design, the problem has not solution in general and therefore, it is necessary to work with a relaxed formulation. Passing to the limit when $\epsilon$ tends to zero we get an asymptotic model where the variable control is now in a Robin boundary condition instead of the diffusion operator. We also present some numerical simulations showing the behavior of the solutions.

我们对空腔壁中两种隔离材料的最佳分布感兴趣，以便在限制最佳隔离材料的数量时获得最佳隔离。我们假设墙的宽度$\epsilon >0$小。这个问题可以表述为某个扩散算子的第一特征值的最小化，其中扩散常数在腔体内为一阶，并且为一阶。$\epsilon$在墙里。正如最佳设计中的常见问题一样，该问题通常无法解决，因此有必要以轻松的方式进行工作。达到极限时$\epsilon$趋于零，我们得到一个渐近模型，其中变量控制现在处于Robin边界条件而不是扩散算子。我们还提出了一些数值模拟，显示了解决方案的行为。