两相材料p-Laplacian能量的最大化
The maximization of the p-Laplacian energy for a two-phase material
摘要：
本文研究了两种扩散材料在有界开集$Ω\subset \mathbb{R}^N$中的最优排列问题，以使能量最大化。扩散问题由$p$ -Laplacian算子建模。众所周知，这类问题一般没有解决方案，因此有必要采用宽松的公式。在本文中，我们得到这样的放松制定使用均匀化理论，即我们取代这两种材料的微观混合物。得到了一些唯一性结果和一组最优性条件。作为结果，我们证明了放松问题的最优解的一些正则性。也就是说，我们表明，通量是在索伯列夫空间$H^1(Ω)^N$和材料的最佳比例是可导出的正交方向的通量。这意味着非松弛问题一般没有解。我们的结果推广了第一作者对Laplace算子的结果。
英文摘要：
We consider the optimal arrangement of two diffusion materials in a bounded open set $Ω\subset \mathbb{R}^N$ in order to maximize the energy. The diffusion problem is modeled by the $p$-Laplacian operator. It is well known that this type of problems 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 $H^1(Ω)^N$ 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.
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