凸约束下的最优多边形
Polygons as optimal shapes with convexity constraint
摘要：
在本文中，我们专注于以下一般形状最优化问题：$$ \min\{J(\Om), \Om convex, \Om\in\mathcal S_{ad}\}, $$其中$\mathcal S_{ad}$是一组二维容许形状和$J:\mathcal{S}_{ad}\to\R$是一个形状功能。利用凸区域集合的一种特殊的参数化方法，我们得到了这类问题的一些极值条件（一阶和二阶）。此外，我们使用这些最优性条件来证明，对于一大类泛函（满足一个类的属性），任何解决这个形状最优化问题是一个多边形。
英文摘要：
In this paper, we focus on the following general shape optimization problem: $$ \min\{J(\Om), \Om convex, \Om\in\mathcal S_{ad}\}, $$ where $\mathcal S_{ad}$ is a set of 2-dimensional admissible shapes and $J:\mathcal{S}_{ad}\to\R$ is a shape functional. Using a specific parameterization of the set of convex domains, we derive some extremality conditions (first and second order) for this kind of problem. Moreover, we use these optimality conditions to prove that, for a large class of functionals (satisfying a concavity like property), any solution to this shape optimization problem is a polygon.
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