论文标题
在凸度约束下形状优化的规律性优化
Regularity in shape optimization under convexity constraint
论文作者
论文摘要
本文关注的是在凸度约束下的一类等速问题的形状优化者的规律性。我们证明，在凸形形状中，周长和扰动项的最小化是c 1,1-groumar。为此，我们定义了拟合到凸度上下文的准少量器的概念，并表明任何此类准少量仪都是C 1,1-rendimular。证明依赖于切割程序，该过程被引入了，以证明与变化背景的计算中相似的规律性结果。使用惩罚方法，我们能够处理卷约束，在这种情况下显示相同的规律性。我们介绍了PDE理论的一些例子，即扰动项是PDE类型的时候，并证明了大量此类示例符合我们的C 1,1-型结果。最后，我们提供了一个反例，表明我们不能期望一般的规律性更高。
This paper is concerned with the regularity of shape optimizers of a class of isoperimetric problems under convexity constraint. We prove that minimizers of the sum of the perimeter and a perturbative term, among convex shapes, are C 1,1-regular. To that end, we define a notion of quasi-minimizer fitted to the convexity context and show that any such quasi-minimizer is C 1,1-regular. The proof relies on a cutting procedure which was introduced to prove similar regularity results in the calculus of variations context. Using a penalization method we are able to treat a volume constraint, showing the same regularity in this case. We go through some examples taken from PDE theory, that is when the perturbative term is of PDE type, and prove that a large class of such examples fit into our C 1,1-regularity result. Finally we provide a counterexample showing that we cannot expect higher regularity in general.
