Optimization problems with fixed volume constraints and stability results related to rearrangement classes

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• 作者：Yichen Liu^a ; ^{yichen.liu07@yahoo.com" class="auth_mail" title="E-mail the corresponding author} ; Behrouz Emamizadeh^b ; ^{Behrouz.Emamizadeh@nottingham.edu.cn" class="auth_mail" title="E-mail the corresponding author} ; Amin Farjudian^c ; ^{Amin.Farjudian@gmail.com" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Trace inequality ; Boundary value problem ; Optimization ; Approximation ; Stability ; Rearrangement theory
• 刊名：Journal of Mathematical Analysis and Applications
• 出版年：2016
• 出版时间：15 November 2016
• 年：2016
• 卷：443
• 期：2
• 页码：1293-1310
• 全文大小：425 K

文摘

The material in this paper has been divided into two main parts. In the first part we describe two optimization problems—one maximization and one minimization—related to a sharp trace inequality that was recently obtained by G. Auchmuty. In both problems the admissible set is the one comprising characteristic functions whose supports have a fixed measure. We prove the maximization to be solvable, whilst the minimization will turn out not to be solvable in general. We will also discuss the case of radial domains. In the second part of the paper, we study approximation and stability results regarding rearrangement optimization problems. First, we show that if a sequence of the generators of rearrangement classes converges, then the corresponding sequence of the optimal solutions will also converge. Second, a stability result regarding the Hausdorff distance between the weak closures of two rearrangement classes is presented.