Regularity for Shape Optimizers: The Degenerate Case
Regularity for Shape Optimizers: The Degenerate Case
复制标题
形状优化器的正则性：退化情况
DOI：
10.1002/cpa.21810
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发表时间：
2017
影响因子：
3
通讯作者：
F. Lin
中科院分区：
文献类型：
--
作者：
D. Kriventsov;F. Lin
我们考虑F(λ1(Ω)，…)的极小元，λN(Ω))+|Ω|)，其中F是每个参数不减的函数，λk(Ω)是ω的第k个狄里克莱特征值。这特别包括仅依赖于前N个特征值中的一部分的函数F，例如经常研究的F=λN。最近证明了极小值的存在，它也是有限周长的有界集合。这里我们证明了极小化Ω的约化边界是由光滑图组成的，并研究了奇点分类的困难。我们的方法是基于近似(“消失的粘性”)的论点，这与直觉相反，允许我们恢复极小化的欧拉-拉格朗日方程，这在其他方面是不可用的。©2019威利期刊公司。
We consider minimizers of F(λ1(Ω),…,λN(Ω))+|Ω|, where F is a function nondecreasing in each parameter, and λk(Ω) is the kth Dirichlet eigenvalue of ω. This includes, in particular, functions F that depend on just some of the first N eigenvalues, such as the often‐studied F=λN. The existence of a minimizer, which is also a bounded set of finite perimeter, was shown recently. Here we show that the reduced boundary of the minimizers Ω is made up of smooth graphs and examine the difficulties in classifying the singular points. Our approach is based on an approximation (“vanishing viscosity”) argument, which—counterintuitively—allows us to recover an Euler‐Lagrange equation for the minimizers that is not otherwise available. © 2019 Wiley Periodicals, Inc.