SIAM Journal on Control and Optimization, Volume 59, Issue 2, Page 1497-1519, January 2021.
We consider the optimal arrangement of two diffusion materials in a bounded open set $\Omega\subset \mathbb{R}^N$ in order to maximize the energy. The diffusion problem is modeled by the $p$-Laplacian operator. It is well known that this type of problem has no solution in general and then that it is necessary to work with a relaxed formulation. In the present paper, we obtain such relaxed formulation using the homogenization theory; i.e., we replace both materials by microscopic mixtures of them. Then we get some uniqueness results and a system of optimality conditions. As a consequence, we prove some regularity properties for the optimal solutions of the relaxed problem. Namely, we show that the flux is in the Sobolev space $H^1(\Omega)^N$ and that the optimal proportion of the materials is derivable in the orthogonal direction to the flux. This will imply that the unrelaxed problem has no solution in general. Our results extend those obtained by the first author for the Laplace operator.

中文翻译：

---

两相材料的$ p $-拉普拉斯能量的最大化

SIAM控制与优化杂志，第59卷，第2期，第1497-1519页，2021年1月。
我们考虑在有界开放集合$ \ Omega \ subset \ mathbb {R} ^ N $中两种扩散材料的最佳排列，以使能量最大化。扩散问题由$ p $ -Laplacian算子建模。众所周知，这种类型的问题通常无法解决，因此有必要采用宽松的公式。在本文中，我们使用均质化理论获得了这种宽松的公式。也就是说，我们用两种材料的微观混合物代替了这两种材料。然后我们得到一些唯一性结果和一个最优条件系统。结果，我们证明了松弛问题的最优解的一些规律性。即，我们表明通量在Sobolev空间$ H ^ 1（\ Omega）^ N $中，并且材料的最佳比例可在与通量正交的方向上导出。这将暗示未放松的问题通常无法解决。我们的结果扩展了第一作者为Laplace运算符获得的结果。